An Introduction to Higher Categorical Algebra
David Gepner
TL;DR
This survey introduces higher categorical algebra, focusing on symmetric monoidal stable $ abla$-categories, ring spectra, modules, and deformation theory within the $ abla$-categorical framework, highlighting developments by Lurie.
Contribution
It provides a comprehensive overview of algebraic structures in $ abla$-categories, emphasizing key concepts like spectra, modules, localization, and deformation theory, based on Lurie's work.
Findings
Introduction of symmetric monoidal stable $ abla$-categories
Explanation of ring spectra and module categories
Overview of localization, completion, and deformation theory
Abstract
This article is a survey of algebra in the -categorical context, as developed by Lurie in "Higher Algebra", and is a chapter in the "Handbook of Homotopy Theory". We begin by introducing symmetric monoidal stable -categories, such as the derived -category of a commutative ring, before turning to our main example, the -category of spectra. We then go on to consider ring spectra and their -categories of modules, as well as basic constructions such as localization, completion, and dualizability. We conclude with a brief account of the cotangent complex and deformation theory.
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Taxonomy
TopicsHomotopy and Cohomology in Algebraic Topology · Algebraic structures and combinatorial models · Advanced Topics in Algebra
An Introduction to Higher Categorical Algebra
David Gepner
David Gepner David Gepner was supported by NSF Grant DMS-1406529 and the Mathematical Sciences Research Institute Program “Derived Algebraic Geometry”.
Contents
1 Introduction
1.1 Higher structures
Higher algebra is the study of algebraic structures which arise in the setting of higher category theory. Higher algebra generalizes ordinary algebra, or algebra in the setting of ordinary category theory. Ordinary categories have sets of morphisms between objects, and elements of a set are either equal or not. Higher categories, on the other hand, have homotopy types of morphisms between objects, typically called mapping spaces. Sets are examples of homotopy types, namely the discrete ones, but in general it doesn’t quite make sense to ask whether or not two “elements” of a homotopy type are “equal”; rather, they are equivalent if they are represented by points which can be connected by a path in some suitable model for the homotopy type. But then any two such paths might form a nontrivial loop, leading to higher automorphisms, and so on. The notion of equality only makes sense after passing to discrete invariants such as homotopy groups.
Since the higher categorical analogue of a set is a space, the higher categorical analogue of an abelian group ought to be something like a space equipped with a multiplication operation which is associative, commutative, and invertible up to coherent homotopy. While invertibility is a property of an associative operation, commutativity is not; rather, it is structure. This is because, in higher categorical contexts, it is not enough to simply permute a sequence of elements; instead, the permutation itself is recorded as a morphism. A commutative multiplication operation must also act on morphisms, so that they may also be permuted, and so on and so forth, provided we keep track of these permutations as still higher morphisms. There are a number of formalisms which make this precise, all of which are equivalent to (or obvious variations on) the notion of spectrum in the sense of algebraic topology.111The overuse of the term “spectrum” in mathematics is perhaps a potential cause for confusion; fortunately, it is almost always clear from context what is meant.
On the one hand, a spectrum is an infinite delooping of a pointed space, thereby providing an abelian group structure on all its homotopy groups (positive and negative); on the other, a spectrum represents a cohomology theory, which is to say a graded family of contravariant abelian group valued functors on pointed spaces satisfying a suspension relation and certain exactness conditions. These two notions are equivalent: the functors which comprise the cohomology theory are represented by the spaces of the infinite delooping. The real starting point of higher algebra is the observation that there is a symmetric monoidal structure on the -category of spectra which refines the tensor product of abelian groups, and that there are many important examples of algebras for this tensor product.
1.2 Overview
Ordinary algebra is set based, meaning that it is carried out in the language of ordinary categories. As mentioned, the higher categorical analogue of sets are spaces, or -truncated spaces if one chooses to work in an -categorical context, and the truncation functors allow us to switch back and forth between categorical levels. In the category of sets, limits and colimits reduce to intersections and unions in some ambient set; in higher category theory, however, these operations must be interpreted invariantly, which implies that a homotopy colimit of sets (viewed as spaces) need not be discrete.
We begin our exposition in Section 2 with some background on the behavior of colimits in higher categories, especially in -categories of presheaves or certain full subcategories thereof (the presentable -categories). We then turn to the Grothendieck construction, which establishes a correspondence between fibrations and functors. The fibration perspective allows for an efficient approach to the theory of (symmetric) monoidal -categories and (commutative) algebras and modules therein, our main objects of interest, at least in the stable setting: a stable -category is a higher categorical analogue of an abelian category, an analogy which we make precise by comparing derived categories of abelian categories with stable -categories via t-structures.
Having equipped ourselves with the basic structures and language, in Section 3 we turn to a more detailed study of spectra and the smash product. An associative (respectively, commutative) ring spectrum is defined as an algebra (respectively, commutative algebra) object in the stable -category of spectra, the universal stable -category. The theory also allows for a notion of (left or right) module object of an -category which is (left or right) tensored over a monoidal -category. We conclude this section with some remarks on localizations of ring spectra, which mirrors the ordinary theory save for the fact that ideals must be interpreted on the level of homotopy groups.
Section 4 is devoted to module theory. In particular, monads appear as an instance of modules, allowing us to address monadicity, which plays a much more important role higher categorically due do the difficultly of ad hoc constructions. Simplicial objects and their colimits, geometric realizations, feature in the construction of the relative tensor product as well as the definition of projective module. We also study the more general class of perfect modules, which are colimits of shifted projective modules in a sense made rigorous by the theory of tor-amplitude, which acts as a substitute for projective resolutions. We also consider free algebras and isolate various finiteness properties of modules and algebras which play important roles in higher algebra.
The final Section 5 deals with deformations of commutative algebras. The formalism of the tangent bundle allows for an elegant construction of the cotangent complex, which governs derivations and square-zero extensions. The Postnikov tower of truncations of a connective commutative algebra is comprised entirely of square-zero extensions, a crucial fact which implies an obstruction theory for computing the space of maps between commutative algebra spectra. The main theorem in that all obstructions vanish in the étale case, from which it follows that the -category of étale commutative algebras over of fixed commutative ring spectrum is equivalent to an ordinary category, namely that of étale commutative algebras over its underlying ordinary ring , a strong version of the topological invariance of étale morphisms.
We conclude this introductory section with some background on various approaches to homotopy coherence and some remarks on higher categorical “set theory”: small and large spaces, categories, universes, etc. The reader who is already familiar with these notions is encouraged to Section 2, or even Section 3 if they are already familiar with stability, presentability, and symmetric monoidal structures.
1.3 Homotopy coherence
To lessen the prerequisites we avoid the use of operads or -operads in this article altogether. Nevertheless, for the sake of putting the theory of higher categorical algebra into historical context, and explaining some of the standard terminology, a few remarks are in order.
A space equipped with a homotopy coherently associative, or homotopy coherently associative and commutative, multiplication operation is traditionally referred to as an -monoid, or -monoid, meaning that it admits an action by an (infinitely homotopy coherently associative) or (infinitely homotopy coherently “everything”, i.e. associative and commutative) operad. If the multiplication operation is invertible up to homotopy, the -monoid is said to be grouplike. An -monoid is grouplike when viewed as an -monoid. These operads were originally constructed out of geometric objects like associahedra, configuration spaces, or spaces of linear isometries.
The -categorical approach prefers to use small combinatorial models for associativity and commutativity, as in Segal’s treatment [36], by incorporating homotopy coherence into the language itself. The result is a significantly more streamlined approach to homotopy coherent algebraic structures, as anticipated in the now extremely influential book of the same name of Boardmann–Vogt [12], which contained the original definition of -category, well before the theory was systematically developed by Joyal and then Lurie. Nevertheless, there is a rich interplay between geometry, topology, and higher category theory, as evidenced by the remarkable cobordism hypothesis, among other things. Even the motivating example of the theory of operads, as originally developed by May [28], namely the little -cubes operad, , collectively form the most important family of -operads, the so-called operads.
As -operads, and ; the former equivalence is easy but the latter is equivalence is hard and requires both the tensor product of Boardmann–Vogt [12] and the additivity theorem of Dunn [15]. Since we will only be concerned with -algebras and -algebras in (symmetric) monoidal -categories, we choose to emphasize the analogy with ordinary algebra by referring to these objects as associative and commutative algebras, respectively. This drastically simplifies the terminology and also allows us to reformulate actions by the associative or commutative -operad, respectively, in terms of functors from a category of ordinals or cardinals (arguably the must basic mathematical objects of all). And while the abstract theory is quite powerful, one should keep in mind that many of the most important examples come from geometry and topology via these more classical constructions.
Heuristically, a -categories are generalizations of (ordinary) categories in which there is a space, instead of a set, of morphisms between any pair of objects. There are a number of ways of making this precise, but suffice it to say that simplicially (or topologically) enriched categories and quasi-categories yield equivalent models: any -category is equivalent to the homotopy coherent nerve of a simplicially enriched category . We will follow the usual notational conventions and sometimes refer to simplically enriched categories as simplicial categories, though this should not be confused with the more general notion of simplicial object in the category of categories.
Remark 1.3.1**.**
The theory of quasi-categories [23] has the distinct advantage that it allows for an easy construction of the -category of functors from an -category to an -category as the exponential
[TABLE]
in the cartesian closed category of simplicial sets. This is a completely combinatorial object: if and are simplicial sets, an -simplex of is natural transformation of functors , so it is completely determined by a compatible family of functions , .
Remark 1.3.2**.**
The chief issue which arises when working with simplicial categories is that the simplicial category of simplicial functors from to is not in general invariant under weak equivalence (simplicial functors which are fully faithful and essentially surjective up to weak homotopy equivalence). To obtain the homotopically correct simplicial category of functors we must replace with a sufficiently “free” version of itself.
Remark 1.3.3**.**
In practice it is usually easier to apply the homotopy coherent nerve functor
[TABLE]
and work in simplicial sets. Here denotes the category of simplicially enriched categories and is the right adjoint of the colimit preserving functor determined by defining to be the simplicial set of partially ordered subsets of .
Theorem 1.3.1**.**
[24, Theorem 2.2.5.1]* The homotopy coherent nerve functor is a right Quillen equivalence. In particular, the homotopy coherent nerve of a simplically enriched category is a quasicategory provided is enriched in Kan complexes (every is a Kan complex).*
Higher categorical algebra is truly homotopical and not just homological in nature, meaning that many of its most important objects simply do not exist within the world of chain complexes or derived categories. The portion of the theory that can be formulated in these terms is differential graded algebra, the abstract study of which employs the language of differential graded categories.
Definition 1.3.4**.**
A differential graded category
is a category enriched over the category of chain complexes of abelian groups. We write for the category of small differential graded categories
Example 1.3.5**.**
Let be an abelian category, or an additive subcategory of an abelian category. Given complexes , there is a natural chain complex of abelian groups of homomorphisms , which in degree is the abelian group with differential given by the formula In this way, acquires a canonical structure of a differential graded category: there is a chain complex of maps between any two objects, suitably compatible with composition. Moreover, the abelian group of maps falls out as the zeroth homology of this complex, so that is the homotopy category of this differential graded category.
Remark 1.3.6**.**
A differential graded category determines a simplicially enriched category via the Dold-Kan correspondence [40], which asserts that there is a suitably monoidal equivalece of categories between connective chain complexes and simplicial abelian groups. Said differently, we may regard a chain complex, provided it is identically zero in negative degrees,222We will use homological grading consistently throughout this article. as a simplicial abelian group. This is done via the functor which associates to such a chain complex and nonempty finite ordinal the abelian group .
Remark 1.3.7**.**
The functor alluded to above admits a lax monoidal structure [40], and lax monoidal functors between enriching monoidal categories allow us to functorially change enrichment. Since the (good) truncation functor and underlying set functor also admit lax monoidal structures, we obtain a composite lax monoidal functor
[TABLE]
and hence a functor from differential graded to simplicially enriched categories. We write for this functor and for the homotopy coherent nerve of the differential graded category . One can define a more explicit differential graded nerve functor , as in [25, Construction 1.3.1.6], but it is equivalent to the homotopy coherent nerve of the associated simplicially enriched category [25, Proposition 1.3.1.17]).
1.4 Terminology and notation
For the purposes of this article, we will always consider spaces from the point of view of -category theory. This means that we will tend to think of a topological space in terms of its associated singular simplicial set (its fundamental -groupoid)
rather than as a point-set object, and that we reserve the right to replace with a homotopy equivalent space (or not even choose a representative of the homotopy type of ). If has the homotopy type of a cell complex then the realization of the singular complex of recovers up to homotopy equivalence; otherwise, the realization of the singular complex of recovers only up to weak homotopy equivalence. As we will never need actual topological spaces (up to homoemorphism) at all anywhere in this article, “space” will always mean “homotopy type” in this sense.
Definition 1.4.1**.**
The -category
of spaces is the homotopy coherent nerve of the simplicially enriched category of Kan complexes, .
Remark 1.4.2**.**
As stipulated by Grothendieck’s homotopy hypothesis, the -category of spaces is equivalent to the -category of -groupoids. Here
denotes the full subcategory consisting of those -categories for which any morphism is an equivalence.
We will also be interested in the -category of pointed spaces,
which can be modeled either internally as pointed spaces or as the homotopy coherent nerve of the simplicial category of pointed Kan complexes.
Remark 1.4.3**.**
The coproduct of a pair of objects and of is computed as the wedge product , the quotient of the disjoint union of and obtained by identifying their basepoints. Via the colimit preserving functor which freely adjoins a basepoint, the cartesian product on extends to the smash product
on . That is, the smash product sits in a cofiber sequence
[TABLE]
and has the property that if and then .
In practice, it is sometimes necessary to be precise about size by bounding various classes of objects by sets of cardinality less than some infinite regular cardinal. For the purposes of this survey article, however, we will gloss over most of these distinctions and employ a very basic version of the theory of Grothendieck universes: objects which exist in our first universe will be called small, objects which exist in the next universe will be called large, and objects which exist only in a still higher universe will be called very large.
As one ascends the higher categorical ladder, the size of the mathematical objects under consideration tends to increase. While a space will always be assumed to be small, an -category will typically be assumed to be large, unless otherwise mentioned. We write
for the large -category of small -categories and
for the very large -category of large -categories.
While some -categorical statements will be formulated in the model of quasicategories, we will often employ the theory without reference to any particular model. From this perspective, a full subcategory is assumed to be closed under equivalences, a cartesian fibration is any functor that is equivalent to a cartesian fibration (see [24, Section 2.4], or Section 2.2 for an overview), and ordinary categories are -categories with discrete mapping spaces.
1.5 Acknowledgements
The modern approach to algebra in higher categorical contexts surveyed in this article is based almost entirely on the groundbreaking work of Jacob Lurie. It is a great pleasure to thank him for his enormous contributions to the subject. Collectively, they comprise new and firm foundations for homotopy coherent mathematics.
