This paper introduces enriched sets as an abstraction of enriched categories, develops their categorical structure, and proposes a method for constructing and mapping between such enriched sets, advancing the theoretical framework of enriched category theory.
Contribution
It defines enriched sets, constructs a category of these sets, and presents a method for building and relating enriched sets from given data, extending enriched category theory.
Findings
01
Enriched sets form a new categorical abstraction.
02
A category of enriched sets is established.
03
A construction method for enriched sets and functors is proposed.
Abstract
We introduce the notion of an enriched set, as an abstraction of enriched categories, and a category of enriched sets. The set of enriched sets is itself described as a set enriched over the category of enriched sets. We introduce a method for the construction of sets enriched over the set of enriched sets from a given enriched set with some addition data, and for "functors" from such enriched sets as should thereby arise to the enriched set of enriched sets.
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TopicsHomotopy and Cohomology in Algebraic Topology · Algebraic structures and combinatorial models · Advanced Algebra and Logic
Full text
Enriched Sets and Higher Categories
Bradley M. Willocks
Abstract.
We introduce the notion of an enriched set, as an abstraction of enriched categories, and a category of enriched sets. The set of enriched sets is itself described as a set enriched over the category of enriched sets. We introduce a method for the construction of sets enriched over the set of enriched sets from a given enriched set with some addition data, and for “functors” from such enriched sets as should thereby arise to the enriched set of enriched sets.
The present work is grown of a desire for a systematic description of methods by which one might reconfigure spaces of one type into spaces of another. To this end, we introduce in the present work ”enriched sets,” abstractions of categories, and a formalism by which they may be reconfigured into related enriched sets (“constellations”) whose arrows are “diagrams” in the original enriched set. We furthermore construct (3.3.19) “functors” from sub-enriched sets of the reconfigured sets to the enriched set of enriched sets, whereby, in loose terms, “an arrow in the reconfigured enriched set is sent to a functor from an arrow category over the domain to an arrow category over the codomain.”
In particular, we’ve constructed in other work a category of pointed categories and pointed correspondences ([18], the “purely categorical” part of which forms the core of this work), and a formalism for the description of a category of “locally affine sheaved spaces” as a subcategory thereof. This category is intended as a domain in which such categories (“geometries”) might be extended and compared. With such geometries contained within a single category, and a plurality of arrows between them (of “non-classical origin”) one might construct various arrow categories, whose objects were arrows between such categories. The intention is that each object x within a geometry should be attached to such an inter-geometric arrow category, and an arrow x→y (a morphism within a particular geometry) might be reconsidered as situated within a larger diagram (a “constellation”) in which the arrow categories of x and y might be mingled, depicting, for example, fibred products u×yx, with u originating from the “other” geometry (e.g. logarithmic structure as in [1], or divided power structure [2]; generally, alternate algebraic structure). Such an association would suggest associating to each x a limit or colimit of the objects u (by the proposition (3.3.12) below). At the same time, one might associate to each x the automorphisms Aut(F) of a forgetful functor ((u→x)↦u), thinking of the Tannakian formalism of [5]. It is hoped that such constructions might be useful in sytematically understanding the relationships between classical algebraic geometry of [7], the various F1 geometries described in [11] (see the paths and bridges section), Berkovich spaces/non-archimedean geometry of [16] or [4], and spaces with modified structure sheaves as in [1] or [3].
Our intention is, that this work should constitute the categorical foundation for processes by which such geometries might be attached to, or subsumed within, enriched categories of diagrams (constellations) constituted possibly of objects and arrows from different geometries, from which they might inherit higher categorical structures (see (3.3.14) or (3.3.19)) and homotopy invariants (from the Tannakian inspiration). Having described, in the other work, [18], a common category for a somewhat general notion of geometry, we would inquire into the “possibilities” regarding homotopy and cohomology theories, hoping, in particular, to extend such notions to alternate geometries, and to compare their different manifestations (we would hope for something like GAGA, [15]).
Thinking of (co)homology, one meets with their plurality, and we are perhaps therefore inclined toward some adaptation of motivic cohomology, or some other formalism by which categories of “schemes” or sheaves of some type (rings classically) might be enveloped by Abelian categories with translations, employing the κ-twists of [18] to replace distinguished objects in Sh(X,Ring) by distinguished objects in some derived category. We hope in the future, based upon the present work (3.3.14), to relate the higher categorical/homotopy data attached to a geometry to the Abelian data attached thereto (to adapt motivic cohomology of [17] or [10] to alternate geometries, using [12] to study the result).
Notation. For that this work is mostly concerned with the consideration of composition laws and their variations, we denote by “f⋅g” the composition of functions f with g, so that f⋅g:x↦f(g(x)). Brackets “┌” and “┐” separate logical statements, where they are much manipulated. Otherwise, notation and general concepts are those of standard category theory ([14], [8]).
2. Enriched Sets
We introduce the notion of sets enriched over a tensor category (A,⊗), which is an abstraction of that of a category, consisting essentially of compositon laws, which assign to each triple (a,b,c)∈S of elements in a set an arrow in A, ∘(a,b,c):h(a,b)⊗h(b,c)→h(a,c). Out approach differs from that of [6] primarily in the use of an extra datum, a “skeleton functor” sk:A⟶B, to replace equality with “equivalence.”
2.1. A Variation on Limits ((sk,e)-limits)
We define a notion of a limit of a functor F:I⟶A, with respect to a “skeleton” functor sk:A⟶B, as the colimit of a domain functor, from a certain arrow category in a fibre product of a pair of categories of functors to A. The construction essentially takes the terminal object in the category of objects over the functor F which are natural after the application of sk.
2.1.1. The Use of dob↓(−)
Recall that Δ(J,A):A⟶HomU−Cat2(0)(J,A) by sending an object c to the c-valued constant functor, and ob(HomU−Cat2(0)(J,A))(F⋅e):⋆⟶HomU−Cat2(0)(J,A) by sending the one arrow in ⋆ to idF⋅e. Recall also that the category ↓(HomU−Cat2(0)(J,A))(Δ(J,A),obHomU−Cat2(0)(J,A)(F⋅e)) of arrows is defined so that its objects are triples (a,α,∅), where a∈Ob(A), α:Δ(J,A)(a)→F⋅e is a natural transformation, and ∅ is the object in the category ⋆ (the category with one arrow). An arrow between (a1,α1,∅)(ϕ,id∅)(a2,α2,∅) is a pair of arrows ((a1ϕa2),id∅)∈Arr(A)×Arr(⋆) for which α2⋅Δ(J,A)(1)(ϕ)=α1.
An isomorphic category is given by forgetting both the ∅ symbol, and the a term (since for any j∈Ob(J), a=dom(α(j)), so that a is determined by α), so that its objects are natural transformations α, where a∈Ob(A) and α:Δ(J,A)(a)→F⋅e. If α1:Δ(J,A)(a1)→F⋅e and α2:Δ(J,A)(a2)→F⋅e, then an arrow α1ϕα2 is an arrow (a1ϕa2)∈Arr(A) for which α2⋅Δ(J,A)(1)(ϕ)=α1
2.1.2. Definition of an (sk,e)-Limit
Consider functors
JeIFAskB.
Consider the set of maps α:Ob(I)→Arr(A) such that sk⋅α defines a natural transformation from a diagonal functor to sk⋅F. Let
[TABLE]
and
[TABLE]
and
[TABLE]
Let ε:P:t=C×ED⟶D be one of the arrows of a fibred product, an arrow in U′−Cat. If For is the functor which takes the object a from an arrow Δ(J,A)(a)⟶F⋅e, an (sk,e)-limit is a colimit of For⋅ε.
This is explained in the following sections.
2.1.2.1. Let P be the full sub-category of the category
[TABLE]
whose objects are natural transformations α, such that for some a∈Ob(A) we have α:Δ(J,A)(a)→F⋅e, such that there exists a natural transformation α~:Δ(I,B)(sk(a))→sk⋅F such that the natural transformation HomU−Cat2(1)(idJ,sk)(α):Δ(J,B)(sk(a))→sk⋅F⋅e given by sending j∈Ob(J) to sk(α(j)):sk(a)→sk(F(e(j))) is equal to the natural transformation HomU−Cat2(1)(e,idB)(α~):Δ(J,B)(sk(a))→sk⋅F⋅e given by sending j∈Ob(J) to α~(e(j)):sk(a)→sk(F(e(j))).
2.1.2.2. Denote by ε:P⟶↓(HomU−Cat2(0)(J,A))(Δ(J,A),ob(HomU−Cat2(0)(J,A))(F⋅e)) the inclusion, given by α↦(a,α,∅). Denote also by p the functor,
[TABLE]
Thus, p is given by sending α↦a, and the fibre of p over any given a∈Ob(A) is the set of natural transformations Δ(J,A)(a)αF⋅e such that for some natural transformation α~:sk⋅Δ(I,A)(a)=Δ(I,B)(sk(a))⟶sk⋅F, one has
[TABLE]
In other words,
[TABLE]
is the full subcategory which contains all objects α such that the image of α in
HomU−Cat2(0)(J,B) under the functor HomU−Cat2(1)(idJ,sk) has a lift to
HomU−Cat2(0)(I,B) by the functor HomU−Cat2(1)(e,idB) (we denote this lift by α~).
2.1.2.3.
Then the (sk,e)-limit of F is the colimit of p, i.e., for any pair (l,λ)∈Ob(A)×Arr(HomU−Cat2(0)(P,A)) for which λ:p→Δ(P,A)(l), we say that (l,λ) is an (sk)−limit(F) iff (l,λ) is a colimit(p) (in the sense in which (λ,l) is a universal arrow in HomU−Cat2(0)(P,A), from p to the constant functor Δ(P,A):A⟶HomU−Cat2(0)(P,A)).
2.1.3. Example
In the above, if either e or sk is an identity functor, then the (sk)-limit of F is the limit (l,λ) of F, if the latter exists.