This article is essentially a summary of the basics of Lurie’s Higher Algebra, with results from Higher Topos Theory and Spectral Algebraic Geometry thrown in as necessary, simplified as much as possible in order to be accessible to the neophyte. Nevertheless, we have tried our best to state definitions and theorems as precisely as possible and provide accurate references for these results (though in the interest of keeping the article as short as possible we have decided not to provide any proofs whatsoever, or even citations for most of the definitions). Of course, Lurie’s work builds on the combined efforts of a great many mathematicians — far too many to list here, and any attempt to do so will inevitably leave omit many valuable contributions.
It is also a pleasure to thank Benjamin Antieau, Tobias Barthel, Jeremiah Heller, Lars Hesselholt, Achim Krause, Tyler Lawson, Haynes Miller, Thomas Nikolaus, Charles Rezk, Markus Spitzweck, and Hiro Tanaka for helpful remarks and comments on earlier versions of this draft. The author would also like to thank the Mathematical Sciences Research Institute for providing an extremely pleasant working environment while much of this article was being written, as well as the National Science Foundation for their generous support.
2 Category theory
2.1 Presheaves and colimits
Definition 2.1.1**.**
A presheaf on an -category is a functor . The -category of presheaves on is the functor -category
[TABLE]
where denotes the -category of (small) spaces.
Remark 2.1.2**.**
Recall [24, Proposition 5.1.3.1] that is equipped with a fully faithful Yoneda embedding ,
given by the formula . Colimits in are computed objectwise as colimits in . Since is cocomplete, in the sense that it admits all small colimits, we see that is as well. In fact, the Yoneda embedding exhibits as the free cocompletion of , in the sense made precise by the following statement.
Theorem 2.1.1**.**
[24, Theorem 5.1.5.6]* Given a small -category and a cocomplete -category , precomposition with the Yonda embedding*
[TABLE]
induces an equivalence between the -categories of functors and colimit preserving functors .
Remark 2.1.3**.**
The actual statement of [24, Theorem 5.1.5.6] is in terms of left adjoint functors . Left adjoint functors preserve colimits, and in this case any colimit preserving functor is a left adjoint.
We will require variants of the above construction in which we freely adjoin only certain types of colimits.
Definition 2.1.4**.**
A map of simplicial sets is cofinal if, for every -category and every -indexed colimit diagram , the induced cone is a colimit diagram. See [24, Proposition 4.1.1.8] for details.
Definition 2.1.5**.**
Let be an infinite regular cardinal. A simplicial set is -small if has fewer than nondegenerate simplices. A simplicial set is -filtered if every map from a -small simplicial set extends to a cone . A simplicial set is sifted if is cofinal.
Example 2.1.6**.**
Filtered simplicial sets are sifted, and the simplicial indexing category is sifted. A functor of -categories preserves sifted colimits if and only if preserves filtered colimits and geometric realizations.
Definition 2.1.7**.**
An object of an -category which admits -filtered colimits is -compact if commmutes with -filtered colimits. An object of an -category which admits geometric realizations is projective if commutes with geometric realizations.
Definition 2.1.8**.**
An indexing class is a collection of simplicial sets which we regard as indexing allowed types of diagrams; this is not standard terminology. Given an indexing class , we write for the full subcategory consisting of those presheaves such that, for all , transforms -indexed colimits in to -indexed limits in .
Remark 2.1.9**.**
The indexing classes which most commonly arise when considering algebraic structures are finite discrete, finite, filtered, and sifted. These classes often come in pairs. A functor of preserves all small colimits if and only if it preserves finite colimits and filtered colimits, or finite discrete colimits (that is, finite coproducts) and sifted colimits.
Proposition 2.1.10**.**
[24, Proposition 5.3.6.2]* Let and be -categories. If admits finite coproducts and admits sifted colimits, precomposition with the Yoneda embedding determines an equivalence of -categories*
[TABLE]
of functors and sifted colimit preserving functors . If admits -small colimits and admits -filtered colimits, precomposition with the Yoneda embedding determines an equivalence of -categories
[TABLE]
of functors and -filtered colimit preserving functors .
2.2 The Grothendieck construction
Given an ordinary category , the theory of categories fibered over developed in [1] characterizes the essential image of the Grothendieck construction
[TABLE]
as the (not full) subcategory of consisting of the fibered categories and their morphisms. Because of the fundamental role the Grothendieck construction plays in higher algebra, we begin with a brief overview of the basic notions of fibered -category theory [24]. Whenever possible we choose to phrase these notions internally inside of itself, with a few exceptions: it is sometimes quite useful to represent an -category by a (marked) simplicial set [24], in particular when specifying diagrams in an -category .
Consider the slice -category : objects of are functors with target , and morphisms of from to are commuting triangles of of the form
[TABLE]
We write for the -category of functors such that .
Definition 2.2.1**.**
Let be a functor333If we are working externally in the ordinary category of quasicategories, we would additionally require to be a categorical fibration [24]; internally, however, this is a meaningless assumption, as the notion of categorical fibration is not stable under categorical equivalence. of -categories. An arrow of is -cartesian
if the induced functor
[TABLE]
is an equivalence of -categories. A -cartesian lift of an arrow in is a -cartesian arrow in such that . A functor is a cartesian fibration if, for every arrow of and every object , there is a -cartesian lift of with target .
Remark 2.2.2**.**
When is clear from context, we will refer to -cartesian arrows simply as cartesian arrows.
A cartesian fibration determines a functor , called the straightening of in [24], basically by taking fibers, which happen to be contravariantly functorial in precisely this case. This operation is inverse to the unstraightening functor, the -categorical analogue of the Grothendieck construction, which “integrates” the functor to a cartesian fibration . There are analogous versions for cocartesian fibrations,
which correspond to covariant functors .
Theorem 2.2.1**.**
[24, Theorem 3.2.0.1]* For any small -category , the unstraightening functor*
[TABLE]
induces an equivalence between the -category and the (not full) subcategory consisting of the cartesian fibrations over and those functors over which preserve cartesian arrows.
The straightening and unstraightening equivalence will be used throughout this article, especially in our definitions of (symmetric) monoidal -category and (commutative) algebra object therein. Even more fundamentally, this equivalence is used in the definition of adjunction.
Definition 2.2.3**.**
Let and be -categories. An adjunction between and is cartesian and cocartesian fibration equipped with equivalences and .
Remark 2.2.4**.**
Given an adjunction, the left adoint is determined by cocartesian lifts of , and the right adjoint is determined by cartesian lifts of . It is possible to construct a unit or counit transformation [24, Proposition 5.2.2.8], and the fact that both and fully faithfully embed into implies that we have equivalences
[TABLE]
for any pair of objects of and of . A functor is a left adjoint if and only if the associated cocartesian fibration is also cartesian, and a functor is a right adjoint if and only if the associated cartesian fibration is also cocartesian.
2.3 Monoidal and symmetric monoidal -categories
Informally, a symmetric monoidal -category is an -category equipped with unit and multiplication maps which are appropriately coherently associative, commutative, and unital. Making this precise requires a means to organize for us the infinite hierarchy of coherence data necessary to assert that the intermediate multiplications , which should be indexed by something like functions from the set with elements to the set with elements, are suitably compatible.
Remark 2.3.1**.**
Actually, as we may first project away some of the factors in the product before we multiply, these operations are indexed by functions of pointed sets. For convenience we allow ourselves to specify a finite pointed set by the cardinality of its elements, writing for the pointed set with basepoint [math] and elements in the complement .
Definition 2.3.2**.**
A morphism in is inert
if, for each element , the preimage consists of a single element. A morphism in is active
if the preimage consists of a single element (necessarily the basepoint).
Example 2.3.3**.**
There are exactly inert maps , namely
[TABLE]
Definition 2.3.4**.**
A symmetric monoidal -category
is a cocartesian fibration which satisfies the Segal condition: for each natural number , the map induced by the inert morphisms , , is an equivalence. A morphism of symmetric monoidal -categories, or a symmetric monoidal functor, from to , is a functor over such that preserves cocartesian edges. The -category of symmetric monoidal -categories is the full subcategory of spanned by the symmetric monoidal -categories.
Remark 2.3.5**.**
The straightening of a symmetric monoidal -category is a functor satisfying the Segal condition. This is the data of a commutative monoid object in , hence the notation . We could take this as the definition of a symmetric monoidal -category, except that in practice these functors are often difficult to write down, an issue which already arises in ordinary category theory: for instance, the tensor product of modules is more naturally defined by a universal property than a specific choice of representative. It is usually easier to avoid making explicit choices altogether by constructing the cocartesian fibration instead, where one may as well take the fiber over to be the -category of all possible choices of the symmetric monoidal product of objects of .
Example 2.3.6**.**
Let be an -category with finite products. Then there exists a cocartesian fibration whose fiber over is the -category of commutative diagrams of the form such that and carries pushout squares in to pullback squares in . In particular, the map is an equivalence in , from which it follows that is an equivalence and therefore that is a symmetric monoidal -category. We refer to this as the cartesian symmetric monoidal structure
on , which exists if and only if has finite products.
Definition 2.3.7**.**
Let be a symmetric monoidal -category. A commutative algebra object of is a section of which sends inert morphisms in to cocartesian morphisms in . The -category of commutative algebra objects in is the full subcategory of the -category of sections of consisting of the commutative algebra objects.
Remark 2.3.8**.**
There is an equivalence . That is, the -category of commutative algebra objects in is equivalent to the -category of commutative monoid objects in .
We will also be interested in monoidal -categories. To set up the theory we need a nonsymmetric analogue on the category of finite pointed sets.
Remark 2.3.9**.**
To keep the prerequisites to a minimum, we purposely avoid using the language of -operads. However, from that perspective, a natural choice of indexing category for monoidal structures is the “desymmetrization” of , the functor whose fiber over is the set of total orderings of . However it will be both more convenient and elementary to simply use simplicial objects instead.
Definition 2.3.10**.**
The functor is defined by identifying the nonempty finite ordinal with the set of “cuts” of the pointed cardinal as follows: we can cut the string before or after any , for a total of possibilities. However, cutting before [math] or after have the same effect (nothing), so we identify the two trivial cuts with the basepoint of .
Definition 2.3.11**.**
An order preserving function is inert
if it is it injective with convex image; that is, .
Remark 2.3.12**.**
The inert morphisms in are the order preserving functions of the form , , .
Definition 2.3.13**.**
A monoidal -category is a cocartesian fibration which satisfies the Segal condition: for each , the map
[TABLE]
induced by the inert morphisms , , is an equivalence. A morphism of monoidal -categories, or a monoidal functor, from to is a functor over such that preserves cocartesian edges. The -category of monoidal -categories is the full subcategory of spanned by the monoidal -categories.
Definition 2.3.14**.**
Let be a monoidal -category. An algebra object of is a section of which sends inert morphisms in to cocartesian morphisms in . The -category of algebra objects in is the full subcategory of the -category of sections of consisting of the algebra objects.
Remark 2.3.15**.**
As in the symmetric case, there is an equivalence : the -category of associative algebra objects in is equivalent to the -category of monoid objects in .
Remark 2.3.16**.**
Using [25, Construction 4.1.2.9] he theory is set up so that a symmetric monoidal -category restricts to a monoidal -category . It follows that we have a forgetful functor .
2.4 Presentable -categories
Definition 2.4.1**.**
Given a small -category , we write for the full subcategory consisting of those functors such that the source of the unstraightening of is -filtered.
Proposition 2.4.2**.**
[24, Proposition 5.3.5.10]* Let be a small -category and an -category with -filtered colimits. There is an equivalence of -categories*
[TABLE]
and hence by Proposition 2.1.10 an equivalence .
Definition 2.4.3**.**
An object of an -category is -compact
if the corepresentable functor commutes with -filtered colimits. We write
for the full subcategory of consisting of the -compact objects.
Definition 2.4.4**.**
An -category is -compactly generated
if admits all small colimits and, writing for the full subcategory consisting of the -compact objects, the canonical map is an equivalence. A presentable -category
is an -category which is -compactly generated for some infinite regular cardinal .
There are dual notions of morphism of presentable -category: namely, those functors which are left (respectively, right) adjoints. We write
(respectively, ) for these -categories. A key point [24, Proposition 5.5.3.13 and Theorem 5.5.3.18] is that the inclusion of subcategories and preserves limits: in fact, given a functor , a cone is limiting in (respectively, ) if and only if the induced cone is limiting in .
Remark 2.4.5**.**
For each infinite regular cardinal we have subcategories (again not full)
(respectively, ) consisting of the -compactly generated -categories and those left (respectively, right) adjoint functors which preserve -compact objects (respectively, -filtered colimits).
Definition 2.4.6**.**
A presentable fibration
is a cartesian fibration such that the straightening factors through the subcategory of presentable -categories and right adjoint functors.
Remark 2.4.7**.**
The adjoint functor theorem
[24, Corollary 5.5.2.9] states that any colimit preserving functor of presentable -categories admits a right adjoint (this is even true more generally, for instance if is only assumed to be cocomplete). Hence can be equivalently described as the subcategory of presentable -categories and left adjoint functors. Dually, writing for the subcategory consisting of the presentable -categories and right adjoint functors, uniqueness of adjoints allows us to construct a canonical equivalence
Definition 2.4.8**.**
A presentable symmetric monoidal -category is a symmetric monoidal -category
such that the underlying -category is presentable and any choice of tensor product bifunctor commutes with colimits separately in each variable.
Theorem 2.4.1**.**
[25, Proposition 4.8.1.17]* Let and be presentable -categories. The subfunctor*
[TABLE]
whose value at consists of those functors which preserve colimits separately in each variable, is corepresented by an object .
Remark 2.4.9**.**
It is straightforward to check, with the definition of as above, that for a presentable -category , there is a canonical equivalence
[TABLE]
where
denotes the -category of left adjoint functors , which is presentable by [25]. Dually, we write
for the -category of right adjoint functors , and note that .
Proposition 2.4.10**.**
[25, Lemma 4.8.1.16]* Let be presentable -categories. Then is a presentable -category, and*
[TABLE]
Theorem 2.4.2**.**
[25, Corollary 4.8.1.12]* The functor extends to a symmetric monoidal functor .*
Remark 2.4.11**.**
The symmetric monoidality of the presheaves functor is an -categorical generalization of the Day convolution product.
If is a small symmetric monoidal -category, then inherits a convolution symmetric monoidal structure [25, Remark 4.8.1.13], given by the formula
[TABLE]
Here the colimit is taken over the -category .
Remark 2.4.12**.**
The fact that is symmetric monoidal implies that the canonical map is an equivalence.
Definition 2.4.13**.**
A space is said to be -truncated if for all and . By convention, a space is said to be -truncated if it is contractible and -truncated if it is empty or contractible.
Definition 2.4.14**.**
An object of an -category is said to be -truncated, ,
if the associated representable functor factors through the full subcategory spanned by the -truncated spaces.
Proposition 2.4.15**.**
[24, Proposition 5.5.6.18]* Let be a presentable -category. The inclusion of the full subcategory of -truncated objects is stable under limits and admits a left adjoint .
Proposition 2.4.16**.**
[25, Example 4.8.1.22]* As an endofunctor of , is idempotent. It therefore determines a localization of with essential image the presentable -categories. In particular, for any presentable -category , we have a canonical equivalence .*
For any presentable -category , the left adjoints of the inclusions for result in a tower of truncations of in , and consequently a comparison map
[TABLE]
Definition 2.4.17**.**
Let be an object of a presentable -category . The Postnikov tower of is the tower of truncations of , regarded as a diagram in .
Definition 2.4.18**.**
Let be a presentable -category. We say that Postnikov towers converge in if the map is an equivalence of -categories.
More concretely, Postnikov towers converge in if, for each object , the map from to the limit of its Postnikov tower is an equivalence.
Example 2.4.19**.**
Postnikov towers converge in the -categories and .
2.5 Stable -categories
Definition 2.5.1**.**
A zero object of an -category is an object which is both initial and final. An -category is pointed if admits a zero object.
Definition 2.5.2**.**
Let be a pointed -category. A triangle in is a commutative square of the form
[TABLE]
where [math] is a zero object of . We typically write for a triangle in , though it is important to remember that a choice of composite and nullhomotopy are also part of the data. We say that a triangle in is left exact if it is cartesian (a pullback), right exact if it is cocartesian (a pushout), and exact if it is cartesian and cocartesian.
Remark 2.5.3**.**
Let be a pointed -category with finite limits and colimits, and consider a triangle in of the form
[TABLE]
Then if the triangle is left exact and is the triangle is right exact. If the triangle is exact, we have equivalences and .
Definition 2.5.4**.**
A stable -category is an -category with a zero object, finite limits and colimits, and which satisfies the following axiom: a commutative square in is cartesian if and only if it is cocartesian (in other words, a pullback if and only if it is a pushout).
Definition 2.5.5**.**
Let and be a functor of -categories which admits finite limits and colimits. A functor is left exact if preserves finite limits, right exact if preserves finite colimits, and exact if preserves finite limits and finite colimits. We write for the (not full) subcategory consisting of the stable -categories and the exact functors.
Proposition 2.5.6**.**
[25, Corollary 1.4.2.11]* Let be a pointed -category. The following conditions are equivalent:*
- (1)
* is stable.* 2. (2)
* admits finite colimits and is an equivalence.* 3. (3)
* admits finite limits and is an equivalence.*
Theorem 2.5.1**.**
[25, Lemma 1.1.2.13]* The homotopy category of a stable -category admits the structure of a triangulated category such that:*
- (1)
The shift functor is induced from the suspension functor . 2. (2)
A triangle in the homotopy category is exact if and only if there exist exact triangles
[TABLE]
in such that , , and are representatives of , and composed with the equivalence , respectively.
Definition 2.5.7**.**
Let be an -category with finite limits. A functor
[TABLE]
is said to be excisive (respectively, reduced) if sends cocartesian squares to cartesian squares (respectively, sends initial objects to final objects). We write
[TABLE]
for the full subcategory of reduced excisive functors .
Definition 2.5.8**.**
Let be an -category with finite limits. The -category of spectrum objects in is the -category
[TABLE]
of reduced excisive functors from finite spaces to .
The reason for the terminology “spectrum object” is that a reduced excisive functor determines, and is determined by, its value on any infinite sequence of spheres of strictly increasing dimension. The most canonical choice is the sequence of all spheres , so that evaluation on this family of spheres induces a map
[TABLE]
which sends the excisive functor to the sequence of pointed objects and equivalences .
Proposition 2.5.9**.**
[25, Remark 1.4.2.25]* Let be an -category with finite limits. The functor*
[TABLE]
induced by evaluation on spheres is an equivalence of -categories.
Remark 2.5.10**.**
Since has finite limits, it has a final object , and so it makes sense to consider the -category of pointed objects in . There is a canonical equivalence .
The main examples of stable -categories are either presentable or embed fully faithfully inside a stable presentable -category as the subcategory of -compact objects for some infinite regular cardinal . Let denote the full subcategory spanned by the stable presentable -categories.
Proposition 2.5.11**.**
[25, Proposition 4.8.2.18]* The inclusion of the full subcategory admits a left adjoint .*
Remark 2.5.12**.**
The left adjoint “stabilization” functor is given by tensoring with the -category of spectra. By virtue of the equivalence , the -category of spectrum objects in is a description of its stabilization.
Corollary 2.5.13**.**
Let and be presentable -categories such that is stable. Then is stable.
Corollary 2.5.14**.**
The symmetric monoidal structure on induces a symmetric monoidal structure on the full subcategory consisting of the stable presentable -categories.
We write for this symmetric monoidal subcategory. Note that this inclusion is lax symmetric monoidal and right adjoint to the symmetric monoidal stabilization functor .
Corollary 2.5.15**.**
The forgetful functor extends to a lax symmetric monoidal functor . In particular, it carries (commutative) algebra objects of to (symmetric) monoidal -categories.
Example 2.5.16**.**
The -category of spectra is a unit of the symmetric monoidal -category of stable presentable -categories. It therefore admits a presentable symmetric monoidal structure, called the smash product.
We will write for any choice of smash product multifunctor.
Remark 2.5.17**.**
Given spectra and , their smash product is usually written is the literature. We follow the convention of [25] and write instead, emphasizing the analogy with the tensor product of abelian groups (or more precisely the derived tensor product of chain complexes).
Remark 2.5.18**.**
By construction, the symmetric monoidal structure on the -category of spectra is compatible with the cartesian symmetric monoidal structure on the -category of spaces via the suspension spectrum functor
[TABLE]
This means that, for spaces , there is a canonical equivalence
[TABLE]
Proposition 2.5.19**.**
[25, Corollary 1.4.4.2]* A stable -category is -compactly generated if and only if admits small coproducts, a -compact generator for some regular cardinal , and (the homotopy category of) is locally small.*
Definition 2.5.20**.**
A Verdier sequence
is a cocartesian square
[TABLE]
in such that [math] is a zero object and the top horizontal map is fully faithful. A semi-orthogonal decomposition
is a Verdier sequence such that the functors and admit right adjoints. It is common to simply write for a Verdier sequence, leaving implicit the requirement that be fully faithful with cofiber .
Remark 2.5.21**.**
Given a Verdier sequence , the fact that is fully faithful implies that the left adjoint functor is fully faithful with right adjoint , and the right adjoint of the functor is also fully faithful. Thus the Verdier sequence is actually a semi-orthogonal decomposition.
2.6 Homotopy groups and t-structures
Unfortunately, the notion of truncation (see Definition 2.4.14 above) considered earlier is poorly behaved in stable -categories , even if happens to be presentable. This is because it is a direct consequence of stability that an object of is -truncated if and only if it is -truncated, so that the only finitely truncated objects at all are the zero objects, all of which are canonically equivalent to one another. Thus any attempt to study itself via the standard obstruction theoretic truncation type methods is doomed to fail.
The notion of a t-structure on a stable -category remedies this situation, providing all kinds of other interesting and potentially effective ways of (co)filtering objects of — ideally even itself — as limits of “Postnikov type” towers of truncation functors , or dually as colimits of “Whitehead type” telescopes of connective cover functors . We emphasize that while a t-structure is structure, it is encoded as innocuously as possible, via a choice of “orthogonal” full subcategories of in the sense made precise below.
First we need some notation. If is a stable -category and is an object of , we also write for a choice of -fold suspension of , . If is a full subcategory of , we write for the full subcategory consisting the objects of the form , where is an object of the full subcategory . Finally, for any pair of objects and of , we write for the abelian group .
Definition 2.6.1**.**
A t-structure on a stable -category consists of a pair of full subcategories and satisfying the following conditions:
- (1)
and ; 2. (2)
If and , then ; 3. (3)
Every fits into an exact triangle of the form with and .
Definition 2.6.2**.**
An exact functor between stable -categories equipped with t-structures is left -exact (respectively, right -exact) if it sends to (respectively, to ). An exact functor is -exact if is both left and right -exact. We set and .
Remark 2.6.3**.**
The data of a t-structure on a stable -category is equivalent to the data of a t-structure on its triangulated homotopy category in the sense originally defined and studied by Beilinson-Bernstein-Deligne [11].
Example 2.6.4**.**
If is a small abelian category, then the bounded derived -category admits a canonical t-structure, where consists of the complexes such that for , and similarly for . If is a Grothendieck abelian category, then the unbounded derived -category admits a t-structure with the same description as the previous example. This stable -category and its t-structure are studied in [25, Section 1.3.5].
Example 2.6.5**.**
If is a connective associative ring spectrum, then the stable presentable -category of left -module spectra admits a t-structure with , the -category of connective left -module spectra. We call this the Postnikov t-structure.
Remark 2.6.6**.**
The mapping space is actually contractible for and . This is not the case for the mapping spectra, as for any pair of objects and in a Grothendieck abelian category (see [25, Proposition 1.3.5.6]).
Proposition 2.6.7**.**
[25, Remark 1.2.1.12 and Warning 1.2.1.9]* The intersection is an abelian category equivalent to the full subcategory of consisting of the discrete objects (that is, [math]-truncated in the sense of Definition 2.4.14).*
Definition 2.6.8**.**
The abelian category is called the heart of the t-structure on , and is denoted .
Example 2.6.9**.**
The hearts of the t-structures in Example 2.6.4 are both equivalence to the abelian category itself. The heart of the t-structure in Example 2.6.5 is , the abelian category of left -modules.
Remark 2.6.10**.**
It turns out that the truncations and are functorial in the sense that the inclusions and admit right and left adjoints, respectively, by [25, Corollary 1.2.1.6].
Definition 2.6.11**.**
Let be a stable -category equipped with a t-structure and let be an object of . There is an equivalence of functors [25, Proposition 1.2.1.10]. The homotopy groups , , are defined via the formula .
Remark 2.6.12**.**
In the stable setting, passing to homotopy groups is a homological functor in the sense that there are long exact sequences
[TABLE]
in whenever is an exact triangle in .
Definition 2.6.13**.**
A t-structure on a stable -category is left complete if the full subcategory of infinitely connected objects consists only of zero objects. Right complete t-structures are defined similarly.
Definition 2.6.14**.**
A t-structure on a stable -category is bounded if the inclusion of the full subcategory of bounded objects is an equivalence. There are analogous notions of right and left bounded.
Many of the most important examples of stable -categories may already be familiar from homological algebra. If is an abelian category with enough projectives, the derived -category of bounded below chain complexes in admits a universal property which characterizes the stable -categories which arise in this way. We will suppose that has enough projective and write for the full subcategory consisting of the projective objects.
Definition 2.6.15**.**
Let be an abelian category with enough projectives. The bounded below derived -category of is the homotopy coherent nerve of the differential graded category (viewed as a simplically enriched category as in Remark 1.3.6) of bounded below chain complexes in (those complexes such that for .
Proposition 2.6.16**.**
[25, Corollary 1.3.2.18 and Proposition 1.3.2.19]* Let be an abelian category with enough projectives. Then is a stable -category, and the full subcategories and consisting of those complexes such that for and for , respectively, comprise a right bounded and left complete t-structure on .*
Theorem 2.6.1**.**
[25, Theorem 1.3.4.4]* Let be an abelian category with enough projectives and the class of quasi-isomorphisms. There is a canonical equivalence of -categories .*
Remark 2.6.17**.**
sits inside as the full subcategory of complexes concentrated in degree zero. Moreover, straightforward homological algebra arguments show that restricts to an equivalence .
Remark 2.6.18**.**
Let be a small abelian category with enough projectives. The category of inductive objects of is again an abelian category with enough projective objects and is equivalent to the large abelian category of product preserving functors from to .
Theorem 2.6.2**.**
[25, Theorem 1.3.3.8]* Let be an abelian category with enough projectives, let be an -category which admits geometric realizations, and let denote the full subcategory consisting of those functors which preserve geometric realizations. Then restriction along the embedding induces an equivalence of -categories*
[TABLE]
between realization preserving functors and functors .
Theorem 2.6.3**.**
[25, Theorem HA.1.3.3.2]* Let be an abelian category with enough projectives, let be a stable -category with a left complete t-structure, and let denote the full subcategory of functors which are right t-exact and send projective objects of to . Then restriction along induces an equivalence of -categories*
[TABLE]
between right t-exact functors which send to and right exact functors .
3 Ring theory
3.1 Spectra
So far we have considered the -category of spectra from two seemingly different but equivalent perspectives: on the one hand, as reduced excisive functors , and on the other, as a unit object of the symmetric monoidal -category of presentable stable -categories. The former is more explicit and yields lots of examples, while the latter is more abstract and suggests a universal property.
Remark 3.1.1**.**
The reader may be wondering what any of this has to do with the classical notion of spectrum in algebraic topology. By definition, , but evaluation on the family of spheres induces an equivalence of -categories
[TABLE]
Remark 3.1.2**.**
Our convention is that we work in the -category of spaces and its pointed variant , so that all (pointed) spaces have the homotopy type of cell complexes. If we wish to model a spectrum, however, it is often convenient to work in a category equipped with a class of weak equivalences and functor which sends the maps in to equivalences in , such as the category of pointed topological spaces. In such models we need only ask that the maps are weak equivalences, a structure which is often referred to as an -spectrum in the literature. In the case we can even require the to be homeomorphisms.
Remark 3.1.3**.**
The forgetful functor preserves limits, meaning we may also form this limit in . Equivalently, is given as the colimit
[TABLE]
in , by virtue of the antiequivalence which takes the adjoint.
Remark 3.1.4**.**
The forgetful functor does not commute with filtered colimits; nevertheless, the -category of finite — or equivalently, in this case, compact — spectra is computed as the filtered colimit of the suspension functor on finite pointed spaces . This equivalence
[TABLE]
in is the -categorical analogue of the Spanier-Whitehead category.
Remark 3.1.5**.**
We have taken the “coordinate free” convention that spectra are reduced excisive functors. Nevertheless, it is often the case in practice that a spectrum is given in terms of an infinite delooping of its infinite loop space , in which case the associated excisive functor is given by the formula
[TABLE]
Here, refers to the fact that , as a commutative -algebra in , is canonically left tensored over , and hence over the symmetric monoidal subcategory as well. Alternatively, this tensor is computed by first applying and then tensoring with in .
Remark 3.1.6**.**
The -category of spectra admits a left and right complete t-structure with heart the category of abelian groups. The identity functor is right exact and so determines a right t-exact functor with image the Eilenberg-MacLane spectra. Strictly speaking, these are the generalized Eilenberg-MacLane spectra: an Eilenberg-MacLane spectrum is a generalized Eilenberg-MacLane spectrum which has homotopy concentracted in a single degree, or equivalently is the image of a chain complex with homology concentrated in a single degree.