If e=idI, then P consists of all natural transformations α:Δ(I,A)(a)→F⋅e=F for which there exists some lift α~:Δ(I,B)(sk(a))→sk⋅F. But α~=HomU−Cat2(1)(sk,idI)(α) would be such a lift. Therefore any α:Δ(I,A)(a)→F has a lift. Furthermore, for any i∈Ob(I), α(i):a→F(i), and for any β:Δ(I,A)(b)→F, and any arrow ϕ:α→β in P, by definition of P, we have that β(i)⋅ϕ=α(i). Therefore, by the definition of a colimit, there exists a unique αl(i):l→F(i) such that for any (Δ(I,A)(a)αF)∈Ob(P), α(i)=αl(i)⋅λ(α). Since each colimit arrow αl(i) is determined by the arrows α(i) which come from natural transformations α, the assignment αl=(i↦αl(i))i∈Ob(I) determines a natural transformation Δ(I,A)(l)→F. Therefore αl∈Ob(P). If the limit of F exists, then it is isomorphic to a terminal object in P, αt∈Ob(P). But by the above argument, this terminal object αt determines a colimit arrow λ(αt):at→l, and being a terminal object in P there is a unique arrow el:l→at=dom(αt(i)) in P. By the definition of terminal objects, el⋅λ(αt)=idat
2.1.4. Lemma
(Inclusion, via right exactness) Given sk,F,e:J→I, ε:P→
↓(Hom(U−Cat2)(0)(J,A))(Δ(J,A),ob(Hom(U−Cat2)(0)(J,A))(F⋅e)), l, and λ as above, suppose further that sk is right exact, and Ob(I)=Ob(J). For each i∈Ob(I)=Ob(J), consider the arrow induced from the colimit l to F(i) by α↦α(i), where (p,α,∅)∈Ob(P) is an object in P. Then this assignment determines an object (l,αl,∅)∈Ob(P).
I.e. the (sk)-limit determines an object,
[TABLE]
in P.
Proof.
If the colimit (l,λ) is sent to the colimit of the forward composition by sk of the dob↓ diagram on P, then arrows from l to it are yet determined by their pullbacks to the components of the forward composition of the colimit diagram, which commute after forward composition.
∎
2.1.5. Lemma
(Uniqueness, via monic) For any sk:A→B,F:I→A∈Arr(U−Cat), for any (l,λ)∈Ob(A)×Arr(HomU−Cat2(0)(P,A)), (P being as above) if the arrow from the (sk)-limit(F) to the product ∏j∈JF(j) induced by the arrows from the previous lemma, i.e. λ∏((j↦λ(((b,f,∅)↦f(j))(b,f,∅)∈Ob(P))))j∈Ob(J))):l→∏j∈Ob(J)F(0)⋅e(0)(j), is monic, then for each (b,f,∅)∈Ob(P), the coproduct arrow b→l is the unique arrow λ′ for which λ(((b,f,∅)↦f(j))(b,f,∅)∈Ob(P))⋅λ′=f(j).
Proof.
Trivial.
∎
2.1.6. Remark
This is the uniqueness of factorization usually associated to limits.
2.1.7. Definition of the Skeleton Functor
Define the category U−SCat∈Ob(U′−Cat) so that
[TABLE]
and for any x,y∈Ob(U−SCat), Hom(U−SCat)(x,y) is the the set
[TABLE]
of isomorphism classes of functors xFy, where [F]=[G] iff F≅G, i.e. iff there exists an isomorphism FαG of functors.
Define the functor
[TABLE]
so that Skel is the identity map on the objects and the quotient map F↦[F] on the arrows.
2.1.8. Example
Consider ϕ,ψ∈Arr(U−Cat), with the same codomain. The (Skel)-limit of the diagram is the subcategory L of dom(ϕ)×U−Catdom(ψ) such that Arr(L)={f∈Arr(dom(ϕ)×U−Catdom(ψ);∃u,v∈Arr(codom(ϕ)),u,v are isomorphisms and u⋅πϕ(f)=πψ(f)⋅v}. Any category with such functors into the two domain categories that the composition of functors on one side is isomorphic to the composition of functors on the other side factors through L via the compositions of the projections with the embedding into the product. By the monic lemma the factorization is unique. However the conclusion of the inclusion lemma might not apply to it, i.e. the two compositions L→dom(ϕ)→codom(ϕ) and L→dom(ψ)→codom(ψ)=codom(ϕ) might not be isomorphic, since I might imagine having two different pairs of arrows (f1,g1), and (f2,g2), such that the isomorphisms u1,v1∈Arr(codom(ϕ)) which form the commuting square u1⋅f1=g1⋅v1 differ from the isomorphisms u2,v2∈Arr(codom(ϕ)) which form the commuting square u2⋅f2=g2⋅v2.
2.1.9. Lemma, for Reduction to the Standard Limit
If
[TABLE]
and the limit arrows are unique then this is the usual limit.
Proof.
Trivial.
∎
2.1.10. Functoriality
An arrow of functors F⋅e→G⋅e which lifts to an arrow of functors sk⋅F→sk⋅G (i.e. an arrow in the fibred product of the two functors HomU−Cat2(1)(e,idB) and HomU−Cat2(1)(idJ,sk) ) induces a map from the (sk,e)-limit of F to that of G, using the colimit map. I.e. α:F→G implies that α(dom(ϕ))⋅F(ϕ)=G(ϕ)⋅α(codom(ϕ)), so that for any arrow β:Δ(J,C)(0)(c)→F⋅e associated to (a,β,∅)∈Ob(P) (notation as in the first definition), HomU−Cat2(1)(e,idA)(α)⋅β also commutes after applying sk (i.e. comes from an arrow in HomU−Cat2(0)(I,B)). Therefore each such a has an arrow into the sk-limit of G from the colimit diagram of the definition, which induces a map from the colimit diagram which determines the sk-limit of F.
2.1.10.1. Given a diagram F:I′⟶HomU−Cat2(0)(I,A), and a choice of an (sk,e)-limit (l(i),λ(i)) for any object i∈Ob(I′), the construction of (2.1.10) determines a function Arr(I′)⟶Arr(A)
2.1.10.2. If for any i∈Ob(I′), the (sk,e)-limit (l(i),λ(i)) is included in P(i) (P(i) being as in the definition of the (sk,e)-limit for F(i)) then (2.1.10.1) determines a functor I′⟶A.
2.1.11. Remark
Roughly speaking, one takes the colimit of the domains of all limit diagrams on the trivial category which, when forwards composed with sk, are the backwards composition by e of an actual limit diagram of sk∘F. Definition (2.3) following this remark is dual to Definition (2.1).
2.1.12. Definition of the (sk)-Colimit
Consider functors JeIFAskB.
2.1.12.1. Let P be the full sub-category of the category
[TABLE]
of arrows, whose objects are given by natural transformations from functor F⋅e
to functor Δ(J,A)(a), i.e. triples (∅,α,a) for varying a∈Ob(A), such that there exists a natural transformation α~ from functor sk⋅F to functor Δ(I,B)(a) such that the natural transformation from sk⋅F to Δ(J,B)(sk(a)) is equal to the natural transformation given by sending j∈Ob(J) to α~(e(j)), i.e. by the set
[TABLE]
[TABLE]
so as to be given by the category of arrows from the diagonal functor to the object functor of F⋅e in the category of functors from J to A.
2.1.12.2. Suppose that ε:P⟶↓(HomU−Cat2(J,A))(Δ(J,A),ob(HomU−Cat2(J,A))(F⋅e)) is the inclusion.
2.1.12.3.
For any sk:A→B,F:I→A∈Arr(U−Cat), codom(F)=dom(sk) implies that any pair (l,λ)∈Ob(A)×Arr(Hom(U−Cat2)(0)(A,U−Set)), (l,λ) is a (sk,e)−colimit(F) iff (l,λ) is a limit (cob↓(Hom(U−Cat2)(0)(J,A))
(ob(Hom(U−Cat2)(0)(J,A))(F⋅e),Δ(J,A))⋅εc).
2.1.13. Lemma
(Inclusion via exactness) Dual to the above.
2.1.14. Lemma
(Uniqueness via epic) Dual to the above.
2.1.15. Example
Consider ϕ,ψ∈Arr(U−Cat), with the same domain. The (Skel)-colimit of the diagram is the category L such that its set of objects is the disjoint union of the objects of the codomain categories and the arrows are the formal compositions of the disjoint union of arrows in Arr(codom(ϕ)), Arr(codom(ψ)), and arrows ea:ϕ(0)(a)→ψ(0)(a), ea−1:ψ(0)(a)→ϕ(0)(a) formally added for each a∈Ob(dom(ϕ))=Ob(dom(ψ)), with the relation generated by requiring that ∀f∈Arr(dom(ϕ)),ϕ(1)(f)⋅edom(f)=ecodom(f)⋅ψ(1)(f). If lϕ:codom(ϕ)→L and lψ:codom(ψ)→L are given by the U−Set coproduct maps then for any lϕ′,lψ′∈Arr(U−Cat) such that l^{\prime}_{\phi}\cdot\phi\cong{{\color[rgb]{.1,.1,.7}l}}^{\prime}_{\psi}\cdot\psi, there is an arrow q:L→codom(lϕ′)=codom(lψ′) such that lϕ′=q⋅lϕ and lψ′=q⋅lψ. If an isomorphism α:lϕ′⋅ϕ→lψ′⋅ψ is specified (or vice versa), then there is a unique q:L→codom(lϕ′) such that Hom(U−Cat2)(1)((iddom(ϕ),q))(1)((a↦ea)a∈Ob(dom(ϕ)))=α (and vice versa).
2.1.16. Lemma
(Reduction) Dual to the above.
2.1.17. Lemma
(Functoriality) An arrow of functors F→G induces a map from the (sk)-colimit of F to that of G, using the limit map.
2.1.18. Remark
Products and coproducts are not affected by sk.
2.2. Definitions regarding Enrichments
We will define weak enrichment of sets and categories.
Sets will be enriched over tensor categories (A,⊗)
and categories over triples (A,⊗,F)
where tensor category (A,⊗) comes with a tensor
functor F:(A,⊗)⟶(Set,×).
A weak enrichment of a set s over (A,⊗)
adds to s a category-like structure, a version of Hom which has values in A
(rather than in sets) but without any associativity or unital requirements. We later introduce, for each functor sk:A⟶B, a category of weakly enriched sets, “associative up to sk,” in that the associativity diagrams are commutative after the functor sk is applied to them. A weak enrichment of a category C over (A,⊗) with respect to a tensor functor (F,ρ):(A,⊗)→(Set,×Set) is
a weak enrichment of the set Ob(C) over (A,⊗) which is compatible with the
HomC, this compatibility being formulated in terms of the tensor functor (F,ρ).
2.2.1. Definition of a Weakly Enriched Set
A
weak enrichment of a set s∈Ob(U−Set) over a tensor category (A,⊗)∈Ob(U−TCat) (a pair consisting of a category A∈Ob(U−Cat and a functor ⊗∈A×U−CatA⟶A)
is a
pair
consisting of a map h:s2→Ob(A)
and a “composition map”
∘:s3→Arr(A)) such that for any a,b,c∈s,
[TABLE]
2.2.2. Definition of the Category of Weak Enrichments
For any (A,⊗)∈Ob(U−TCat),
the category of (A,⊗)-enriched sets WE(A,⊗)∈Ob(U−Cat) has as objects weak enrichements of sets S=(s,hS,∘S),
and for two weak enrichments S and T an arrow
f:S=(s,hS,∘S)→T=(t,hT,∘T) is a pair of functions
f=(f1:s→t,f2:s2→Arr(A))
such that the following hold.
2.2.2.1. ∀a,b∈s,f2(a,b):hS(a,b)⟶hT(f1(a),f1(b)), and
i.e. the compositions commute with the arrows defining a “functor from S to T”.
2.2.3. Lemma
The above construction, of WE(A,⊗), extends to a functor WE:U−TCat⟶U′−Cat, from the category of tensor categories to the category of categories.
For any functor of tensor categories (F,ρ):(A,⊗A)→(B,⊗B) define a functor WE(F,ρ):WE(A,⊗A)→WE(B,⊗B) from the category of weak enrichments over (A,⊗A) to that of (B,⊗B) as follows.
2.2.3.1.
It sends an object S=(s,h,∘)
of
WE(A,⊗A)
to the triple
F(S)=(s,h′,∘′)
where for
a,b,c∈s,
[TABLE]
2.2.3.2.
It sends an arrow
ϕ:S=(s,hs,∘s)→(t,ht,∘t)=T in WE(A,⊗A)
(here s2∋(a,b)↦ϕ(a,b)∈Arr(A))
to the arrow F(ϕ):F(S)→F(T) that sends
(a,b)∈s2 to F(ϕ(a,b)))∈Arr(B).
2.2.4. Definition of an Weakly Enriched Category
A weak enrichment of a category C with respect to a tensor functor (A,⊗)(F,ρ)(U−Set,×U−Set) is a quadruple (C,h,∘,ϕ) such that C∈Ob(U−Cat) is a category, h and ∘ define a weak enrichment of the set Ob(C), and ϕ:Ob(C)2→Arr(U−Set) is a function, such that
2.2.4.1. For any a,b∈Ob(C), ϕ(a,b):F(h(a,b))→HomC(a,b) is an isomorphism;
of hom sets in C is given by the weak enrichment, i.e.
[TABLE]
2.2.5. Definition of the Category of Weakly Enriched Categories
The category
WECat(F,ρ) of categories weakly enriched over a tensor category
(A,⊗) with respect to a tensor functor
(F,ρ):(A,⊗)→(U−Set,×U−Set), has objects which are
categories (C,h,∘,ϕ) weakly enriched over (A,⊗).
An arrow
f:(C,hC,∘C,ϕ)→(D,hD,∘D,ψ)
consists
of a functor
(f0,f1):C⟶D
and a function f2:Ob(C)2⟶Arr(A),
such that
2.2.5.1. (f0,f2):(Ob(C),hC,∘C)→(Ob(D),hD,∘D) is an arrow of weak enrichments of sets;
i.e. the functor agrees with that implied by the enrichment.
2.2.6.
One can construct a functor from the category of tensor categories over the tensor category of sets U−TCat/(U−Set,×U−Set) to the category of categories, i.e.
[TABLE]
in analogue to the construction of Lemma 2.2.3, as follows. For any arrow (Φ,ρ):(F,ρF)⟶(G,ρG) of tensor categories (F,ρF):(A,⊗A)⟶(U−Set,×U−Set) and (G,ρG):(B,⊗B)⟶(U−Set,×U−Set) over (Set,×U−Set) define a functor
WECat0(F,ρF)⟶WECat0(G,ρG).
2.2.6.1. It is defined on an object (C,h,∘,ϕ)∈Ob(WECat0(F,ρ)) by
[TABLE]
2.2.6.2. It is deifined on arrows (F,F2):(C,hC,∘C,ϕ)→(D,hD,∘D,ψ) by
[TABLE]
2.2.7. Definition of Two Forgetful Functors
Define the following two functors.
2.2.7.1. For any tensor functor (F,ρ):(A,⊗)⟶(U−Set,×U−Set), the forgetful functor ForWE(dom(F,ρ))WE(F,ρ):WECat(A,⊗,F)⟶WECat(A,⊗) from the category of weakly enriched categories with respect to (F,ρ) to weakly enriched sets with respect to (A,⊗) is the functor given by passing from a category C to its set of objects Ob(C). More precisely, it is defined on an object (C,h,∘,ϕ)∈Ob(WECat(F,ρ)) by
[TABLE]
and on an arrow (f,f2)∈Arr(WECat(F,ρ)) by
[TABLE]
2.2.7.2. The forgetful functor from the category of weakly enriched categories to the category of categories ForCatWE(F,ρ):WECat(F,ρ)⟶U−Cat is the functor which forgets the enrichment structure, returning the underlying category. I.e. it sends a weakly enriched category (C,h,∘,ϕ) to C.
2.2.8. Definition of the Category WE(sk)(A,⊗)
For any sk:A⟶B∈Arr(Cat), define the category WE(sk)(A,⊗)∈Ob(Cat) of ((A,⊗),sk)-enriched sets.
are the pairs of maps of sets (F0,F1)∈Arr(Set)2 such that F0:S→T and F1:S2→Arr(A) and
2.2.8.2.1.
For any a,b∈S,F1(a,b)∈HomA(hS(a,b),hT(F0(a),F0(b)).
2.2.8.2.2. F1 respects composition after applying sk, i.e. for any a,b,c∈S,
[TABLE]
holds.
2.2.8.3. For any (sk)-associator α, we define the subcategory WEAssoc(sk,α)(A,⊗)⊆WE(sk)(A,⊗), informally WEAssoc(sk)(A,⊗), to be the full subcategory whose objects are (sk)-associative enriched sets.
2.2.9. Remark
Roughly speaking, F0 is the map between objects of enriched sets, and F1:hS→hT∘F is the “natural transformation of hom functors,” (there are no non-trivial arrows in S). This means that applying the “functor,” (F0,F1), then composing in T, versus composing in S and then applying the functor, gives two arrows in A, such that sk of one arrow is equal to sk of the other.
2.2.10. Lemma
WE(sk)(A,⊗) is a category.
Proof.
The issue is composition. Given composible arrows
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Starting from the result of application of the functor sk to the arrow which uses the composition ∘U,
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by functoriality of ⊗
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by functoriality of sk
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by (G0,G1)∈Arr(WE(sk)(A,⊗)),
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by (F0,F1)∈Arr(WE(sk)(A,⊗)),
[TABLE]
∎
2.2.11. Lemma
If (A,⊗) has products, then so does WE(sk)(A,⊗). The product is functorial.
2.2.12. Definition of (sk)-Associativity
Consider a tensor category (A,⊗)∈Ob(U−ATCat) with a functor sk:A→B and an (sk)-associator α:Ob(A)3→Arr(A). An (A,⊗)-enriched set (S,h,∘)∈Ob(WE(A,⊗)) is said to be (sk,α)-associative if for any a,b,c,d∈S,
[TABLE]
[TABLE]
i.e. the standard self-consistency diagram (pentagram) for the enriched composition ∘ is required to commute after applying the functor sk
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If the associator α is understood, then we will write “(sk)-associative”.
2.2.13. Definition of WEAss(A,⊗)(sk,α)
Suppose that (A,⊗) has an associator α (see LABEL:ATCat). Define WEAss(A,⊗)(sk,α) to be the full subcategory of WE(sk)(A,⊗) generated by enriched sets (S,hS,∘S)∈Ob(WE(sk)(A,⊗)) which are (sk,α)-associative. If the associator is understood, then we will denote this by “WEAss(A,⊗)(sk)”.
2.3. Enrichment of HomWE(sk)(A,⊗)(I,C)
Consider a tuple of functors {pi:Ii⟶A}i=1n. Suppose that for each i∈{1,...,n}, the colimit colimpi∈Ob(A) exists, with universal arrows ei(xi):pi(xi)→colimpi. Suppose that the colimit of the functor ⊗i=1npi:∏i=1nIi⟶A defined by (xi)i=1n↦⊗i=1npi(xi) is also an object in A. Consider the arrow (colim⊗i=1npi→⊗i=1ncolimpi)∈Arr(A) induced by (xi)i=1n↦⊗i=1nei(xi); i.e. by tensoring the universal arrows together. The following lemma states that under certain conditions on the pi, the above defines a natural transformation with respect to arrows of functors ϕi:pi→qi.
The (A,⊗)-enrichment of the hom-sets in WE(sk)(A,⊗) involves such colimits, and the definition of the composition requires that the above arrows should be isomorphisms. This means that the “forward and backward composition functors” to be introduced in lemma 2.3.7 below are determined by the arrows between products ∏hS(...)→∏hT(...).
2.3.1. Lemma on the Naturality of τ
Suppose that {Fi,Gi:Ii⟶A}i=1n are functors, and {(FiϕiGi)}i=1n are arrows of functors. Suppose that for each i∈{1,...,n}, PFi⊆↓(HomU−Cat2(1)(Ji,A))(Δ(Ji,A),Fi∘εi) and PGi⊆↓(HomU−Cat2(1)(Ji,A))(Δ(Ji,A),Gi∘εi) are subcategories, where εi:Ji⊆Ii is the subcategory with only identity arrows.