Definition 3.1.7**.**
A cohomology theory
is a -graded family of functors and natural isomorphisms satisfying the following exactness conditions:
- (1)
For any cofiber sequence integer the sequence of abelian groups is exact.
- (2)
For any (possibly infinite) wedge decomposition and integer , the homomorphism is an isomorphism.
Remark 3.1.8**.**
These are the Eilenberg-Steenrod axioms [16] for a (generalized) cohomology theory.
The original formulation included the dimension axiom,
which required that for all . One can show without much difficulty that the cohomology theories which satisfy the dimension axiom are necessarily of the form , which is to say cohomology with coefficients in the abelian group . This axiom was disregarded as interesting “generalized” cohomology theories were discovered.
Remark 3.1.9**.**
The Brown representability theorem
[13] asserts that any suitably left exact functor is representable. In particular,
[TABLE]
for some pointed space , and the resulting sequence of pointed spaces can be chosen so that the suspension isomorphisms
are induced by equivalences in (in fact the loop space structure induces the group structure on the represented functor, and any such group structure arises in this way). This is how spectra arose in practice.
Remark 3.1.10**.**
By choosing a representing spectrum, any cohomology theory in the classical sense as above gives rise to a cohomological functor in the -categorical sense, by which we mean a limit preserving functor . One can show that any such functor is necessarily a right adjoint, so that the -categories of spectra and cohomological functors are canonically equivalent:
[TABLE]
Similarly, , so that any cohomological functor of pointed spaces extends uniquely to a cohomological functor of spectra.
Remark 3.1.11**.**
A cohomological functor induces a functor which necessarily factors through the triangulated homotopy category of spectra. There is a version of Brown representability for triangulated categories which are compactly generated in the appropriate sense, due to Neeman [30]. This is not a corollary of the corresponding formal result for compactly generated stable -categories, namely that is the full subcategory consisting of the cohomological functors. Indeed, triangulated categories aren’t always homotopy categories of stable -categories and don’t necessarily even admit finite limits or colimits; rather, the requisite exactness properties are encoded by the triangulated structure.
Remark 3.1.12**.**
Much of the classical algebraic topology literature models spectra as a full subcategory of local objects inside of a larger category. Our definition of spectra is as the full subcategory of (reduced) excisive functors. The excisive approximation [24, Example 6.1.1.28] of a reduced functor is given by the formula
[TABLE]
As any finite space admits a cell decomposition, a reduced excisive functor is determined by its values on spheres; in fact, any sequence of spheres with whenever . Moreover, for any reduced , we have maps
[TABLE]
and hence maps . This motivates the next definition.
Definition 3.1.13**.**
A prespectrum consists of an -indexed collection of pointed spaces and structure maps .
To organize prespectra into an -category , it will be useful to write
[TABLE]
for the “shifted” suspension and loops functors and , where we take the convention that is contractible. Then is left adjoint to , and the -category of prespectra is defined by forming either of following the pullbacks :
[TABLE]
Remark 3.1.14**.**
A prespectrum gives rise to a sequence of representable functors which collectively represent a cohomology theory if and only if the map is an equivalence. The “diagonal” map identifies with the full subcategory of consisting of the equivalences, and the iterated pullback square
[TABLE]
exhibits as the equalizer of the endofunctors .
Remark 3.1.15**.**
This inclusion admits a left adjoint spectrification functor . Indeed, if is a prespectrum, the -space of the associated spectrum is given by the formula
[TABLE]
The equivalence is induced by the maps , which becomes the equivalence after passing to the colimit. This is essentially the same formula as in Remark 3.1.12.
Remark 3.1.16**.**
The -category of prespectra is quite useful in practice. As a source of examples, a pointed space evidently determines a suspension presprectrum whose associated spectrum is . Since the spectrification functor preserves colimits, the -category of prespectra can be used as a tool for computing colimits and smash products of spectra.
3.2 The smash product
The construction of a symmetric monoidal model category of spectra was a major foundational problem in the subject for quite some time. One issue is that there isn’t an obvious candidate for the smash product of prespectra; instead, given prespectra and , their smash product is most naturally indexed on the poset ; that is, . Adams’ theory of “handicrafted smash products” [3] shows that any cofinal choice of poset results in a prespectrum representing the smash product, and verifies that this procedure descends to a symmetric monoidal structure on the homotopy category of spectra. However this is insufficient for many purposes, especially as the homotopy category doesn’t admit even the most basic sorts of limits and colimits like pullbacks and pushouts.
Another more significant issue is already apparent in homological algebra: in the derived -category of chain complexes of modules over a commutative ring, the derived tensor product is really only defined up to quasi-isomorphism, so there’s no philosophical reason to expect this to lift to a symmetric monoidal structure on the ordinary category of chain complexes. The first resolutions of this problem in homotopy theory were the symmetric spectra of Hovey-Shipley-Smith [20] and the -modules of Elmendorf-Kriz-Mandell-May [17].
There are morphisms of -idempotent objects of
[TABLE]
necessarily symmetric monoidal, which enable us to calculate the tensor product in these presentable -categories. Specifically, writing for unique symmetric monoidal left adjoint functor, we deduce that
[TABLE]
for any finite collection of pointed spaces . The analogous result remains true for unpointed spaces by addition of a disjoint basepoint.
Remark 3.2.1**.**
Using the description of spectra as the limit of tower associated to the endofunctor , we obtain maps
[TABLE]
by projection to the factor. The reason for this notation is that we have equivalences ; indeed, if is a spectrum, . The collection of functors , or any infinite subset thereof, form a conservative family of functors in . They admit left adjoints which factor through .
Remark 3.2.2**.**
By [24, Proposition 6.3.3.6], any spectrum admits a canonical presentation by desuspended suspension spectra
[TABLE]
in which the maps correspond to the composites
[TABLE]
Using the fact that , we can write down explicit formulas for the spaces in the smash product of any finite sequence of spectra.
Remark 3.2.3**.**
Another characterization of the -fold smash product functor is as the derivative of the -fold cartesian product functor or its coreduction, the -fold smash product functor . See [25, Example 6.2.3.28] for further details.
Remark 3.2.4**.**
If is a fixed spectrum, the functor which sends to preserves colimits and therefore, according to the adjoint functor theorem, admits a right adjoint. This right adjoint is the mapping spectrum functor , which admits the following description: if is a spectrum, then is the spectrum given by the formula
[TABLE]
with structure maps
[TABLE]
where the first equivalence uses the fact that is a stable -category and therefore that the composite functor is equivalent to the identity.
In order to be able to work effectively with spectra, we need ordinary algebraic invariants such as homotopy groups. One way to obtain the homotopy groups of a spectrum is via the Postnikov t-structure, defined as follows.
Definition 3.2.5**.**
Let denote the full subcategory of spectra consisting of those objects such that is contractible.
Remark 3.2.6**.**
The functor is corepresented by the unit object of , which is compact. It therefore preserves limits and filtered colimits; furthermore, limits and filtered colimits of contractible spaces are contractible. Hence the inclusion preserves limits and filtered colimits and therefore, by the adjoint functor theorem, it admits a left adjoint. We write for the left adjoint followed by the right adjoint, so that any spectrum admits a natural unit map . The fiber of the unit map then determines an endofunctor , the connective cover. We write for the essential image of and .
Proposition 3.2.7**.**
[25, Proposition 1.4.3.6]* The pair of full subcategories and are the connective and coconnective parts of a t-structure on . Moreover, this t-structure is left and right complete, and its heart is canonically equivalent to the category of abelian groups.*
Remark 3.2.8**.**
The spectra that lie in the full subcategory are called the connective spectra, and we will often write in place of .
Remark 3.2.9**.**
The homotopy groups of a spectrum , defined via the t-structure on , collectively form a -graded abelian group
[TABLE]
Viewing as a sequence of pointed spaces equipped with equivalences , we can also obtain the homotopy groups of via the homotopy groups of the pointed spaces . Specifically, the nonnegative homotopy groups of are the homotopy groups of the underlying infinite loop space; that is, for ,
[TABLE]
The negative homotopy groups of , on the other hand, are the homotopy groups of a sufficiently high delooping; that is, for ,
[TABLE]
More generally, , where for we take to mean the homotopy type of the space .
Definition 3.2.10**.**
Let and be spectra. The -homology of , , is the graded abelian group . Dually, the -cohomology of , , is the graded abelian group .
Remark 3.2.11**.**
If is a pointed space, we write and . If is a spectrum representing a generalized cohomology theory, this recovers the -cohomology groups of the pointed space .
3.3 Associative and commutative algebras
Definition 3.3.1**.**
An associative ring spectrum is an algebra object of the monoidal -category of spectra. A commutative ring spectrum is a commutative algebra object of the symmetric monoidal -category of spectra.
We write and for the -categories of associative and commutative ring spectra, respectively.
Remark 3.3.2**.**
As the notation suggests, we often refer to associative or commutative ring spectra as associative or commutative algebra spectra, or just associative or commutative algebras when the symmetric monoidal -category of spectra is clear from context. The latter terminology is especially convenient in relative contexts, such as when we wish to work over444Over means over , reversing the direction of the arrows. Algebraically this means working under , but the terminology is influenced by geometric intuition. a fixed base commutative ring spectrum , where the corresponding notion is that of an associative or commutative -algebra (spectrum here is implicit).
Remark 3.3.3**.**
Recall that a commutative algebra object of the symmetric monoidal -category of spectra is a section
[TABLE]
of the cocartesian fibration such that is cocartesian whenever is inert. We will typically write in place of and refer to as the algebra object. The value of on the active map , pushed forward to the fiber over via this map, is the multiplication .
Similarly, an associative algebra object is a section
[TABLE]
of the restricted (along the cut map ) fibration which carries inert arrows to cocartesian arrows. We will typically also write for the underling spectrum of and refer to as the associative ring spectrum.
Remark 3.3.4**.**
A section amounts to a diagram of the form
[TABLE]
in . If it is an algebra object, the inert maps in force to decompose as the -fold product of under the equivalence , where we identify with as usual. Restricting to the active maps and pushing everything forward to the the fiber over via the inert maps, and writing , , for the images of , , , we obtain a diagram
[TABLE]
in , encoding exactly the maps one would expect from an algebra object. The commutative case is similar but more complex due to the permutations.
In certain situations, some of which we will encounter later, it is convenient to work with only the connective spectra. Recall that denotes the full subcategory consisting of the spectra whose negative homotopy groups vanish for all . In other words, as full subcategory of spectra, where is defined via the standard Postnikov t-structure. Since the tensor product of connective spectra is again connective, we actually obtain as a symmetric monoidal subcategory. Thus we have -categories and of connective associative and commutative algebra spectra.
Proposition 3.3.5**.**
[25, Proposition 7.1.3.19]* Postnikov towers converge in the presentable -categories , , and .*
Definition 3.3.6**.**
An -monoid is an algebra object of the symmetric monoidal -category of spaces. An abelian -monoid is a commutative algebra object of . An -monoid, or abelian -monoid, is said to an -group, or abelian -group, if the ordinary discrete monoid is a group.
Remark 3.3.7**.**
We have -categories and of -monoids and -groups, as well as their abelian variants and . The group completion of an -monoid is the left adjoint of the fully faithful inclusion , and is given by the formula .
Remark 3.3.8**.**
The reader might be wondering why we did not define associative and commutative ring spectra as homotopy coherently associative and commutative ring objects in the -category of spaces. The primary reason is that this only yields the connective ring spectra, and nonconnective spectra, even ring spectra, such as topological K-theory, are among our most important examples. Nevertheless it is true, and a good sanity check, that we have an equivalence , which induces equivalences and , where the latter notions are defined as in [18] (among other algebraic theories, such as semirings and their variants).
Definition 3.3.9**.**
A homotopy associative ring spectrum is an associative algebra object in the monoidal homotopy category of spectra. A homotopy commutative ring spectrum is a commutative algebra object in the symmetric monoidal homotopy category of spectra.
Remark 3.3.10**.**
Homotopy categories are ordinary categories. Hence there are no coherences to specify, and a homotopy associative ring spectrum is the data of a ring spectrum equipped with a unit map and a multiplication map which are associative and unital in the sense that the diagrams
[TABLE]
commute up to homotopy (a choice of homotopy is not part of the data). As in ordinary algebra, commutativity in this case is a property of a homotopy associative ring spectrum . While the theory of homotopy associative or commutative ring spectra plays an important role in algebraic topology, the categories of left (or right) modules over homotopy associative ring spectra are poorly behaved, lacking basic structure such as finite limits and colimits.
Proposition 3.3.11**.**
[25, Proposition 7.2.4.27]* The -categories and are compactly generated. Moreover, they are generated under sifted colimits by compact projective objects.*
Definition 3.3.12**.**
A connective associative ring spectrum is said to be locally of finite presentation if is compact as an object of . A connective commutative ring spectrum is said to be locally of finite presentation if is a compact as an object of .
3.4 Left and right modules
An associative (respectively, commutative) algebra is something that exists in a monoidal (respectively, symmetric monoidal) -category . Classically it was common to take to be the symmetric monoidal category of abelian groups, or (left) -modules for a commutative ring . Recall that a monoidal -category can be regarded as a cocartesian fibration over (equivalently a functor ) satisfying the Segal condition, the underlying category of is the value at the ordinal , and the simplicial object can be regarded as a categorical bar construction on , using its monoidal structure.
The ordinary bar construction generalizes as follows: if admits a left -action and admits a right -action, then we may form a simplicial object which in degree is equivalent to . The -categories and which arise in this way are said to be left and right tensored over , respectively; this is the structure which, when given an algebra object of , allows us to define the notions of left and right -module object of and . Such -categories are themselves cocartesian fibrations over , and hence over , but they satisfy a slight variant of the Segal condition.
Definition 3.4.1**.**
Let be a monoidal -category. An -category left tensored over is a cocartesian fibration together with a morphism of cocartesian fibrations over which satisfies the following relative version of the Segal condition: for each natural number , the map
[TABLE]
induced by and the inclusion of the final vertex is an equivalence.
Remark 3.4.2**.**
The morphism of cocartesian fibrations over is an example of a left action object of . More precisely, we write for the full subcategory consisting of those morphisms of cocartesian fibrations which satisfy the relative Segal condition of Definition 3.4.1. As in [25, Notation 4.2.2.5], there is a functor which sends the left action object to the monoidal -category .
Remark 3.4.3**.**
The underlying -category of the left tensored -category is the fiber over , which we will typically denote . There are equivalences for all . The Segal condition at gives an equivalence and the inclusion of the initial vertex induces a left action morphism .
Remark 3.4.4**.**
Dually, there is an entirely analogous notion of right tensored -category whose definition instead involves the inclusions of the initial vertices, with the action coming from the inclusion of the final vertices.
Remark 3.4.5**.**
A monoidal -category is canonically left (respectively, right) tensored over itself. Roughly, restricting the cocartesian fibration along the right (respectively, left) cone functor which sends to (respectively, to ) yields a cocartesian fibration whose fiber over is equivalent to . The morphism of cocartesian fibrations is obtained as in [25, Example 4.2.2.4]. See [25, Variant 4.2.2.11] for details and a comparison to the operadic approach.