2.3.1.1. Suppose that the functors pFi:PFi⟶A and pGi:PGi⟶A are as in the conditions of the limit inclusion lemma (i.e. colim(pFi) determines an object in PFi, with the analogue holding for Gi)
2.3.1.2. Define an arrow of sets τ():∏i=1nOb(HomU−Cat2(1)(Ii,A))→Arr(A) so that for any (Hi)i=1n∈∏i=1nOb(HomU−Cat2(1)(Ii,A)), τ((Hi)i=1n)):colim(⨂i=1npHi)→⨂i=1ncolim(pHi) is the universal arrow for the colimit induced by the assignment (where λ(i) is the natural transformation defining the colimit of pHi)
[TABLE]
2.3.1.3. If u=u(p):colim(⨂i=1npFi)→colim(⨂i=1npGi) is the universal arrow for the colimit induced by the assignment
[TABLE]
then
[TABLE]
I.e., τ:colim(⨂i=1npi)→⨂i=1ncolim(pi) is “natural at (ϕi)i=1n.”
Proof.
By the monic arrow condition, the arrows involved are situated above the products, ∏a∈Ob(Ii)Fi(a), so that the arrows ⊗i=1na(i)→⊗i=1nlGi are pure tensors respecting the arrows ϕi.
∎
The following lemma defines a weak enrichment of the set HomWE(sk)(A,⊗)(C,D). To any “(A,⊗)-functors” Φ,Ψ:C→D, one attaches a category P, and defines the hom object between Φ and Ψ to be a colimit of a certain functor P⟶A. Roughly speaking, P keeps track of all arrows into to the product ∏x∈Ob(C)hD(Φ(x),Ψ(x)) which respect the composition with any arrows “coming from some hC(x,y),” after one applies sk. P is a full sub-category of the category of arrows over ∏x∈Ob(C)hD(Φ(x),Ψ(x)). The objects of P are all arrows (aπ∏x∈Ob(C)hD(Φ(x),Ψ(x)), such that for any x,y∈Ob(C), for any (t0thC(x,y))∈Arr(A), tensoring π with t, projecting to the y-component ∏x∈Ob(C)hD(Φ(x),Ψ(x))→hD(Φ(y),Ψ(y)), and composing in D is (sk)-equal to tensoring t with π, projecting to the x-component, and composing in D. One defines p:P⟶A to be the functor which remembers the domain of a given arrow. One associates to Φ and Ψ the object colim(p)∈Ob(A) (assuming that the colimit exists).
One composes, i.e. defines, for all (A,⊗)-functors Φ,Ψ,Ξ, an arrow
[TABLE]
by taking the inverse of the arrow colim(PΦ,Ψ⊗PΨ,Ξ)→(colimPΦ,Ψ)⊗(colimPΨ,Ξ) (that this is an isomorphism is assumed), and recognizing colim(PΦ,Ψ⊗PΨ,Ξ) as an object in PΦ,Ξ by using the composition in D and the projection for the products to define arrows PΦ,Ψ(x)⊗PΨ,Ξ(y)→∏x∈Ob(C)hD(Φ(x),Ξ(x)). As an object in PΦ,Ξ, colim(PΦ,Ψ⊗PΨ,Ξ) has assigned to it an arrow into colimPΦ,Ξ, which is defined to be the hom object assigned to Φ and Ξ. One composes this colimit arrow with the inverse of the first arrow to define the composition arrow.
2.3.2. Lemma on the Enrichment of HomWE(sk)(A,⊗)(C,D)
Suppose that (A,⊗) has a symmetrizer and associator for the tensor.
to be the full subcategory generated by objects (i.e. arrows aπ∏x∈Ob(C)hD(Φ(x),Ψ(x)) in A) such that for any x,y∈Ob(C),(t0thC(x,y))∈Arr(A),
[TABLE]
[TABLE]
Then define hˉWE(sk)(A,⊗)1(C,D)(Φ,Ψ) to be the colimit of the domain object functor p:P⟶A defined by (a,f)↦a.
2.3.2.2. Suppose that the compostion on D is (sk)-associative, and the arrows u=u(p):colim⊗i=1npi→⊗i=1ncolimpi are isomorphisms, defined as in the previous lemma, and p={(1,pΦ,Ψ)}∪{(2,pΨ,X)}:{1,2}→Arr(Cat). Then define the composition ∘WE(A,⊗)(sk)(C,D)(Φ,Ψ,X)∈Arr(A) by taking it to be the composition of the colimit arrow e:colim(−1⊗−2)→hWE(A,⊗)(sk)(C,D)(Φ,X) associated to the object (colim(−1⊗−2),ϕ)∈Ob(PΦ,X) determined by the arrow
[TABLE]
induced by sending any given ((a,f),(b,g))∈Ob(PΦ,Ψ×CatPΨ,X) to the product arrow given to the assignment
i.e. part i. gives the hom objects and part ii. gives the composition.
2.3.2.4.1. The enriched set hˉWE(A,⊗)(sk)(C,D) is (sk)-associative.
2.3.2.4.2. If (A,⊗) has a unit I such that ∘D is (Yo(0)opp(I))-associative, then so does hˉWE(A,⊗)(sk)(C,D).
2.3.3. Remark
The enrichment on HomWE(sk)(A,⊗)(C,D), i.e. the objects h(Φ,Ψ) defined in the previous lemma for (A,⊗)-functors Φ and Ψ, were initially constructed as (sk)-equalizers. I believe that the present construction can also be realized as an (sk)-equalizer, but by use of a diagram containing arrows of the form [(Homfun(A)∘((−⊗J)×idA),∘∘(π⊗idJ))]∈Arr(Ω), and with restrictions on A.
2.3.4. Definition of the Enriched Arrows Functor
If (A,⊗)∈Ob(TCat) has coproducts, then define
[TABLE]
by (S,h,∘)↦∐s,t∈Sh(s,t) and (F0,F1)↦∐s,t∈SF1(s,t).
2.3.5. Remark
The functor sk is not referred to in this definition. Arrˉ(A,⊗) is the “enriched arrow functor.”
2.3.6. Lemma
Arrˉ(A,⊗) is faithful.
The following lemma concerns the self-enrichment of the category WEAssoc(sk)(A,⊗). The enriched hom set defined in the previous lemma is denoted by “hˉWE(A,⊗)(sk)(B,C).” Part (i) of the following lemma defines the “forward composition/pushforward functor,”
hˉWE(A,⊗)(sk)(B,C)→hˉWE(A,⊗)(sk)(B,D). Part (ii) defines the “backward composition/ pullback functor,” hˉWE(A,⊗)(sk)(C,D)→hˉWE(A,⊗)(sk)(B,D). Part (iii) states that one can use these to define an arrow (hˉWE(A,⊗)(sk)(B,C)×hˉWE(A,⊗)(sk)(C,D)→hˉWE(A,⊗)(sk)(B,D))∈Arr(WEAssoc(sk)(A,⊗)) which gives the enriched composition in WEAssoc(sk)(A,⊗).
2.3.7. Lemma on Composition Functors
Given (sk)∈Arr(Cat), for any (F:C→D), (G:B→C)∈Arr(WEAssoc(sk)(A,⊗)),
is induced by ∏a∈Ob(B)hC(Ψ1(a),Ψ2(a))→∏a∈Ob(B)hD(F⋅Ψ1(a),F⋅Ψ2(a)), which induces a functor PhˉWE(A,⊗)(sk)(B,C)(Ψ1,Ψ2)⟶PhˉWE(A,⊗)(sk)(B,D)(F⋅Ψ1,F⋅Ψ), so that an arrow is induced from the colimit of the first diagram (phˉWE(A,⊗)(sk)(B,C)(Ψ,Ψ2) to the colimit of the second phˉWE(A,⊗)(sk)(B,D)(F⋅Ψ1,F⋅Ψ).
is induced by ∏a∈Ob(C)hD(Φ1(a),Φ2(a))→∏a∈Ob(B)hD(Φ1⋅G(a),Ψ2⋅G(a)), which is the product map induced by the assignment (a↦πG(a)).
These are analogues to the usual forward and backward functors associated to composition on either end of a functor category Hom(B,C).
2.3.7.3. From an arrow of functors α:×A→⊗, the previous two constructions, and the product structure, construct an arrow in WESet(A,⊗)
[TABLE]
(Not unique. The choice corresponds with the choice, of the path F1⋅G1→F1⋅G2→F2×G2, versus the path F1⋅G1→F2⋅G1→F2×G2).
2.3.7.4. Defining ∘ˉ:Ob(WEAss(A,⊗)(sk))3↦Arr(WEAss(A,⊗)(sk)) by sending (B,C,D)∈Ob(WEAssoc(sk)(A,⊗) to the arrow in (iii), where WEAssoc(sk)(A,⊗)⊆WEAssoc(sk)(A,⊗) is defined to the
[TABLE]
is an (WEAss(sk)(A,⊗),×WEAssoc(sk)(A,⊗)-enriched set, whose composition is (Ob)-
associative and (sk⋅Arrˉ(A,⊗))-associative.
Proof.
Parts i. and ii. consist only in checking for (sk)-commutativity so that the constructions can be made. Part iii., states that for any C,D,E∈Ob(WE(A,⊗)), for any Φ1,Φ2,Φ3∈Ob(hˉWE(A,⊗)(sk)(C,D)), for any Ψ1,Ψ2,Ψ3∈Ob(hˉWE(A,⊗)(sk)(D,E),
[TABLE]
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given that colim(p)∈P with a monic arrow into the relevant product, and that ∀f,g:x→∏i∈Iyi, ∀i∈I,sk(πi⋅f)=sk(π⋅g)⟹sk(f)=sk(g).
All arrows between the objects hˉWE(A,⊗)(sk)(C,D)→hˉWE(A,⊗)(sk)(C′,D′) commute with monic arrows hˉWE(A,⊗)(sk)(C,D)→∏c∈Ob(C)hD(F(c),G(c)).
After taking the inverse of the isomorphism ⊗colimpi←colim⊗pi (that this is an isomorphism is assumed), these maps are determined by the arrows Ψi∗ and Φi∗. On the components of the product Φi∗ come from identity arrows and Ψi∗ from Ψ(a,b).
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(one side is Φ3∗⊗Φ3∗⊗Ψ1∗⊗Ψ1∗ and the other is Φ2∗⊗Φ3∗⊗Ψ1∗⊗Ψ2∗). The Φ3∗⊗Φ3∗⊗Ψ1∗⊗Ψ1∗ side is
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The Φ2∗⊗Φ3∗⊗Ψ1∗⊗Ψ2∗ side is
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By definition of Phˉ(C,D)(Ψ1,Ψ2), in particular, “commutativity” of the composition with any arrow going through a hom object of C, the two arrows are (sk)-equal.
∎
2.3.8. Remark
On underlying “objects” this is the usual composition (e.g. 1-composition, of functors).
2.3.9. Lemma
For any arrow of functors ρ:×A→⊗A, for any Sˉ,Tˉ∈
Ob(WEAssoc(sk)(A,⊗)), such that the diagrams determining the enrichment on hˉ(Sˉ,Tˉ) satisfy the P-colimit inclusion condition, we construct two arrows of enriched sets
[TABLE]
[TABLE]
by the following. We explicitly describe ηl. For any (s,ϕ)∈S×HomWEAssoc(sk)(A,⊗)(Sˉ,Tˉ)≅Ob(Sˉ×hˉ(Sˉ,Tˉ)),
[TABLE]
and for any (s,ϕ),(t,ψ)∈S×HomWEAssoc(sk)(A,⊗)(Sˉ,Tˉ)≅Ob(Sˉ×hˉ(Sˉ,Tˉ)), if q:hhˉ(Sˉ,Tˉ)(ϕ,ψ)→∏s0∈ShT(ϕ(s0),ψ(s0)) is the arrow in A given by the colimit universality, determining an object in P of the enrichment,
[TABLE]
Define ηr so that
[TABLE]
2.3.10. Definition
WE with initial object. For any (A,⊗), for any initial object ∅A∈Ob(A), we make the following definitions.
2.3.10.1. For any two (sk)-commutative arrows of functors λA:∅A⊗IdA→IdA←IdA⊗∅A:ρA we define WE(sk,λA,ρA)(A,⊗)⊆WEAssoc(sk)(A,⊗) to be the full subcategory generated by enriched sets Sˉ=(S,hS,∘S)∈Ob(WEAssoc(sk)(A,⊗) such that for any a,b,c∈S,
[TABLE]
2.3.10.2. For any associator α so that (A,⊗,α)∈U−ATCat, for any two (sk)-arrows of functors λA:∅A⊗IdA→IdA←IdA⊗∅A:ρA such that
[TABLE]
[TABLE]
and
[TABLE]
[TABLE]
[TABLE]
i.e., which are sk-associative, we define WE(sk,α,λA,ρA)(A,⊗)⊆WE(sk,α)(A,⊗) to be the full subcategory generated by enriched sets Sˉ=(S,hS,∘S)∈Ob(WEAssoc(sk)(A,⊗) such that for any a,b,c∈S,
[TABLE]
and
[TABLE]
2.3.11. Example
U−Set, taking the initial object to be the empty set. The product of the empty set with any set is empty, and so the equality is trivial.
2.3.12. Example
n−Cat, taking the initial object to be the empty category. As above.
2.3.13. Example
Pointed Set. The initial object is the set with one element, {∅}, the one element distinguished. In this case one would require that composition of any element with the distinguished element should return the original element, as an identity arrow.
2.3.14. Lemma
The category WE(sk,α,λA,ρA)(A,⊗) of (A,⊗)-enriched sets with an initial object has coproducts.
2.3.15. Definition
For any tensor category (A,⊗) with an action of an initial object λA:∅A⊗IdA→IdA←IdA⊗∅A:ρA, define a functor Bar(λA,ρA):A⟶WE(sk)(A,⊗), on objects by
[TABLE]
[TABLE]
[TABLE]
and on arrows by
[TABLE]
so that it takes the arrow ϕ itself for the non-trivial (non-initial) part of the enrichment.
2.3.16. Definition
For any (A,⊗,α)∈Ob(ATCat), for any functor sk:A⟶B, for any (sk)-action (λA,ρA) of an initial object ∅A∈Ob(A), we define the category Unit(sk)(A,⊗) and the functor
[TABLE]
by the following.
2.3.16.1. The category Unit(sk)(A,⊗) has for objects a sort of class of monads, arrows μ:a⊗a→a in A, and for arrows arrows f:a→b which “respect the multiplication.”
[TABLE]
[TABLE]
[TABLE]
2.3.16.2. The functor Δ−UnitλA,ρA):=(Δ−Unit0,Δ−Unit1) is defined on objects, so that for any n∈N, for any (a,μ)∈Ob(Unit(A,⊗))
[TABLE]
[TABLE]
[TABLE]
[TABLE]
[TABLE]
and on arrows, so that for any (e,f)∈Arr(Δinj×Unit(A,⊗)),
[TABLE]
[TABLE]
[TABLE]
2.3.17. Lemma
If the action (λA,ρA) is (α)-associative then Δ−Unit(λA,ρA) (and
Bar(λA,ρA)) factors through WE(sk,α)(A,⊗), i.e. produces (sk)-associative enriched sets.
2.3.18. Lemma
If the product structure (A,×A) has a unit (I0,λ0,ρ0), then for any arrow μ0, that (I0,μ0)∈Ob(Unit(sk)(A,⊗)), i.e. that μ gave a sort of monad structure to I0, would imply that (WE(sk)(A,⊗),×WE(sk)(A,⊗)) had a unit, given by the enriched set ({∅},h0∘0) with one element, single hom object h0=I0, and composition ∘0(∅,∅,∅)=μ).
2.3.19. Lemma
If A has an initial object with an action λA,ρA, and F:A⟶U−Set is representable, and preserves coproducts, then the functor
[TABLE]
is representable.
2.3.20. Definition. of Enriched Sets with Units
For any tensor category (A,⊗) for any functor sk:A⟶B, for any (A,⊗)-unit*(sk)* (I,λA,ρA) we make the following definitions.
(i). Define a category WE(sk,I,λA,ρA)(A,⊗)∈Ob(U′−Cat) so that its objects are enriched sets with unit data,
[TABLE]
[TABLE]
[TABLE]
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[TABLE]
and its arrows are those of WE(sk)(A,⊗) which preserve the unit, so that for all Sˉ=(S,hS,∘S,iS),Tˉ=(T,hT,∘T,iT)∈Ob(WE(sk,I,λA,ρA)(A,⊗)),
[TABLE]
[TABLE]
(ii). Define the forgetful functor ForWE(sk)(A,⊗)WE(sk,I,λA,ρA)(A,⊗):WE(sk,I,λA,ρA)(A,⊗)⟶
WEAssoc(sk)(A,⊗) by forgetting the unit data, (S,hS,∘S,iS)↦(S,hS,∘S).
2.3.21. Lemma. Pre-Curry
For any (A,⊗) with (sk)-unit(A,⊗) data (I,λA,ρA), and (sk)-unit(A,×A) data (I0,λ0A,λ0A) for any arrow c∈HomA(I0,I) in A we have the following.
2.3.21.1. Define an arrow Cur0 of sets by the following. For any arrow of (A,⊗)-enriched sets
[TABLE]
there is an arrow of (A,⊗)-enriched sets
[TABLE]
[TABLE]
defined by the following. For all s∈S the object
[TABLE]
is an arrow in WEAssoc(sk)(A,⊗) for some maps of sets F0′ and F1′, such that for all t,t′∈T,
[TABLE]
and
[TABLE]
For all s1,s2∈S, the arrow F1′(s1,s2):hS(s1,s2)→hˉ(F0′(s1),F0′(s2)) in A is the colimit arrow given by the enrichment lemma diagram to the object (hS(s1,s2),π,∅) in the arrow category by the arrow
[TABLE]
determined by the map
[TABLE]
2.3.21.2. If the P-colimits are objects of P, then for any two arrows (objects in the hom enriched set)
[TABLE]
[TABLE]
define an arrow
[TABLE]
[TABLE]
by assigning, to each s∈S the colimit arrow
[TABLE]
assigned to the P-object given by the projection
[TABLE]
whence we obtain a P-arrow
[TABLE]
whence the colimit arrow
[TABLE]
The pair
[TABLE]
[TABLE]
is an arrow of (A,⊗)-enriched sets.
2.3.21.3. If the P-colimits are objects of P, with monic arrows into the relevant products, then this is a natural transformation of functors
[TABLE]
2.3.22. Remark
One would expect, that if the tensor structure were the product structure, then the arrow c of the previous should be the identity.
2.3.23. Lemma
Let (A,⊗) be a tensor category with a functorial product ×A:A×A⟶A, and F:I⟶A, G:J⟶A any functors with limits (lF,λF) and (lG,λG). Temporarily define the arrow
[TABLE]
to be that which is naturally induced by the assignment to each (i,j)∈Ob(I×J) of the arrow t:×(F(i),F(j))→colim(F)×Acolim(G) such that
[TABLE]
If τ is an isomorphism, then for any Sˉ,Tˉ,Uˉ∈Ob(WE(sk)(A,⊗),
[TABLE]
2.3.24. Definition. of the Skeleton Quotient
For any (A,⊗,α)∈Ob(ATCat), for any functor sk:A⟶B, for any (sk)-unit (I,λA,ρA)∈Ob(A)×Arr(HomU−Cat2(1)(A,A))×Arr(HomU−Cat2(1)(A,A)), such that the arrows of functors
[TABLE]
[TABLE]
are isomorphisms, we define the functor
[TABLE]
by the following. Temporarily define Q∈Ob(U′−Cat) so that
[TABLE]
and for any Sˉ=(S,hS,∘S),Tˉ=(T,hT,∘T)∈Ob(WEAssoc(sk)(A,⊗),
[TABLE]
hold, where the equivalence relation R is temporarily defined so that for any Sˉ,Tˉ∈Ob(WEAssoc(sk)(A,⊗)), for any F,G∈HomWEAssoc(sk)(A,⊗)(Sˉ,Tˉ), F∼RG iff there exist
[TABLE]
[TABLE]
such that
[TABLE]
and
[TABLE]
are both true. We define SK(A,⊗,sk,α,I,λA,ρA) to be the quotient functor, so that
[TABLE]
and
[TABLE]
Generally, when the accompanying data are understood, we will informally write skˉ=SK(A,⊗,sk,α,I,λA,ρA).
3. Higher Categories and Constellations
According to the literature ([9],[13]) one desires that the higher categorical structures ought not to satisfy equalities, but equivalences of some sort. We use the “skeleton functors” and (sk)-limits throughout to implement this. The first section defines n-categories, and the enriched set of n-categories, by the usual inductive intuition. The second section introduces a few book-keeping notions. The enriched set of n-categories is not associative in a satisfactory sense.