Definition 3.4.6**.**
Let be a monoidal -category and let be an -category left tensored over . A left module object of is a map such that the composite is an algebra object of and, if is an inert map in such that , is a cocartesian morphism of . We write
[TABLE]
for the full subcategory consisting of the left module objects of .
Remark 3.4.7**.**
In the special case in which is equivalent to , regarded as being left tensored over itself via , we simply write in place of .
Remark 3.4.8**.**
There is an evident notion of right module object of an -category right tensored over a monoidal -category.
Definition 3.4.9**.**
A left module spectrum is a left module object of the monoidal -category of spectra.
We write for the -category of left module spectra.
Remark 3.4.10**.**
The -category of left module spectra comes equipped with a forgetful functor which returns the algebra object in the definition of a left module object. Given an associative algebra spectrum , we write for the fiber over of the forgetful functor. We also have a forgetful functor obtained by pulling back along the forgetful functor . We will sometime simply write for this forgetful functor, when it is clear from context that we are only considering left modules over commutative algebra spectra.
3.5 Localization
In this section we briefly review the theory of localization of ring spectra. Recall that a multiplicatively closed subset , containing the unit, of an associative ring is said to satisfy the left Ore condition if, for or every pair of elements and there exist elements and such that , and if is an element such that for some element then there exists an element such that .
Remark 3.5.1**.**
If is an associative algebra spectrum, any homogeneous element can be represented by a “degree ” left -module map which is unique up to homotopy.
Definition 3.5.2**.**
Let be an associative algebra spectrum and a set of homogeneous elements satisfying the left Ore condition. A left -module spectrum is -local if, for every element , left multiplication by induces an isomorphism . We write for the full subcategory consisting of the -local left -module spectra.
Proposition 3.5.3**.**
[25, Remark 7.2.3.18]* The inclusion of the full subcategory of -local objects admits a left adjoint , the -localization functor.*
Remark 3.5.4**.**
The theory of Bousfield localization generalizes that of Ore localization. In the stable setting, this is the data of a left adjoint functor of stable presentable -categories such that the right adjoint is fully faithful. The kernel of is the stable presentable subcategory consisting of those objects such that in . An instance of this construction is the localization of the -category of spectra at a fixed spectrum : in this case, a spectrum is -acyclic if and a spectrum is -local if every map from an -acyclic object is null. While it turns out that Bousfield localization preserves (commutative) algebra structures [25, Proposition 2.2.1.9], it is quite difficult to control the localization in this generality, which is why we focus on the Ore localization instead.
Remark 3.5.5**.**
It is possible to use the left Ore condition to give a reasonable explicit description of the homotopy groups of the left Ore localization , for and a set of homogeneous elements satisfying the left Ore condition. See [25, Construction 7.2.3.19 and Proposition 7.2.3.20] for details.
Definition 3.5.6**.**
Let be an associative algebra spectrum and a set of homogeneous elements. A map of associative algebra spectra exhibits as the left Ore localization of at if, for every associative algebra spectrum , is fully faithful with image those such that is invertible in for all .
Theorem 3.5.1**.**
[25, Proposition 7.2.3.27]* Let be an associative algebra spectrum and a set of homogeneous elements satisfying the left Ore condition. The localization admits the structure of an associative algebra equipped with an algebra map such that, for any associative algebra spectrum , precomposition with is fully faithful*
[TABLE]
with image those such that is invertible for all .
Remark 3.5.7**.**
If is a set of homogeneous elements satisfying the left Ore condition, the canonical map is an isomorphism of graded rings, where denotes the localization as graded rings.
Remark 3.5.8**.**
If is an associative algebra spectrum such that the graded ring is graded commutative, then the left Ore condition on a multiplicative subset is automatically satisfied.
There is an analogous statement for commutative algebra spectra.
Definition 3.5.9**.**
Let be a commutative algebra spectrum and a set of homogeneous elements. A map of commutative algebra spectra exhibits as the localization of at if, for every commutative algebra spectrum , is fully faithful with image those such that is invertible in for all .
Theorem 3.5.2**.**
[25, Example 7.5.0.7]* Let be a commutative algebra spectrum and a multiplicative set of homogeneous elements. The localization admits the structure of a commutative algebra equipped with a commutative algebra map such that, for any commutative algebra specrum , precomposition with is a fully faithful functor*
[TABLE]
with image those such that is invertible for all .
Remark 3.5.10**.**
Given a commutative algebra spectrum and an arbitrary subset of homogeneous elements of , we often write in place of , where denotes the multiplicative closure of in . Indexing the elements of by some ordinal , so that , we have equivalences of commutative algebra spectra , where the infinite tensor product is defined to be the filtered colimit of the finite tensor products.
A map of ordinary commutative rings is a Zariski localization if there exists a finite collection of elements , defining basic Zariski open sets , such that is isomorphic to the product map as objects of . There is a similar notion for ring spectra.
Example 3.5.11**.**
Let be a commutative ring spectrum and suppose given an element , which we regard as an -linear map via the equivalence . Then the filtered colimit
[TABLE]
is a -idempotent left -module: that is, the relative tensor product (see Section 4.2) is an equivalence . If follows that the associated Bousfield localization of is given the formula , and that has the following universal property: is contractible if and empty otherwise.
Example 3.5.12**.**
Let be a commutative ring spectrum and consider the affine scheme . The structure sheaf is determined by its values on the basic open sets , . Localizing at elements of allows us to enhance to a sheaf of commutative ring spectra on (the topological space of) by the formula . This is the affine spectral scheme .
Definition 3.5.13**.**
Let be a commutative ring spectrum. Then is local if is local, in the sense that there exists a unique maximal ideal . Equivalently, is local if, for any , either or is a unit.
4 Module theory
4.1 Monads
Given an -category , the -category of endofunctors of is canonically a monoidal -category with respect to composition. Viewing as a simplicial set and as simplicial sets, where composition is already a functor on the nose, we obtain a simplicial model for this monoidal -category, which moreover comes equipped with a strict left action on via the evaluation pairing .
Definition 4.1.1**.**
Let be an -category. A monad on is an algebra object of the monoidal -category of endofunctors of .
Remark 4.1.2**.**
We write for the underlying endofunctor of the monad. The unit and multiplication maps are usually denoted and . There are also homotopy coherent higher multiplications.
Remark 4.1.3**.**
Let be a monad on an -category . As is left tensored over via the evaluation map , it makes sense to consider left -module objects in (these are often referred to instead as -algebras, especially in ordinary category theory). We write for the -category of left -modules.
Example 4.1.4**.**
Let be a functor of -categories which admits a left adjoint . Then the composite functor admits a canonical structure of a monad on . Here, the unit is the unit of the adjunction, and the multiplication is induced by the counit . The formalism of adjunctions allows one to fill in all the coherences in a unique (up to a contractible space of choices) way.
Remark 4.1.5**.**
Let be a functor of -categories which admits a left adjoint and let denote the resulting monad on , as in the example above. Then factors as the composite , where is the functor which sends the object to the left -module , and is the forgetful functor.
Definition 4.1.6**.**
Let be a functor of -categories which admits a left adjoint and let be the resulting monad on . Then is monadic over if the induced map is an equivalence.
Knowing when a functor is monadic is quite important: for instance, monadic functors are conservative, and they exhibit the source as a kind of -category of modules in the target. Fortunately there is a recognition principle for monadic functors. To state it we’ll need the following notions.
Definition 4.1.7**.**
The category has an object for each integer and an arrow for each nondecreasing function with which composes in the obvious way. The subcategory has the same objects but only those arrows such that . Note that we may identify with the full subcategory of consisting of those objects with ; in fact, the category parametrizes augmented simplicial objects.
Definition 4.1.8**.**
An augmented simplicial object is split if extends to a functor . A simplicial object is split if it extends to a split augmented simplicial object. Finally, given a functor , an (augmented) simplicial object of is -split if is split as an (augmented) simplicial object of .
The monadicity theorem is a higher categorical analogue of the Barr-Beck Theorem. The result plays a considerably more important role higher categorically due to the difficulty of producing explicit constructions in this context.
Theorem 4.1.1**.**
[25, Theorem 4.7.3.5]* Let be a functor of -categories which admits a left adjoint . Then is monadic over if and only if is conservative, admits colimits of -split simplicial objects, and preserves colimits of -split simplicial objects.*
4.2 Relative tensor products
We now consider the -category of left modules over a base commutative ring spectrum (which could be the sphere itself, in the absolute case). To generalize ordinary algebra, we’d like to have a notion of (commutative) -algebra spectrum. In order to make this notion precise, we need a (symmetric) monoidal structure on the -category of left -modules.
Remark 4.2.1**.**
Using the language of -operads, these results can be refined to produce -monoidal -categories of left -modules when is only an -algebra spectrum. See [25] for the details of this approach.
As in ordinary algebra, the -category of (either left or right) modules for an associative ring spectrum will not carry a symmetric monoidal structure, which has as the unit and commutes with colimits in each variable, unless the algebra structure on extends to commutative algebra structure. Nevertheless, given a left -module and a right -module , we may form the iterated tensor products
[TABLE]
The algebra structure on and the left and right module structures on and organize these into a simplicial spectrum with
[TABLE]
Definition 4.2.2**.**
The relative tensor product is a spectrum equivalent to the geometric realization of the simplicial spectrum :
[TABLE]
Remark 4.2.3**.**
No noncanonical choices were involved in this construction, and indeed the relative tensor product can be shown to extend to a functor
[TABLE]
which preserves colimits separately in each variable. We therefore obtain a morphism in . By Morita theory, such a map is determined by a left -module, which in this case is itself.
Left adjoint functors between -categories of modules determine, and are determined by, bimodules. More precisely, if and are associative ring spectra, an -bimodule determines a functor via the relative tensor product, and conversely. We will avoid the theory of -bimodules by simply using the equivalent -category .
Theorem 4.2.1**.**
[25, Theorem 7.1.2.4]* Let be associative algebra spectra. The relative tensor product induces an equivalence of -categories*
[TABLE]
Remark 4.2.4**.**
The above equivalence sends the left -module to the functor and a left adjoint functor to , viewed as a left -module via its right -action.
Remark 4.2.5**.**
The right adjoint of the left adjoint functor is .
The following version of Morita theory, originally due to Schwede-Shipley [35], is a convenient recognition principle for -categories of modules.
Theorem 4.2.2**.**
[25, Theorem 7.1.2.1]* Let be a stable presentable -category and let be an object of . Then is compactly generated by if and only if the functor is an equivalence.*
Remark 4.2.6**.**
If is a map of associative ring spectra, then we may view as a left -module, in which case the resulting left adjoint functor is the basechange functor, with right adjoint the forgetful functor. Note that the forgetful functor preserves colimits, as they are detected on underlying spectra, so that this is the same as tensoring with the left -module , i.e. . In particular, there is a further right adjoint .
Example 4.2.7**.**
Suppose that is a localization of . Then the forgetful functor is fully faithful with essential image the -local left -module spectra, namely those such that .
4.3 Projective, perfect, and flat modules
We now study the -categories for an associative ring spectrum.
Remark 4.3.1**.**
The Morita theory of the previous section can be used to identify stable -categories of the form for an associative algebra spectrum , where now for some compact generator of . It follows that is compactly generated. Moreover, a stable -category is of the form if and only if admits a compact generator, and is the -completion of its full subcategory of compact objects.
Definition 4.3.2**.**
Let be an associative ring spectrum. A left -module is perfect if is a compact object of .
Definition 4.3.3**.**
Let be an associative ring spectrum. A left -module is free if is a (possibly infinite) coproduct of (unshifted) copies of , viewed as a left module over itself.
Definition 4.3.4**.**
A free left -module is finitely generated if it is equivalent to a finite coproduct of copies of .
Just as in ordinary algebra, there are other notions of “freeness” corresponding to various forgetful-free adjunctions. For instance, if is a left -module and to a ring map, then is often referred to as the “free” left -module associated to the left -module , even though is rarely a free -module (although it will be of course if was actually a free left -module).
Using the long exact sequence on homotopy groups, it is straightforward to check that a map of connective left -modules (over a connective associative ring ) is surjective if and only if is connective.
Definition 4.3.5**.**
Let be a connective associative ring spectrum. A left -module is projective if it is connective and projective as object of the -category of connective left -modules, in the sense that the corepresented functor preserves geometric realizations (that is, colimits of simplicial diagrams). See [24, Definition 5.5.8.18] for details.
Remark 4.3.6**.**
It is unreasonable to ask for a left -module to be projective as an object of the -category itself, as the only projective objects of this -category are the zero objects. Indeed, suppose that is a projective object of . For any -module N, we can write the suspension of as the geometric realization of the simplicial -module
[TABLE]
But , so for all left -modules and therefore .
Proposition 4.3.7**.**
[25, Proposition 7.2.2.6]* Let be a stable -category with a left complete t-structure and let be an object. The following conditions are equivalent:*
- (1)
* is projective (as an object of ).*
- (2)
For every , .
- (3)
For every and every integer , .
- (4)
For every and every integer , .
- (5)
For every exact triangle in , the map is surjective.
Proposition 4.3.8**.**
[25, Corollary 7.2.2.19]* Let be a map of connective associative algebra spectra such that is an isomorphism. The basechange functor restricts to an equivalence*
[TABLE]
on homotopy categories of projective objects. In particular, the [math]-truncation map induces an equivalence .
Proposition 4.3.9**.**
[25, Proposition 7.2.2.7]* Let be a connective associative algebra spectrum and a projective left -module. Then there exists a free left -module such that is a retract of . If additionally is finitely generated, we may take to be finitely generated as well.*
While the notion of projectivity really only makes sense over connective ring spectra, the notion of flatness makes sense over arbitrary ring spectra.
Definition 4.3.10**.**
Let be an associative ring spectrum. A left -module spectrum is said to be flat over if is a flat left -module, in the sense of ordinary algebra, and for each integer , the canonical map
[TABLE]
is an isomorphism.
Nevertheless, over a connective associative ring spectrum , the notion of flatness behaves in a manner more similar to that of ordinary algebra. For example we have the following important generalization of Lazard’s theorem.
Theorem 4.3.1**.**
[25, Theorem 7.2.2.15]* Let be a connective associative algebra spectrum and a connective left -module. The following conditions are equivalent:*
- (1)
* is flat.* 2. (2)
* is a filtered colimit of finitely generated free left -modules.* 3. (3)
* is a filtered colimit of finitely generated projective left -modules.* 4. (4)
The functor given by is left -exact. 5. (5)
If is a discrete right -module then is discrete.
Remark 4.3.11**.**
A free (respectively, projective) left -module is a filtered colimit of finitely generated free (respectively, projective) left -modules. Thus the statement of Lazard’s theorem remains true (albeit less precise) if we disregard finite generation.