The third section introduces constellations, certain constructions of enriched sets, which are under certain conditions associative. We give, in the “Lens” theorem, a formalism for the construction of arrows of (WE(sk)(A,⊗),×)-enriched sets from individual constellations to the enriched set of (A,⊗)-enriched sets.
3.1. n-Categories
An n-Category is defined inductively as an object in the category of (n-1)-enriched categories. n-Categories with their basic structures are inductively defined, referring to each other (and therefore inseparable).
Here, for any n∈N the category of (n)-categories U(n)−Cat
is the category of sets that are weakly enriched over the category
U(n−1)−Cat of (n−1)-categories.
3.1.2. Definition of n-Category
Assuming that we have defined these objects for all integers ≤n we define them for n+1.
3.1.2.1.
The “forgetful”, or “objects” functor
is defined on n-categories and it takes an n-category (an enriched set) to the underlying set
By the usual units on objects and ρu(n) and λu(n) on the hom objects.
3.1.2.8.
Define, for any C,D∈Ob((U,n+1)−Cat), the statement
(C,D) are (n+1)equivalent0⟺
There exist F:C→D, G:D→C, such that for any (c1,c2)∈Ob(C), (d1,d2)∈Ob(D), F(1)(c1,c2) and G(1)(d1,d2) are (n)-equivalences, and (G⋅F,idC),(F⋅G,idD) are (n+1)-equivalent1.
3.1.2.9.
Define, for any C,D∈Ob((U,n+1)−Cat), for any F,G∈Hom(U,n+1)−Cat)(C,D), the statement
(F,G) are (n+1)−equivalent1⟺
There exist
[TABLE]
[TABLE]
,
such that the various arrows
[TABLE]
[TABLE]
given by the composition of ∘ˉ, the arrow I(n)ˉ→hˉWE((U,n)−Cat,×(n))(sk(n))(C,D)(F,G) associated to ϕ or ψ (see 2.3.2 part(iv).) and a unit arrow (λu(n) or ρu(n)), are (n)-equivalences0 (Slightly loose usage. Adapt part (8).) (i.e. they (sk(n))-invert one another).
Roughly speaking there are (sk(n))-natural transformations between F and G, which induce forward and backward compostion functors by the unit and enrichment lemma, which are (n)-equivalences, and such that ϕ∘ψ and ψ∘ϕ induce (n)-equivalent functors to the identities for the respective hom objects.
3.1.2.10.
Define the (n)-skeleton functor sk(n) as a quotient functor
[TABLE]
where Q is the category defined by
[TABLE]
[TABLE]
where [F](n)eq=[G](n)eq iff (F,G) are (n)-equivalent.
to be the category of sets (sk(n))-associatively enriched over over the category of (n)-categories
3.1.3.
Parts (ii) and (iii) of the following lemma give construction for limits and colimits in WE(A,⊗), to be applied to the (co)limits appearing in the construction of the enriched hom sets.
3.1.4. Lemma on Limits and Colimits in WE(A,⊗)
For any (A,⊗)∈Ob(TCat),
3.1.4.1. For:WE(sk)(A,⊗)⟶WE(term∘sk)(A,⊗) is faithful, where term:codom(sk)⟶⋆ is the functor whose codomain is the terminal category. I.e. one forgets that one had had a composition requirement.
3.1.4.2. The limit of F:I⟶WE(A,⊗) can be constructed by the limit of the underlying sets and (ai,bi)i∈I↦limF1′, where F1′:I⟶A is defined on objects by
[TABLE]
3.1.4.3. For any F:I⟶WE(sk)(A,⊗), if τ:colim∘⊗→⊗∘(colim×Catcolim) is an isomorphism where hom objects hF(0)(i)(x,y) are concerned, then the colimit can be similarly constructed, by
[TABLE]
i.e. taking the colimit of all hom objects below both i and j. Define composition by the arrow induced by tensoring the colimit arrows assigned to ([(a,i)],[(b,j)]) and ([(b,j)],[(c,k)]), composed with the inverse of τ.
3.1.5. Remark
The explicit description of limits and (co)limits is applied to verify in the following lemma the isomorphism required for part (ii) of 2.3.2.
3.1.6. Lemma
∀n∈N, colim⋅×(n)→×(n)⋅(colim×Catcolim) is an isomorphism.
Proof.
On the level of sets, this is the isomorphism given by [(ai,bj)]↦([ai],[bj]). By the previous lemma the product of enriched sets is given by taking the products of their hom objects, so that τn+1:colim∘×(n+1)→×(n+1)∘(colim×Catcolim) is determined by τ0 on underlying set and τn on hom objects. By induction, τn is for any n an isomorphism.
∎
The “meaning” of the following theorem consists in the special cases of parts (iii) and (iv) of 2.3.7.
3.1.7. Theorem on (U,n)−Cat
The category (U,n)−Cat is weakly enriched over itself. I.e. ((U,n+1)−Cat,hˉWE((U,n)−Cat,×(n))(sk(n),∘ˉ(n))∈Ob(WE((U,n+1)−Cat,×(n+1)). The hom set agrees with that given by applying the objects functor Ob=F(n) to the hom n-category, i.e. Ob∘Homˉ(U,n)−Cat≅Hom(U,n)−Cat.
Proof.
One must check that the constructions of 2.3.2 (see part(ii)) and 2.3.7 can be applied at each step.
sk(n)-associativity is part of the definition of (U,n)−Cat. The isomorphism of the previous lemma is the only other requirement.
∎
3.1.8. Remark
The restriction of WE((U,n)−Cat,×(n)) to the subcategory of (sk(n))-associative enrichments is necessary for the construction of the hom set enrichment, which is necessary for the definition of the next skeleton functor, sk(n+1)).
3.1.9. Remark
That (U,n+1)−Cat as an enriched set is sk(n+1)-associative (and therefore properly an (n+2)-category) was expected, but not yet clear to me. By part (iv) of 2.3.7 it is associative with respect to the objects functor and sk(n)∘Arrˉ((U.n)−Cat,×(n)), i.e. it is sk(n)-associative with respect to each hom object (n-category). The difficulty seems to be in inferring, from the arrows giving the equivalences within the hom objects, arrows giving equivalences from without. I suspect that this should be easier to do for particular types of n-categories.
3.1.10. Example
(2)−Cat∈Ob(WE((2)−Cat,×(2))). The skeleton is used at the level of the hom objects, so that only the usual skeleton, sk(1), is seen in this case. The objects are enriched sets.
[TABLE]
where the composition is (sk)-associative, where sk=sk(1):Cat⟶Q is the quotient functor determined by identifying isomorphic arrows (functors). The arrows are arrows of enriched sets
[TABLE]
respecting composition after the application of (sk).
By the Hom-enrichment construction one associates to any C,D∈Ob((2)−Cat), Φ,Ψ∈Hom((2)−Cat)(C,D), the category PΦ,Ψ of all arrows (xf∏c∈Ob(C)hD(Φ(c),Ψ(c)) satisfying the (sk)-commutativity requirement. p:PΦ,Ψ⟶Cat is the functor defined by ((x,f)↦x)(x,f)∈Ob(PΦ,Ψ). By definition hˉ2−Cat(a,b)(Φ,Ψ):=colimPΦ,Ψ
The description of the enrichment on (2)−Cat requires, for any (C,D,E)∈O, an arrow
[TABLE]
representing composition. That the above is an arrow in (2)−Cat, interpreted, means that for any Φ1,Φ2,Φ3∈Hom((2)−Cat)(C,D),Ψ1,Ψ2,Ψ3∈Hom((2)−Cat)(D,E),
[TABLE]
[TABLE]
[TABLE]
where
[TABLE]
[TABLE]
where ∘ˉ(C,D) denotes the enriched composition in hˉ2−Cat(C,D). I.e., there is a function (arrow of sets) α:Ob(dom(F))=Ob(dom(G))→Arr(Cat) defining a natural isomorphism between the functors F and G.
3.1.11. Proposition
If the P-colimit inclusion condition is satisfied for (U,n)−Cat, regarding the construction of the hom enrichment, then it is satisfied for (U,n+1)−Cat as well. I.e., the two arrows colimp⊗e0→∏c∈Ob(C)hD(Φ(c),Ψ(c)), one from right composition and the other from left composition, are (n+1)-equivalent.
Proof.
The forgetful functor is at each step given by the objects functor. In this case, P is given by all arrows (aπ∏c∈Ob(C)hD(Φ(c),Ψ(c)))∈Arr((U,n)−Cat), such that for any arrow (e0ehC(x,y))∈Arr((U,n)−Cat) into a hom object in C, the two arrows (if ⊗=×(n))
[TABLE]
one given by composition with e0 on one side and the other by composition on the other, are sk(n)-equivalent. Therefore a choice of an (n+1)-equivalence of (n+1)-functors is still a choice of
[TABLE]
[TABLE]
where hˉWE((U,n)−Cat,×(n))(sk(n))(r,l) is itself by construction a colimit of the domain object functor
[TABLE]
[TABLE]
By the inclusion condition for the n case the hom object assigned to r and l has a monic arrow into the product of hom objects hD(r(0)(x),l(0)(x)). By the isomorphism of the previous lemma and the construction of the colimit in WE(sk)(A,⊗) in the lemma before that, an arrow of functors ϕ∈Ob(hˉWE((U,n)−Cat,×(n))(sk(n))(lcolimp,rcolimp))) is a map of sets
[TABLE]
[TABLE]
[TABLE]
Claim - That a choice argument implies the existence of a natural isomorphism ϕ from the natural isomorphism ϕi.
∎
3.2. Addresses
We introduce the notion of an address, which is sequence of hom objects, each nested within the previous by the n-categorical enrichment. It is essentially a book-keeping tool, meant to record the “location of a k-arrow within an n-category.”
3.2.1. Definition of the Empty n-Category
∅U(1)−Cat:=(∅,∅,∅,∅,∅)∈Ob(U−Cat)=Ob(U(1)−Cat) is the empty category, and ∀n∈N,∅U(n+2)−Cat:=(∅U(1)−Cat,∅,∅)∈Ob(U(n+2)−Cat) is the empty (n+2)-category.
3.2.2. Definition of Addresses
We define two address functions, one for objects in (U,n)−Cat and one for arrows.
3.2.2.1. For any n∈N, fAddU(n)0:Ob(U(n)−Cat)→U′ is defined to be the function which sends an n-category x∈Ob(U(n))−Cat) to the set of functions α:{1,...,j}→U′ such that for any k∈{1,...,j}, where j∈{0,...,n},
[TABLE]
[TABLE]
[TABLE]
[TABLE]
For any n∈N, AddU(n)0:Ob(U(n)−Cat)→U′ is the function which sends an n-category x as above to the set of functions α:{0,...,j}→U′ such that there exist a,b,C,h,∘ for which α=(a(k),b(k))k∈{0,...,j} and (a(k),b(k),C(k),h(k),∘(k))∈fAddU(n)0(x).
These assign to each n-category its set of “(full) addresses,” being sequences
(a(i),b(i),C(i),h(i),∘(i)) such that (a(i+1),b(i+1)) is a pair of objects in the base category C(i) of the (n−i−1)-category associated to the previous pair (a(i),b(i)) by the enrichment. fAdd refers to the former list and Add to the truncated latter.
The “length,” ∣α∣=∣(a,b)∣, will denote its order as a set.
3.2.2.2. For any n∈Ob(N), AddU(n)1:Arr(U(n)−Cat)→U′ is defined to be the function which sends ϕ∈Arr(U(n)−Cat) to a function
[TABLE]
defined inductively, by requiring that
[TABLE]
and that for any (a,b)∈AddU(n)0(dom(ϕ)), for any ϕˉ∈Arr(U(n−∣(a,b)∣)−Cat), S(a,b):=ϕˉ iff there exists (a0,b0)∈AddU(n)0(dom(ϕ)) such that
[TABLE]
and there exists ψ=((f0,f1),f2)∈Arr(U(n−∣(a,b)∣+1)−Cat), such that
[TABLE]
This associates to every arrow of n-categories a function which sends an address for the domain category to the arrow of (n−k)-categories assigned to it by the original arrow.
3.2.3. Remark
That the above definition consists of two maps, one for n-categories and the other for arrows of n-categories, suggests some functor giving an alternate description of n-categories.
3.2.4. Definition of the Functors IncU(n)−CatU(m)−Cat and ForU(n)−CatU(m)−Cat
For any n,m∈N\{0} such that n<m, define functors IncU(n)−CatU(m)−Cat:U(n)−Cat⟶U(m)−Cat and ForU(n)−CatU(m)−Cat:U(m)−Cat⟶U(n)−Cat inductively, by the following.
so that IncU(n+1)−Cat:=(Inc0U(n+1)−Cat,Inc1U(n+1)−Cat):U(n+1)−Cat⟶U(n+2)−Cat.
Now temporarily define IncU(1)−Cat:U−Cat⟶U(2)−Cat to be the functor which sends a category C to the 2-category with enrichment hC(a,b):=(HomC(a,b),{idf;f∈HomC(a,b)},...) given by attaching only identity arrows. Define
[TABLE]
[TABLE]
3.2.4.2. Similarly, for any x=(C,h,∘)∈Ob(U(m+1)−Cat),
[TABLE]
and for any ϕ=(ϕ0,ϕ2)∈Arr(U(m+1)−Cat),
[TABLE]
so that ForU(m+1)−Cat:t=(For0U(m+1)−Cat,ForU(m+1)−Cat)1):U(m+1)−Cat⟶U(m)−Cat.
Now temporarily define ForU(2)−Cat:U(2)−Cat⟶U−Cat to be the functor which forgets the enrichment. Define
[TABLE]
[TABLE]
3.2.5. Lemma
∀n∈N, U(n+1)−Cat has products and coproducts.
Proof.
For products, by induction on n. At the base take the usual product category. For any tuple (xi)i∈S, (yi)i∈S, use the inductive step to take the product ∏i∈ShCi(xi,yi).
For coproducts, at the base take the usual coproduct category (objects are the disjoint union. Hom∐i∈S((a,j),(b,k)) is for j=k, and HomCj(a,b) for j=k). If n≥1, then for the enrichment, h∐i∈SCi((a,j),(b,k)) is ∅U(n)−Cat for j=k, and hCj(a,b) for j=k.
∎
3.2.6. Definition of Products and Coproducts
∏U(n)−Cat and ∐U(n)−Cat will be functions
⋃S∈UHomU′−Set(S,Ob(U(n)−Cat))⟶Ob(U(n)−Cat), the canonical constructions described in the previous lemma’s proof.
3.2.7. Definition of the Restricted Simplicial Sets
Define Δ∈Ob(U−Set) to to be the simplicial category, i.e. its objects are finite ordered sets and its arrows are order-preserving functions.
For any n∈Ob(U−Set), define the category Δ(n):=Δ\({j∈N;j≤n−1},≤N)=↓(Δ)(ob(Δ)(({j∈N;j≤n−1},≤N)),id(U−Cat)(Δ)). This is the arrow category under the set with n elements.
3.2.8. Definition of Primitive Arrows
∀n∈N, ∀f∈Arr(Δ), f is primitive iff ∣∣dom(f)∣−∣codom(f)∣∣=1. ∀ϕ=(f,e,id∘)∈Arr(Δ(n)), ϕ is primitive iff f is primitive.
3.2.9. Lemma
Any arrow in Δ or Δ(n) is a composition of primitive arrows.
3.2.10. Lemma on a Pseudo-Simplicial Structure on (U.n)−Cat
For any n∈N, there exists a unique
[TABLE]
such that for any ϕ=(f,id({1,...,n},≤))∈Arr(Δ(n)), f is primitive implies the following.
3.2.10.1. If f injective, then ρ(1)(ϕ):U(∣dom(f)∣)−Cat⟶U(∣codom(f)∣)−Cat is defined on objects by
[TABLE]
iff
[TABLE]
and for any full address α=(a,b,Cα,hα,∘α)∈fAddU(∣dom(f)∣)((C,h,∘)), for any k∈{1,...,∣dom(f)∣}, f(k+1)=f(k)+2 implies
[TABLE]
[TABLE]
[TABLE]
[TABLE]
The functor ρ1(ϕ) is defined on arrows by
[TABLE]
iff
[TABLE]
and for any address α=(a,b)∈AddU(∣codom(f)∣)(dom(G)), for any k∈{1,...,∣dom(f)∣},
[TABLE]
[TABLE]
3.2.10.2. If f is surjective, then ρ1(ϕ):U(∣dom(f)∣)−Cat⟶U(∣codom(f)∣)−Cat is defined on objects by
[TABLE]
iff
[TABLE]
and for any full address α=(a,b,Cα,hα,∘α)∈fAddU(∣codom(f)∣)0(D), for any k∈N, f(k+1)=f(k) implies
[TABLE]
where
[TABLE]
[TABLE]
The functor ρ1(ϕ) is defined on arrows by
[TABLE]
iff
[TABLE]
and for any full address α=(a,b,Cα,hα,∘α)∈fAddU(∣codom(f)∣)0(C), for any k∈N, f(k+1)=f(k) and ∣α∣=k+1 imply
[TABLE]
I.e. if f is injective, delete the k-th step, replacing it with the coproduct of all n-k-1 categories appearing in the enriched homs there. If surjective, add a step, a base category with only one object, leaving its enriched hom as that which had preceded it.
3.2.11. Lemma on Representing the k-Arrows Functor
Adopt the notation of (3.2.10). Then for any arrow f∈Arr(Δ(n)) if the functor R:(U,n)−Cat⟶(U,∣f∣)−Cat given by requiring that ρ(f)=(⋅,R,(U,∣codom(f)∣)−Cat), then the functor
[TABLE]
is representable.
3.2.12. Remark
I expect there to be some enriched version of this.
3.2.13. Lemma
Adjunction of functors given to opposite pairs of primitive arrows by ρ.
3.2.14. Conjecture on (sk)-associativity for a Subcategory of n−Cat
For any n∈N, for any Bˉ=(B,hB,∘ˉ)∈Ob((WE(U,n)−Cat,×(n))), if there exists C∈Ob(U−Cat), and
[TABLE]
satisfying the following, properties, then Bˉ is sk(n)-associative.
3.2.14.2. For any address β=(Bi,hi,∘ˉi,ai,bi)i∈{1,...,k}∈Add(Bˉ), for some c,d∈Ob(C), the functor
[TABLE]
is faithful, and
[TABLE]
is an equivalence of categories, where “∣β∣” denotes the order of β as a set of pairs, i.e. the number of categories or pairs of objects appearing in the sequence.
3.2.14.3. The functors Φ(β) agree with the composition given by the Hom enrichment lemma, (2.3.7), up to natural isomorphism. Explanation follows.
3.2.14.3.1. Let there be three addresses β,β1,β2∈Add(Bˉ), such that
[TABLE]
and
[TABLE]
i.e. the addresses β1 and β2 correspond to a triple a1(k+1),b1(k+1)=a2(k+1),b2(k+1)∈Ob(Ck) in the underlying category for one of the hom objects, composed to yield β3.
3.2.14.3.2. Then there is a natural isomorphism of functors
[TABLE]
where ∘ˉCat is that of (Enrichment, 2.3.7) for (U,2)−Cat.
3.3. Constellations
We describe, in this section, a method by which WE(sk)(A,⊗)-enriched sets can be constructed from associative (A,⊗)-enriched sets. Given an assignment, to each pair of objects in Sˉ∈Ob(WEAssoc(sk)(A,⊗)), of an (A,⊗)-enriched set, we define the hom object attached to a particular pair to be the enriched set of (A,⊗)-functors which send distinguished elements to the pair. The composition is the composition of a restriction functor with a Kan extension functor which would extend the domains of the two component arrows. Such an enriched set is called a constellation of Sˉ. Under certain conditions, constellations are associative.
We also construct “Lens functors,” (A,⊗)-arrows T:L→WE(skˉ)(WE(sk)(A,⊗),×), from sub-enriched sets L⊆Stell(Sˉ,...) of constellations to the enriched set of enriched sets. If a constellation is associative, then the sub-enriched sets generated by such functors would inherit the associativity.