Remark 4.3.12**.**
The Tor spectral sequence has -page
[TABLE]
and converges to the homotopy groups of the relative tensor product. If or is flat over , vanishes for , the spectral sequence degenerates at the -page, and is calculated as graded tensor product of and over .
Remark 4.3.13**.**
As a consequence we observe that if and are both flat over , then their tensor product is again flat over . Since the unit object of is flat, we see that the full subcategory inherits the structure of a symmetric monoidal -category.
In order to calculate in the -category of left -modules, it is useful to have something analogous to a projective resolution. If is connective, the theory of tor-amplitude plays this role, giving a means of construct any perfect -module inductively, in a finite number of steps, as an iterated cofiber of maps from shifted finitely projective left -modules.
Definition 4.3.14**.**
Let be a connective commutative ring spectrum. A left -module has tor-amplitude contained in the interval if for any discrete left -module , for . If such integers and exist, is said to have finite tor-amplitude.
If is an -module, then has tor-amplitude contained in if and only if is a complex of left -modules with tor-amplitude contained in in the ordinary sense. Note, however, that this definition differs from that in [2, I 5.2] as we work homologically as opposed to cohomologically.
Proposition 4.3.15**.**
[7, Proposition 2.13]* Let be a connective associative algebra spectrum and let and be left -module spectra.*
- (1)
If is perfect then has finite tor-amplitude. 2. (2)
If and has tor-amplitude contained in then the left -module has tor-amplitude contained in . 3. (3)
If and have tor-amplitude contained in then the fiber and cofiber of a map have tor-amplitude contained in and . 4. (4)
If is perfect with tor-amplitude contained in then is connective and . 5. (5)
If is perfect with tor-amplitude contained in then is equivalent to for a finitely generated projective left -module . 6. (6)
If is perfect with tor-amplitude contained in then there exists an exact triangle with finitely generated projective and perfect with tor-amplitude contained in .
Remark 4.3.16**.**
If additionally is a connective commutative algebra spectrum and and are left -modules such that has tor-amplitude contained in and has tor-amplitude contained in , then has tor-amplitude contained in .
4.4 Tensor powers, symmetric powers, and free objects
Given a map of associative algebra spectra , the basechange functor (corresponding to the left -module ) is left adjoint to the forgetful functor (corresponding to the left -module ). Indeed, the counit of the adjunction is induced from the left -module action map , and the unit of the adjunction is induced from the left -module unit map . In particular, is the free left -module on the left -module , by virtue of the equivalence
[TABLE]
Remark 4.4.1**.**
Observe that need not be free as a left -module in the sense of Definition 4.3.3 unless is free as a left -module.
Remark 4.4.2**.**
The forgetful functor is conservative and preserves all small limits and colimits. Hence it is monadic, and exhibits as the -category of left modules for the monad .
The forgetful functors preserve limits, so it is natural to ask whether or not they admit left adjoints. Using presentability and the adjoint functor theorem (see Remark 2.4.7), this is the case if and only if they preserve -filtered colimits for some regular cardinal . As we might expect from algebraic kinds of categories, they preserve all filtered colimits (as well as geometric realizations), so the left adjoints exist, and are instances of free algebra functors [25]. As is ordinary algebra, it is useful to consider the relative situation, so we work over a base commutative ring spectrum .
Proposition 4.4.3**.**
[25, Corollaries 3.2.2.4 and 3.2.3.2]* The forgetful functors preserve small limits and sifted colimits.*
Basically by construction, these forgetful functors are also conservative, hence monadic by the Barr-Beck-Lurie Theorem 4.1.1. This means that the free-forgetful adjunctions exhibit and as -categories of left modules for their respective monads. The underlying endofunctors of these monads are just in ordinary algebra, the tensor and symmetric algebra constructions.
Remark 4.4.4**.**
Straightening the symmetric monoidal -category to a functor and restricting to the active maps for each , we obtain -fold tensor power functors . That is, , where the tensor product is taken over .
Proposition 4.4.5**.**
[25, Proposition 4.1.1.18]* Let be a left -module. The free associative -algebra on is the tensor algebra*
[TABLE]
Remark 4.4.6**.**
This only describes the underlying left -module of the free associative -algebra on . The multiplication
[TABLE]
is given by concatenation of tensor powers. This is still only a small, but important, piece of the homotopy coherently associative algebra structure.
The free commutative algebra functor, also known as the symmetric algebra, uses the symmetric power functors , given by the formula
[TABLE]
where the tensor product is taken in the symmetric monoidal -category of left -module spectra. Here the subscript refers to the homotopy quotient of by the permutation action of the symmetric group , which is to say that quotient in the -categorical sense. This is to distinguish from more strict versions of quotients by group actions which arise through either ordinary categorical models of the -category , where it can make sense to ask for an ordinary categorical quotient, or in the context of equivariant homotopy theory, where there are several notions of fixed points.
Proposition 4.4.7**.**
[25, Example 3.1.3.14]* Let be a left -module. The free commutative -algebra on is the symmetric algebra*
[TABLE]
Remark 4.4.8**.**
While it isn’t terribly difficult to describe the multiplication on explicitly, organizing all of the higher multiplications in a coherent manner seems difficult without abstract machinery. The theory of operadic left Kan extensions [25] is one way to make this precise.
Example 4.4.9**.**
In the case where , we have . This is often denoted , as we have that on homotopy groups. As the notation suggests, the free tensor algebra on one generator in degree zero, , happens to be a commutative -algebra spectrum, though it is not the free commutative -algebra on one generator in degree zero. Instead we have that , which is sometimes denoted in order to distinguish it from . In this case, is the coproduct of the -homologies of the symmetric groups .
Remark 4.4.10**.**
If is a -algebra then, for all , the map is an isomorphism. This follows from the Atiyah-Hirzebruch spectral sequence and the vanishing of group cohomology in characteristic zero by Maschke’s theorem. Hence the map
[TABLE]
obtained by taking the homotopy quotient is an equivalence.
Remark 4.4.11**.**
Away from characteristic zero the map is typically not an isomorphism. In particular, the map is rarely an equivalence. Nevertheless, the fact that admits a commutative algebra structure (although it is not free as a commutative algebra) implies, by basechange along the commutative algebra map , that there is always a commutative -algebra map , which is an equivalence when is a -algebra and not typically otherwise.
There are other sorts of free algebra functors as well. The forgetful functor fails to be monadic because it isn’t conservative; it is, however, monadic on the full subcategory of connective spectra, the obvious subcategory on which it is conservative. The resulting monad is a higher categorical analogue of the free abelian group monad.
Definition 4.4.12**.**
Let be an -group (respectively, -monoid). The group ring (respectively, monoid ring) is the associative ring spectrum , with -algebra structure induced from that of via the symmetric monoidal functor .
Remark 4.4.13**.**
Group and monoid rings are a rich source of examples of ring spectra. For instance, toric varieties, which are locally modeled on the group and monoid rings like and , are combinatorial enough that they descend from the integers to the sphere, giving basic examples of spectral schemes such as the projective space [27, Construction 5.4.1.3].
Definition 4.4.14**.**
The general linear group of a ring spectrum is the -group of left -module automorphisms of .
Example 4.4.15**.**
Let , the free abelian -monoid on one generator. Then is the free commutative algebra spectrum on one generator in the sense that, via the equivalences
[TABLE]
specifying a commutative algebra map is the same as specifying a point of , the image of the generator of .
Example 4.4.16**.**
Let be the abelian -group given by the infinite loop space of the sphere . Under the equivalence between the -categories of connective spectra and abelian -groups, it follows that is the free abelian -group on one generator, or equivalently the group completion of the free abelian -monoid on one generator of the previous example.
By adjunction, for any commutative algebra spectrum , we have equivalences
[TABLE]
where the -group is equivalently the subspace of the -monoid consisting of the invertible components; that is, it fits into the pullback square
[TABLE]
Since corepresents the functor which sends the commutative algebra to its space of units, the “derived scheme” can be regarded as derived version of the multiplicative group scheme.
Example 4.4.17**.**
Any abelian -group with determines a “connective spectral abelian group scheme” with underlying ordinary scheme , the multiplicative group. While the previous example represents the functor of units, it is sometimes necessary to consider more “strictly commutative” derived analogues of . At the other extreme, one can consider as an -group, in which case the spherical group ring is a flat commutative -algebra such that, for any commutative ring spectrum ,
[TABLE]
This space of “strict units” of plays a central role in elliptic cohomology [26].
Thom spectra ([38], [29], [4]) are spectra which occur as quotients of the sphere by the action of a group. Again, it is somewhat more useful to have the version relative to a fixed commutative ring spectrum . By construction the -group is the universal group which acts on by -linear maps, so any -group over acts as well. The group homomorphism deloops to a map of pointed spaces .
Example 4.4.18**.**
The equivalence and compatible families of -group maps and give -group maps and . The Thom spectra and are the homotopy quotients and . While defined in an apparently abstract and formal way, there’s a surprising connection to geometry: is the bordism ring of unoriented manifolds, is the bordism ring of stably almost complex manifolds. The analogues for other tangential structures hold; for instance, is the stably framed bordism ring.
4.5 Smooth, proper, and dualizable objects
Definition 4.5.1**.**
A symmetric monoidal -category is closed if, for any object of , the right multiplication by functor admits a right adjoint .
Remark 4.5.2**.**
This allows for the construction of function objects of , naturally as a functor . In particular, there are natural equivalences
Definition 4.5.3**.**
Let be a closed symmetric monoidal -category. A dual of an is an object of the form , where denotes a unit object.
As duals are uniquely determined, we will write , or , for a dual of .
Remark 4.5.4**.**
There is a canonical evaluation map
[TABLE]
any map corresponding to the identity of under the equivalence .
Definition 4.5.5**.**
An object of a closed symmetric monoidal -category is dualizable if there exists a coevaluation map such that the compositions
[TABLE]
are equivalent to the identity.
We write for the full subcategory consisting of the dualizable objects of . It always contains at least one object, any unit object .
Example 4.5.6**.**
Let be a commutative ring spectrum and consider the closed symmetric monoidal -category of left -module spectra. Stability forces the full subcategory of dualizable objects to be closed under finite limits, colimits, and retracts [19, Theorem 2.1.3]. Since is compactly generated, it follows that the full subcategories of compact and dualizable objects coincide.
Given a commutative algebra object of , we may form the -category of left -module objects of . For instance, . Of course, , so . It is useful to consider the relative version of this: given a commutative ring spectrum , the relative tensor product equips the presentable -category with the structure of a commutative algebra object.
Definition 4.5.7**.**
Let be a commutative ring spectrum. An -linear -category is a left -module in .
We write for the -category of -linear categories and -linear functors (left -module maps).
Remark 4.5.8**.**
Since is stable, we have .
Remark 4.5.9**.**
is in fact closed symmetric monoidal: if and are -linear -categories, the -category of -linear functors (that is, -module morphisms in ) from to is again an -linear -category (a -module in ). The dual
[TABLE]
of is the -category of -linear functors from to . Hence an -linear -category is dualizable if there exists a coevaluation map such that the compositions
[TABLE]
are equivalent to the identity.
Proposition 4.5.10**.**
Let be a commutative ring spectrum. Then is a rigid symmetric monoidal -category; that is, all objects are dualizable.
Remark 4.5.11**.**
Consider the subcategory of compactly generated stable presentable -category and left adjoint functors which preserve compact objects. Then is equivalent, via the functor which restricts to the full subcategories of compact objects, to the -category of idempotent complete (that is, all idempotents split) small stable -categories and exact functors. The inverse equivalence is given by , which when restricted to , factors through the subcategory . Since stability and idempotent completeness are properties of small -categories, and a functor of small stable -categories is exact if and only if it is right exact, we deduce that is equivalent to a full subcategory of (see [24, Proposition 5.5.7.8] for details). It follows from [25, Proposition 4.8.1.4], using [25, Remark 2.2.1.2], that inherits a symmetric monoidal structure which is compatible with the symmetric monoidal structure on or . The -category is a commutative algebra object in , so we have a subcategory of compactly generated -linear categories and colimit and compact object preserving functors. Similar arguments show that inherits the structure of a symmetric monoidal -category from .
Proposition 4.5.12**.**
[7, Proposition 3.5]* An -algebra is compact in if and only if is compact in .*
Remark 4.5.13**.**
An object is dualizable in if and only if it is dualizable in and the evaluation and coevaluation morphisms lie in the (not full) subcategory .
Definition 4.5.14**.**
A compactly generated -linear category is proper if its evaluation map is in ; it is smooth if it is dualizable in and its coevaluation map is in . An -algebra is proper if is proper; it is smooth if is smooth.
Remark 4.5.15**.**
The property of being compact, smooth, or proper in is invariant under Morita equivalence.
Remark 4.5.16**.**
If is an -algebra, then is proper if and only if is a perfect -module. Indeed, the evaluation map is the map
[TABLE]
that sends to . Similarly, is smooth if and only if the coevaluation map
[TABLE]
which sends to the -module , exists and is in . So we see that is smooth if and only if is perfect as an -module.
Proposition 4.5.17**.**
[7, Lemma 3.9]* If is a smooth -linear category then for some -algebra spectrum .*
Proposition 4.5.18**.**
[25, Proposition 7.3.5.8]* Let be a commutative ring spectrum and an associative -algebra. If is compact in then is smooth. If is a smooth and proper then A is compact in .*
Corollary 4.5.19**.**
[25, Corollary 7.3.5.9]* Let be a commutative ring spectrum and let be an associative -algebra. Then A is smooth and proper if and only if it is compact as an object of and .*
Remark 4.5.20**.**
The noncommutative cotangent complex is the fiber
[TABLE]
of the multiplication map. Thus, in the noncommutative setting, is a smooth -algebra if and only if is a perfect left -module.
4.6 Nilpotent, local, and complete objects
In this section we fix a commutative ring spectrum and a finitely generated ideal . We write and, for any left -module , , regarded as a left -module via the commutative algebra map .
Remark 4.6.1**.**
There does not seem to be a good notion of ideal in the -category of commutative ring spectra. Rather, it seems that for most part, which is relevant are ideals in the discrete ring or graded ring . The problem is that the cofiber of an -module map need not admit a commutative multiplication, or even any multiplication at all. A good example is the Moore spectrum for a prime, which does not carry an associative algebra structure; in fact, doesn’t even support a unital binary multiplication map (see [34, Proposition 4]).
Remark 4.6.2**.**
The nilpotence theorem of Devinatz-Hopkins-Smith [14] states that if is a homotopy associative ring spectrum, then the kernel of the map consists entirely of nilpotent elements. This generalizes the Nishida nilpotence theorem [31], which states that every element of of positive degree is nilpotent. It is used to show that the only homotopy associative ring spectra with the property that is a graded field are (extensions of) the Morava K-theory spectra at the prime (suppressed from the notation) at height . Here , for a generator in degree , and . By [6], there is an essentially unique associative -algebra structure on , at least at odd primes.
Remark 4.6.3**.**
Given a left -module , the Bousfield localization of the -category of spectra at sits in a Verdier sequence
[TABLE]
where denotes the full subcategory of those spectra such that (the kernel of the multiplication by map ). Since the inclusion of the full subcategory evidently preserves colimits, it follows that this is a semi-orthogonal decomposition. It is a recollemont precisely when the right adjoint inclusion preserves colimits, which implies that the localization is smashing: .
Definition 4.6.4**.**
Let be a commutative ring spectrum, let be a finitely generated ideal, and let be an associative -algebra.
- (1)
A left -module is -nilpotent if .
- (2)
A left -module is -local if for every -nilpotent .
- (3)
A left -module is -complete if for every -local .
Remark 4.6.5**.**
Taking homotopy groups, we see that is -nilpotent if each , , is an -nilpotent left -module. Equivalently, is -nilpotent if every element of each is annihilated by some power of .
Remark 4.6.6**.**
It suffices to test -locality only on those -nilpotent objects which are also compact as left -modules. This is because the fully faithful inclusion of the -nilpotent left -modules is a colimit preserving functor of compactly generated stable -categories [27, Proposition 7.1.1.12], which implies that it admits a right adjoint . For instance, if is principle, then is a compact generator of and is generator of which is compact as an object of but not typically as an object of . In particular, .
Remark 4.6.7**.**
The fully faithful inclusion of the -local objects admits both a left adjoint and a right adjoint . The existence of the left adjoint follows from the fact that preserves limits, essentially by definition, and filtered colimits by the previous remark. The existence of the right adjoint follows from the fact that is exact and so it also preserves finite colimits, hence all colimits. It follows that the -localization functor is given by tensoring with .
Theorem 4.6.1**.**
[27, Theorem 7.3.4.1]* A left -module is -complete if each homotopy group is a derived -complete left -module, in the sense that .*
Remark 4.6.8**.**
The inclusion of the -complete objects preserves limits, again by construction, and -filtered colimits for . Indeed, this follows from the fact that we need only test completeness on a generator of , and this generator is -compact in for some .
Proposition 4.6.9**.**
[27, Proposition 7.2.4.4 and Proposition 7.3.1.4]* The fully faithful inclusions and induce Verdier sequences*
[TABLE]
which are semi-orthogonal decompositions of .
Remark 4.6.10**.**
In the stable setting, a recollement
is the data of a fully faithful inclusion of a stable subcategory which admits both a left and a right adjoint [8]. An example is the inclusion of the -local objects. The composite is an equivalence of categories with inverse . A stable recollement determines a fracture square, a cartesian square
[TABLE]
of endofunctors of . For , the arithmetic square is the cartesian square
[TABLE]
obtained by completing a spectrum at all primes and rationalization.
Theorem 4.6.2**.**
[27, Proposition 7.4.1.1]* Let be a commutative ring spectrum and let a finitely generated ideal. Suppose given a map of associative -algebras , and consider the commutative square*
[TABLE]
in . If is an equivalence, this square is cartesian.
5 Deformation theory
5.1 The tangent bundle and the cotangent complex
The cotangent complex formalism is an instance of the fiberwise stabilization of a presentable fibration. Given a pair of presentable fibrations and , we write for the full subcategory of those functors over which admit a left adjoint such that is an equivalence in for every object , where denotes any choice of unit transformation exhibiting the adjunction.
Remark 5.1.1**.**
This is the precise condition needed to ensure that restricts to a right adjoint on fibers over any object of , or even after pullback along any morphism .
Definition 5.1.2**.**
A stable envelope of a presentable fibration is a presentable fibration equipped with a morphism
[TABLE]
of presentable fibrations over which exhibits as the fiberwise stabilization of over in the following sense: if is a stable presentable fibration, the induced map is an equivalence.
Definition 5.1.3**.**
A tangent bundle of a presentable -category is a stable envelope of the target fibration .
Remark 5.1.4**.**
The fiber of over the object is the stabilization of the fiber of the target fibration .
Remark 5.1.5**.**
The tangent bundle admits an explicit construction as the -category of unreduced excisive functors
[TABLE]
In this case, the morphism of presentable fibrations over is given by evaluating at the arrow ; that is, .
Let be a presentable -category, and consider the commutative triangle
[TABLE]
where is the target fibration. A relative version of the adjoint functor theorem implies that admits a left adjoint, allowing for the following construction.
Definition 5.1.6**.**
The absolute cotangent complex functor
is the composition , where the first map is the diagonal (constant) embedding and the second is left adjoint to . We will write for the value of at the object of , and refer to as the (absolute) cotangent complex of .
Remark 5.1.7**.**
The cotangent complex has a long history. Its first incarnation was as the sheaf of Kähler differentials, and a derived version of this was introduced by Berthelot and Illusie ([2], [21], [22]). Around the same time, Quillen ([33], [32]) and André ([5]) defined a derived version in the context of (simplicial) commutative rings, and later Basterra and Mandell ([9], [10]) developed the theory in the more general context of commutative algebra spectra, where they refer to it as topological André-Quillen homology.
Remark 5.1.8**.**
Let be an object of . The identification of the fiber of the tangent bundle over with is such that corresponds to the image of under .
Remark 5.1.9**.**
The diagonal embedding is left adjoint to the evaluation at . Hence the cotangent complex functor is left adjoint to the composite functor
[TABLE]
Definition 5.1.10**.**
The relative cofiber over functor
[TABLE]
is the functor which sends the morphism in to the pushout
[TABLE]
where is any zero object of the fiber of over .
Definition 5.1.11**.**
The relative cotangent complex functor is the composition
Remark 5.1.12**.**
A zero object of a fiber of need not be a zero object of itself. Given a morphism in , the relative cofiber of is an object of the fiber of over .
We write for the value of the relative cotangent complex functor on an object of the -category of arrows in .
Remark 5.1.13**.**
By construction, the relative cotangent complex of a morphism fits into a relative cofiber sequence
[TABLE]
in . This induces an actual cofiber sequence in the -category . Here is a straightening of the restriction of along .
Remark 5.1.14**.**
It follows that the commutative square in
[TABLE]
associated to a pair of composible morphisms and in is cocartesian, hence a relative cofiber sequence since in .
Definition 5.1.15**.**
The tangent correspondence of a presentable -category is the cocartesian fibration associated to the cotangent complex functor .
Remark 5.1.16**.**
The tangent correspondence is also a cartesian fibration since is a left adjoint.
Remark 5.1.17**.**
The cocartesian fibration associated to a functor can be constructed as the pushout
[TABLE]
in which the left vertical map is the inclusion at . The functor has fiber over [math] and the full subcategories and , and over the unique map the functor ; this is evidently a cocartesian fibration as we can push forward objects of along via . Thus a functor amounts to the data of a functor and a natural transformation from to . If is also a cartesian fibration, a unit transformation induces a canonical functor .
Remark 5.1.18**.**
A derivation in is a morphism such that the composite is the identity and is constant. More concretely, a derivation in is a morphism in from an object in the fiber over [math] to an object in the fiber of over (see [25, Remark 7.4.1.2]). The -category of derivations in is the pullback
[TABLE]
in which the right vertical map is induced by the functor and the bottom horizontal map is adjoint to the identity of .
5.2 Derivations and square-zero extensions
We now specialize to the case in which is the presentable -category of commutative algebra spectra.
Remark 5.2.1**.**
All algebras will be assumed commutative for the remainder of this article. If is a commutative algebra object there is a canonical equivalence , hence a canonical equivalence , so we needn’t distinguish left and right module structures in the commutative case. We thus write
in place of and .
Theorem 5.2.1**.**
[25, Corollary 7.3.4.14]* The functor which sends the pair to exhibits as a tangent bundle of . In particular, for any commutative algebra spectrum , we have an equivalence .*
We will write
for a functor which corresponds to . We use this notation because is the formula for the split square-zero extension in ordinary algebra, though we have obtained its augmented commutative algebra structure through abstract stabilization techniques.
Remark 5.2.2**.**
Let be a commutative ring spectrum and an -module. Then the split square-zero augmented commutative -algebra structure on is square-zero in the sense that the compositions
[TABLE]
are null whenever .
Remark 5.2.3**.**
On homotopy groups, the split square-zero extension is an ordinary split square-zero extension of graded commutative rings. In other words, for any pair of elements and of , the multiplication on is given by the expected formula
[TABLE]
Remark 5.2.4**.**
Any split square-zero extension , viewed as an augmented commutative -algebra, is canonically a spectrum object in the pointed -category . Indeed, in degree , , and the map
[TABLE]
is an equivalence, as is a zero object of and
[TABLE]
is a cartesian square of . Moreover, as is stable, the functor factors through . There is even an evident map back in the order direction given by taking the fiber, which sends the spectrum object to .
In ordinary commutative algebra, a derivation over is an -module map satisfying the Leibniz rule
[TABLE]
Instead of using elements, we could instead define a derivation as a section of the projection , taken in the category of commutative -algebras. Replacing with we obtain the following notion.
Definition 5.2.5**.**
Let be a commutative algebra spectrum and an -module. A derivation from to is a section in of the projection .
Remark 5.2.6**.**
This is a special case of the notion of derivation introduced in the previous section. Here we avoid explicit mention of the tangent correspondence by pulling back to the fiber of via the split square zero extension functor .
Remark 5.2.7**.**
The space
of derivations is the fiber
[TABLE]
over the identity . The composite of the commutative algebra map with the second projection is a map of spectra which we will abusively refer to as the derivation.
Remark 5.2.8**.**
For any connective commutative algebra and left -module , there is an equivalence That is, the cotangent complex corepresents the functor .
Remark 5.2.9**.**
Using the tangent correspondence formalism from the previous subsection, the -category
of derivations has objects derivations and morphisms commutative squares of the form
[TABLE]
with a commutative algebra map and an -module map.
Let be a commutative ring spectrum, an -module, and a derivation with associated section of the projection . By construction, is a morphism of commutative algebras.
Definition 5.2.10**.**
A map of commutative algebra spectra is a square-zero extension
of by the -module if there exists a derivation and a cartesian square in of the form
[TABLE]
Remark 5.2.11**.**
There is a functor
which sends the derivation to the map , where is a pullback
[TABLE]
in , and whose essential image consists of the square-zero extensions.
Remark 5.2.12**.**
Note that, if is a square-zero extension of by , then the fiber sequence of -modules implies that we also have a fiber sequence and hence a canonical equivalence . Hence derivations give rise to square-zero extensions of by . This is why we use instead of itself.
Remark 5.2.13**.**
As the name suggests, square-zero extensions are actually square-zero. That is, the fiber has the property that the -fold symmetric power map is null for any .
Definition 5.2.14**.**
A morphism in is an -small extension
if has homotopy concentrated in degrees and the multiplication map is nullhomotopic. A derivation is -small if the associated square-zero extension is -small.
Theorem 5.2.2**.**
[25, Theorem 7.4.1.26]* The composition*
[TABLE]
is fully faithful with essential image the full subcategory of consisting of the -small extensions (here is as in Remark 5.2.11 above).
Corollary 5.2.15**.**
Any -small extension is a square-zero extension.
One of the primary source of examples of square-zero extensions comes from the Postnikov tower of a connective commutative algebra .
Proposition 5.2.16**.**
[25, Corollary 7.4.1.28]* Let . For each , the map*
[TABLE]
exhibits as a square-zero extension of by .
Remark 5.2.17**.**
The fact that the Postnikov tower is composed of square-zero extensions is one of the main reasons why the cotangent complex plays such an important role in spectral algebra and geometry. The space of maps
[TABLE]
between connective commutative algebra spectra and decomposes as the limit of the space of maps between their truncations, and for any we have a pullback diagram
[TABLE]
This implies that the fibers of are accessible via infinitesimal methods: arguing inductively up the Postnikov tower, we are reduced to understanding spaces of derivations from , an -linear question concerning maps .
5.3 Deformations of commutative algebras
Given a connective commutative algebra spectrum and a square-zero extension of by a connective -module , it is natural to study the space of deformations of the square-zero extension to a connective commutative -algebra . That is, we wish to understand the space of cocartesian squares in of the form
[TABLE]
(it turns out that is automatically also a square-zero extension). As in ordinary algebra, this reduces to a module theoretic question concerning the cotangent complex: such a commutative -algebra exists if and only if the map induced by a derivation classifying factors through the absolute cotangent complex of .
Definition 5.3.1**.**
Let be a commutative algebra spectrum, a square-zero extension of by an -module , and a commutative -algebra. A deformation of to
is a commutative -algebra equipped with an equivalence of commutative -algebras.
Remark 5.3.2**.**
We need not assume that is flat over , as this is the case if and only if is flat over . Indeed, if is flat, then the basechange is flat by a spectral sequence argument. Conversely, the fact that any deformation of along a flat map results in a flat map follows from a tor-amplitude argument.
Remark 5.3.3**.**
In the connective case, just like in ordinary commutative algebra, there are cohomological obstructions to the existence and uniqueness of deformations.
Given an -linear map with a connective -module, the associated square-zero extension of by is again connective. Given a morphism of connective commutative algebra spectra , we obtain a map , and a deformation of along exists if and only if factors as a composite
[TABLE]
of and a -module map . In this case, the deformation is recovered as the square-zero extension . Notice, though, that such a factorization exists if and only if the composite
[TABLE]
is null, so that a deformation exists if and only if the obstruction class in corresponding to this map vanishes. The set of equivalence classes of deformations is a torsor for the group ; in particular, it might be empty, or only admit an element locally on .
Definition 5.3.4**.**
A derivation is said to be connective if is a connective commutative algebra spectrum and is an connective -module.
Let denote the subcategory consisting of the connective derivations and those morphisms of connective derivations
[TABLE]
such that the induced map is an equivalence. Similarly, let denote the subcategory consisting of those objects such that both and are connective and those morphisms the squares
[TABLE]
which are cocartesian in ; in other words, is an equivalence. When restricted to , the square zero extension functor of Remark 5.2.11 factors through .
Theorem 5.3.1**.**
[25, Theorem 7.4.2.7]* The composition*
[TABLE]
factors through the subcategory , and the resulting functor is a left fibration (a cocartesian fibration with -groupoid fibers).
Proposition 5.3.5**.**
[25, Proposition 7.4.2.5]* For any connective derivation , induces an equivalence*
[TABLE]
where denotes the subcategory defined above.
5.4 Connectivity results
A universal derivation
is any derivation which is corresponds to an equivalence . Given a map of commutative ring spectra ,
[TABLE]
is a commutative square in . Taking vertical cofibers, we obtain map of -modules which is adjoint to a map of -modules . We will write for this map.
Theorem 5.4.1**.**
[25, Theorem 7.4.3.1]* Let be a morphism in such that for some . Then .*
Example 5.4.1**.**
We note the elementary fact that if then, for any natural number , and consequently as well. Thus if is the projection to , it is straightforward to show that . The general case is obtained from this special case by connectivity and induction arguments.
Proposition 5.4.2**.**
[25, Corollary 7.4.3.2]* Let be a map of connective commutative algebra spectra such that is -connective for some . Then the relative cotangent complex is -connective, and the converse holds provided that is an isomorphism.*
Remark 5.4.3**.**
The absolute cotangent complex of a connective commutative algebra spectrum is itself connective. This follows immediately from the previous proposition since the cofiber of the unit map is connective.
Corollary 5.4.4**.**
Let be a map of connective commutative algebra spectra. Then is an equivalence if and only if is an isomorphism and the relative cotangent complex vanishes.
Remark 5.4.5**.**
Let be a map of connective commutative algebra spectra such that is -connective for some . The induced map factors as the composite
[TABLE]
and the equivalence , together with the connectivity of and , imply that is -connective. We also have an exact triangle
[TABLE]
in which and are -connective, so that is -connective as well. It follows that the cofiber of is -connective..
Proposition 5.4.6**.**
[25, Lemma 7.4.3.8]* Let A be a connective commutative algebra spectrum. There are canonical isomorphisms of -modules*
[TABLE]
Remark 5.4.7**.**
Let be a map of connective commutative algebra spectra such that is -connective. Tensoring the exact triangle with the -module , we obtain an exact triangle
[TABLE]
which exhibits as a -connective map. Composing with , we obtain a -connective map .
Proposition 5.4.8**.**
[25, Proposition 7.4.3.9]* Let be a morphism in . Then is connective and .*
Theorem 5.4.2**.**
[25, Theorem 7.4.3.18]* Let be a morphism in . If is locally of finite presentation over , is a perfect -module. The converse holds provided is of finite presentation over .*
5.5 Classification of étale maps
Recall that a map of discrete commutative rings is said to be étale if is a finitely presented flat commutative -algebra such that the multiplication map is the projection onto a summand. Geometrically, this is the algebraic analogue of a (not necessarily surjective) covering space: there exists a commutative ring and a cartesian square of schemes of the form
[TABLE]
Étale maps of commutative rings are smooth of relative dimension zero [37, Lemma 10.141.2]. In particular, there exists a presentation of as a commutative -algebra of the form , where the are a sequence of elements of such that the image of the Jacobian matrix of partial derivatives is invertible in .
Definition 5.5.1**.**
A map of commutative algebra spectra is étale
if exhibits as a finitely presented commutative -algebra such that is a flat -module and is étale.
Ideally, we would like to be able to calculate the space of étale maps between commutative algebra spectra and in terms of the set of étale maps between their underlying discrete commutative algebras and . Since an étale map induces an étale map , we can address this question directly by studying the space of étale lifts of an étale morphism . The main result of this section, one of the major results of higher algebra, is that the space of such lifts is contractible.
Proposition 5.5.2**.**
[25, Remark 7.5.1.7]* Given a commutative triangle*
[TABLE]
in , if is étale, then is étale if and only if is étale. In particular, any map between étale commutative -algebras is automatically étale.
Remark 5.5.3**.**
Let be a map of connective commutative ring spectra. Then is étale if and only if each is étale.
Remark 5.5.4**.**
Since an étale map of discrete rings is smooth of relative dimension zero, its module of relative Kähler differentials vanishes, and one can show using ordinary algebraic methods that its relative cotangent complex also vanishes. This begs the question of whether or not the relative cotangent complex of an étale map of commutative algebra spectra vanishes. Using flatness, one reduces to the connective case, so that is also connective, and if there’s a least with . This is a contradiction: by connectivity considerations as in the previous section,
[TABLE]
Hence for any étale map .
Remark 5.5.5**.**
Said differently, this means that the absolute cotangent complex functor satisfies étale basechange:
if is an étale map of commutative algebra spectra, then the exact triangle
[TABLE]
together with the vanishing of the relative cotangent complex implies that .
Let denote the (not full) subcategory consisting of the connective derivations and those morphisms of connective derivations
[TABLE]
such that the induced map is an equivalence and the commutative algebra map is étale. Let denote the (not full) subcategory consisting of the (connective) commutative algebra spectra and the étale maps.
Remark 5.5.6**.**
The forgetful functor is a cocartesian fibration such that, for each , the fiber is an -groupoid (i.e., it is a left fibration). This is because an étale morphism of derivations is cocartesian: given connective derivations and and a map , we have an equivalence , so if we obtain an equivalence . In particular, for any connective derivation , the forgetful functor induces an equivalence
Remark 5.5.7**.**
Consider the functor which sends the derivation to the square-zero extension . Compositing with the restrictions to the set of vertices of , we obtain functors and such that restricts to the left fibration . For a given connective derivation , and induce functors
[TABLE]
such that is an equivalence by Proposition 5.3.5 and and is an equivalence by the remark above. We obtain the following corollary.
Corollary 5.5.8**.**
Let be a square-zero extension of connective commutative ring spectra. The relative tensor product
[TABLE]
induces an equivalence from the -category of étale commutative -algebras to the -category of étale commutative -algebras.
Theorem 5.5.1**.**
[25, Corollary 7.5.4.3]* For any commutative algebra spectrum , induces an equivalence .*
Remark 5.5.9**.**
Note that any étale commutative -algebra is automatically in since the flatness condition implies that must be discrete: if . Hence , and the theorem asserts that the -category of étale commutative -algebras is equivalent to the ordinary category of étale commutative -algebras.
Remark 5.5.10**.**
If is a commutative ring spectrum, the structure of the -category of étale commutative -algebras implies that the small étale site of is equivalent to the small étale site of the discreet ring , and the analogous result holds for the small Zariski sites of Example 3.5.11. These facts form the cornerstones of spectral algebraic geometry, as treated in [27] and [39].
Remark 5.5.11**.**
There are robust notions of spectral scheme and Deligne-Mumford stack. Versions of the Artin representability
theorem for these higher categorical objects are formulated and proved as [27, Theorems 18.1.0.1 and 18.1.0.2], providing necessary and sufficient conditions for a functor to be represented by such an object. These conditions are surprisingly straightforward: the ordinary stack must be represented by an ordinary scheme or Deligne-Mumford stack, must admit a cotangent complex, must preserve limits of Postnikov towers, and must preserve pullbacks of diagrams of the form in in which both of the maps and surjective on with nilpotent kernel.
Index
-
-group Definition 3.3.6
-
-monoid Definition 3.3.6
-
abelian -group Definition 3.3.6
-
abelian -monoid Definition 3.3.6
-
active map Definition 2.3.2
-
adjoint functor theorem Remark 2.4.7
-
affine spectral scheme Example 3.5.12
-
algebra object Definition 2.3.14
-
associative ring spectrum Definition 3.3.1
-
bounded t-structure Definition 2.6.14
-
Brown representability theorem Remark 3.1.9
-
cartesian
-
arrow Definition 2.2.1
-
cartesian symmetric monoidal structure Example 2.3.6
-
closed
-
symmetric monoidal -category Definition 4.5.1
-
coevaluation map Definition 4.5.5
-
cofiber sequence
-
relative Remark 5.1.13
-
cohomological functor Remark 3.1.10
-
cohomology theory Definition 3.1.7
-
commutative algebra object Definition 2.3.7
-
commutative ring spectrum Definition 3.3.1
-
connective
-
spectrum Remark 3.2.8
-
connective cover Remark 3.2.6
-
cotangent complex
-
absolute Definition 5.1.6
-
relative Definition 5.1.11
-
Day convolution product Remark 2.4.11
-
deformation
-
obstruction classes Remark 5.3.3
-
of commutative algebra spectra Definition 5.3.1
-
derivation Definition 5.2.5
-
universal §5.4
-
derived -category Definition 2.6.15
-
unbounded Example 2.6.4
-
differential graded category Definition 1.3.4
-
dimension axiom Remark 3.1.8
-
Dold-Kan correspondence Remark 1.3.6
-
dual Definition 4.5.3
-
dualizable object Definition 4.5.5
-
Eilenberg-MacLane spectrum Remark 3.1.6
-
generalized Remark 3.1.6
-
Eilenberg-Steenrod axioms Remark 3.1.8
-
étale
-
basechange Remark 5.5.5
-
map Definition 5.5.1
-
evaluation map Remark 4.5.4
-
exact
-
functor Definition 2.5.5
-
excisive approximation Remark 3.1.12
-
excisive functor Definition 2.5.7
-
unreduced Remark 5.1.5
-
fibration
-
cartesian Definition 2.2.1
-
cocartesian §2.2
-
finitely generated
-
free module Definition 4.3.4
-
flat module Definition 4.3.10
-
free
-
algebra §4.4
-
associative -algebra Proposition 4.4.5
-
commutative -algebra Proposition 4.4.7
-
left module §4.4
-
module Definition 4.3.3
-
function object Remark 4.5.2
-
fundamental -groupoid §1.4
-
general linear group Definition 4.4.14
-
Grothendieck construction §2.2
-
group completion Remark 3.3.7
-
group ring Definition 4.4.12
-
heart of a t-structure Definition 2.6.8
-
homological functor Remark 2.6.12
-
homotopy coherent nerve Remark 1.3.3
-
homotopy groups Definition 2.6.11
-
inert map
-
in and Definition 2.3.11
-
left complete t-structure Definition 2.6.13
-
left exact
-
functor Definition 2.5.5
-
left module
-
object Definition 3.4.6
-
spectrum Definition 3.4.9
-
left Ore condition §3.5
-
left -exact functor Definition 2.6.2
-
left tensored -category Definition 3.4.1
-
local commutative ring spectrum Definition 3.5.13
-
local object Definition 3.5.2
-
localization Definition 3.5.9
-
functor Proposition 3.5.3
-
left Ore Definition 3.5.6
-
Zariski §3.5
-
locally of finite presentation Definition 3.3.12
-
monad Definition 4.1.1
-
monadic functor Definition 4.1.6
-
monoid ring Definition 4.4.12
-
monoidal -category Definition 2.3.13
-
monoidal functor Definition 2.3.13
-
Morita theory §4.2
-
-small extension Definition 5.2.14
-
-truncated
-
object Definition 2.4.14
-
space Definition 2.4.13
-
perfect module Definition 4.3.2
-
pointed -category Definition 2.5.1
-
Postnikov t-structure Example 2.6.5
-
Postnikov tower §2.4
-
presentable -category Definition 2.4.4
-
presentable fibration Definition 2.4.6
-
presentable symmetric monoidal -category Definition 2.4.8
-
presheaf -category Definition 2.1.1
-
prespectrum Definition 3.1.13
-
projective
-
object Proposition 3.3.11
-
projective module Definition 4.3.5
-
projective object Definition 2.1.7
-
recollement Remark 4.6.10
-
reduced functor Definition 2.5.7
-
relative tensor product Definition 4.2.2
-
representability theorems Remark 5.5.11
-
right complete t-structure Definition 2.6.13
-
right exact
-
functor Definition 2.5.5
-
right module
-
object Remark 3.4.8
-
right -exact functor Definition 2.6.2
-
right tensored -category Remark 3.4.4
-
semi-orthogonal decomposition Definition 2.5.20
-
sifted
-
simplicial set Definition 2.1.5
-
smash product
-
of spaces Remark 1.4.3
-
of spectra Example 2.5.16
-
Spanier-Whitehead -category Remark 3.1.4
-
spectral sequence Remark 4.3.12
-
Atiyah-Hirzebruch Remark 4.4.10
-
spectrum object Definition 2.5.8, Definition 2.5.8
-
split augmented simplicial object Definition 4.1.8
-
square-zero extension Definition 5.2.10
-
split §5.2
-
stable -category Definition 2.5.4
-
stable envelope Definition 5.1.2
-
straightening §2.2
-
suspension isomorphism Remark 3.1.9
-
symmetric algebra Proposition 4.4.7
-
symmetric monoidal
-
functor Definition 2.3.4
-
symmetric monoidal -category Definition 2.3.4
-
-exact functor Definition 2.6.2
-
t-structure Definition 2.6.1
-
tangent bundle Definition 5.1.3
-
tangent correspondence Definition 5.1.15
-
tensor algebra Proposition 4.4.5
-
tensor product
-
of presentable -categories Theorem 2.4.1
-
Thom spectra §4.4
-
tor-amplitude Definition 4.3.14
-
triangle Definition 2.5.2
-
exact Definition 2.5.2
-
left exact Definition 2.5.2
-
right exact Definition 2.5.2
-
truncation
-
associated to a t-structure Remark 2.6.10
-
functor Proposition 2.4.15
-
unstraightening §2.2
-
Verdier sequence Definition 2.5.20
-
Yoneda embedding Remark 2.1.2
-
zero object Definition 2.5.1
-
-compact object Definition 2.1.7, Definition 2.4.3
-
-compactly generated Definition 2.4.4
-
-filtered simplicial set Definition 2.1.5
-
-small simplicial set Definition 2.1.5
-
-spectrum Remark 3.1.2
The reference list from the paper itself. Each links out to its DOI / PubMed record.
- 1[1] Revêtements étales et groupe fondamental . Lecture Notes in Mathematics, Vol. 224. Springer-Verlag, Berlin-New York, 1971. Séminaire de Géométrie Algébrique du Bois Marie 1960–1961 (SGA 1), Dirigé par Alexandre Grothendieck. Augmenté de deux exposés de M. Raynaud.
- 2[2] Théorie des intersections et théorème de Riemann-Roch . Lecture Notes in Mathematics, Vol. 225. Springer-Verlag, Berlin-New York, 1971. Séminaire de Géométrie Algébrique du Bois-Marie 1966–1967 (SGA 6), Dirigé par P. Berthelot, A. Grothendieck et L. Illusie. Avec la collaboration de D. Ferrand, J. P. Jouanolou, O. Jussila, S. Kleiman, M. Raynaud et J. P. Serre.
- 3[3] J. F. Adams. Stable homotopy and generalised homology . University of Chicago Press, Chicago, Ill.-London, 1974. Chicago Lectures in Mathematics.
- 4[4] Matthew Ando, Andrew J. Blumberg, David Gepner, Michael J. Hopkins, and Charles Rezk. An ∞ \infty -categorical approach to R 𝑅 R -line bundles, R 𝑅 R -module Thom spectra, and twisted R 𝑅 R -homology. J. Topol. , 7(3):869–893, 2014.
- 5[5] Michel André. Homologie des algèbres commutatives . Springer-Verlag, Berlin-New York, 1974. Die Grundlehren der mathematischen Wissenschaften, Band 206.
- 6[6] Vigleik Angeltveit. Uniqueness of Morava K 𝐾 K -theory. Compos. Math. , 147(2):633–648, 2011.
- 7[7] Benjamin Antieau and David Gepner. Brauer groups and étale cohomology in derived algebraic geometry. Geom. Topol. , 18(2):1149–1244, 2014.
- 8[8] Clark Barwick and Saul Glasman. A note on stable recollemonts. Available at https://arxiv.org/abs/1607.02064 .