3.3.1. Lemma. Enriched Yoneda
Let (A,⊗) be a tensor category with an (sk)-associator α and (sk)-units λu and ρu. If an object a∈S admits an (sk)-identity, I→hS(a,a), then any “(sk)-natural transformation of functors” hS(−,a)→hS(−,b) is given by an associated arrow I→hS(a,b). In detail, for any tensor category (A,⊗) with (sk)-associators and an (sk)-unit,
[TABLE]
[TABLE]
for any enriched set (S,hS,∘S), for any two objects, a,b∈S,
[TABLE]
if Φ is a map of sets assigning to each c∈S an arrow hS(c,a)→hS(c,b) in A, satisfying a certain naturality condition, and if the arrow idA:I→hS(a,a) is an (sk)-identity of the object A (in the sense that sk(∘(a,a,x)⋅(idhS(x,a)⊗ida)ρu)=sk(idhS(x,a))),
[TABLE]
[TABLE]
[TABLE]
[TABLE]
then for any c∈S, Φ(c) is indicated by Φ(a) and the identity.
[TABLE]
3.3.2. Definition of Enriched Adjoints
The first part defines one-sided adjoints. The second side requires an associator, α, and defines two-sided adjoints.
3.3.2.1. We define, for a tensor category (A,⊗) an ladjunction(F),
[TABLE]
[TABLE]
to be a map Φ, from pairs of objects in the opposing domains to Arr(A),
[TABLE]
[TABLE]
which satisfies a naturality condition,
[TABLE]
and ┌sk(∘dom(G)(F(a),F(b),c)⋅(F(1)(a,b)⊗idhdom(G)(F(b),c))⋅(idhdom(F)(a,b)⊗Φ(b,c)))=sk(Φ(a,c)⋅∘dom(F)(a,b,G(c)))┐ and
┌sk(Φ(a,c)−1⋅∘dom(G)(F(a),F(b),c)⋅(F(1)(a,b)⊗idhdom(G)(F(b),c)))=sk(∘dom(F)(a,b,G(c))⋅(idhdom(F)(a,b)⊗Φ(b,c)−1))┐┐┐┐┐ and ┌┌Φ is an radjunction(F,G)┐⟺
(symmetric condition, respecting composition on the other side)┐┐┐┐┐┐┐
with a symmetric definition for an radjunction(F,G) on the other side.
3.3.2.2. For a tensor category (A,⊗) with an (sk) associator α,
[TABLE]
[TABLE]
[TABLE]
we similarly define an adjunction(F,G) to be a map of sets Φ,
[TABLE]
[TABLE]
[TABLE]
[TABLE]
[TABLE]
[TABLE]
[TABLE]
[TABLE]
[TABLE]
[TABLE]
[TABLE]
[TABLE]
[TABLE]
[TABLE]
[TABLE]
[TABLE]
3.3.3. Definition of Enriched Kan Extensions
. The first part defines functorial Kan extensions, as adjoints to restriction “functors” i∗, where a restriction functor is understood to be as in the the lemma on composition functors (2.3.7). The second part defines pointwise Kan extensions.
3.3.3.1. For a tensor category with skeleton and associator,
[TABLE]
for any arrows i,i∗ of enriched sets for which i∗ is the “restriction” of i,
any pair (K,Φ) of arrows of sets are said to be left or right Kan extensions (Lan or Ran) accordingly as they serve as adjunctions to the restriction arrow i∗.
[TABLE]
[TABLE]
[TABLE]
3.3.3.2. For any tensor category (A,⊗), for any F∈Hom(WEAssoc(sk)(A,⊗))(dom(i),
codom(F)), Fˉ∈HomWEAssoc(sk)(A,⊗)(codom(i),codom(F)),Φ∈Arr(U′−Set), we say that (Fˉ,Φ) is a Lan(A,⊗)(sk)(i,F) if it satisfies above requirements, involving a single object on the right side. We say that ((ˉF),Φ) is a Ran(A,⊗)(sk)(i,F) in the analogous case.
3.3.4. Definition of a Constellation
Suppose that WE(sk)(A,⊗) has cofibres. Given, (i) an enriched set Sˉ=(S,hS,∘S), and (ii) maps of sets e1,e2,e3,i1,i2, so that (ii.i) the e-maps assign to any triple r,s,t∈S a triple of (A,⊗)-arrows ei(r,s,t) for which all three the same codomain , and (ii.ii) the i-maps assign to each s,t∈S distinguished objects in the enriched sets dom(e2(r,s,t))=dom(e1(s,t,r))=dom(e3(s,r,t) for arbitrary r∈S (requiring this we implicitly assign to each pair (s,t) this enriched set dom(e2(r,s,t))), we define an enriched set on S, over WEAssoc(sk)(A,⊗) by associating to each s,t∈S the enriched set of (A,⊗)-functors dom(e3(s,r,t)→Sˉ which send distinguished elements of the domain to a and b. Composition is given by Kan extensions.
In detail, for any tensor category (A,⊗) with skeleton functor sk and associator α, for any (A,⊗)-enriched set Sˉ,
[TABLE]
[TABLE]
for any arrows of sets e1,e2,e3 (with their domains and codomains marked by Iˉ and Jˉ), which assign to tuples of S arrows of WEAssoc(sk)(A,⊗), and i1,i2, which distinguish elements within the enriched sets Iˉ, (in which case we say that “e1,e2,e3,i1,i2 are constellation data for Sˉ”)
[TABLE]
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[TABLE]
[TABLE]
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[TABLE]
we say that any enriched set (S,hˉ,∘ˉ) is a constellation(Sˉ,e1,e2,e3,i1,i2) iff
its hom objects are sets of (A,⊗)-arrows respecting the distinguished objects,
[TABLE]
[TABLE]
[TABLE]
and the composition arrow ∘ˉ(a,b,c) is given by the restriction along e3(a,b,c) of a Kan extensions function. The arrow
e3∗:hˉWEAssoc(sk)(A,⊗)(Jˉ(a,b,c),Sˉ)→hˉWEAssoc(sk)(A,⊗)(Iˉ(a,c),Sˉ) below is as in the
HomWEAssoc(sk)(A,⊗) enrichment lemma.
[TABLE]
[TABLE]
[TABLE]
[TABLE]
[TABLE]
[TABLE]
and ┌┌(S,hˉ,∘ˉ) is an r-constellation(A,⊗,α)(sk)(Sˉ,e1,e2,e3,i1,i2)┐⟺ (Analogous, but with Ran)┐┐┐┐┐┐┐┐┐┐┐┐┐┐
3.3.5. Lemma
If A has products and an initial object ∅A∈Ob(A), then for any S1,S2,S3∈Ob(WEAssoc(sk)(A,⊗)), the coproduct map
[TABLE]
associated to S1⊔S2 “extends” to an arrow
[TABLE]
[TABLE]
i.e. there exists an arrow ⊔(S1,S2)(S3) such that the object functor sends it to the underlying map on sets, Ob(sk,A,⊗)(⊔(S1,S2)(S3)))=⊔0(S1,S2)(S3).
3.3.6. Lemma
If (A,⊗) has an (sk)-unit, I, then any two Kan extensions are (sk)-equivalent, in the sense of (2.3.24).
3.3.7. Lemma
For any f,g∈Arr(WEAssoc(sk)(A,⊗)),skˉ(f∗⋅g∗)=skˉ((g⋅f)∗), where f∗⋅g∗,(g⋅f)∗:hˉWEAssoc(sk)(A,⊗)(codom(g),x)→hˉWEAssoc(sk)(A,⊗)(dom(f),x) are as in the enrichment lemma.
3.3.8. Proposition
Suppose that the tensor category (A,⊗) with (sk)-associator α has an (sk)-unit I and skˉ:WEAssoc(sk)(A,⊗)⟶Q is the quotient functor given by identifying such (A,⊗)-functors as should admit pairs of equivalences between them, given by elements of the sets given by applying Yo(A)opp(I) to the hom objects in hˉWEAssoc(sk)(A,⊗)(c,d), as in (2.3.24). Then for any Sˉ,e1,e2,e3,i1,i2, etc. as in the previous definition, (i)⟹(ii).
(i). For any a,b,c,d∈S, for any diagram (ε:D⟶WEAssoc(sk)(A,⊗))∈Arr(U′−Cat) in WEAssoc(sk)(A,⊗) which includes a subcategory D of the form
[TABLE]
[TABLE]
[TABLE]
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[TABLE]
for any colimit (LD,λD) of D, for any arrows of enriched sets
[TABLE]
[TABLE]
[TABLE]
[TABLE]
with functions ΦλD(I(a,b)⊔J(b,c,d)),Φe1(a,b,d)⊔e2(a,b,d)),ΦλD(J(a,b,c)⊔I(c,d)),Φe1(a,c,d)⊔e2(a,c,d))∈
Arr(U′−Set), such that each of the pairs
[TABLE]
[TABLE]
is a Lan respectively of the arrows subscribed in its components, we have that
[TABLE]
and
[TABLE]
(ii). The enriched set (S,hˉ,∘ˉ)∈Ob(WE(skˉ)(WEAssoc(sk)(A,⊗),×WEAssoc(sk)(A,⊗))) is (skˉ)-associative.
3.3.9. Remark
The condition (i) of the previous proposition expresses the notion that one can take both Kan extensions, necessary for the composition hˉ(a,b)×hˉ(b,c)×hˉ(c,d)→hˉ(a,d), at the same time. The composition, when done in either either order, should therefor result in a restriction of a functor whose domain is an (A,⊗)-set glued together from the I and J sets.
3.3.10. Example
n−Cat. The Yoneda functor Yo(n−Cat)opp(I(n)) of the unit for the product ×(n) differs from the objects functor (given by Yo(n−Cat)opp(Inoenrich), where Inoenrich, has one object, enriched by the empty category (even without a unit). Unless the category of n-functors is restricted to those preserving the identity, recording idempotent higher arrows.
3.3.11. Remark
Given an arrow (F:Sˉ→Tˉ) of enriched sets, one might define an arrow (Fˉ:Stell(Sˉ,e,...)→Stell(Tˉ,e,...))∈Arr(WE(skˉ)(WEAssoc(sk)(A,⊗),×WEAssoc(sk)(A,⊗))), between the left constellations (the data ej,ik being the same), if the “pushforward” F∗:hˉ(Iˉ,Sˉ)→hˉ(Iˉ,Tˉ) commutes with the Kan extensions used in the compositions.
3.3.12. Proposition
For any s:S×S→Arr(WEAssoc(sk)(A,⊗)), t12,t3:S×S×S→Arr(WEAssoc(sk)(A,⊗)),
3.3.12.1. For any two sets of constellation data for Sˉ, (e1,e2,e3,ie1,ie2),(f1,f2,f3,if1,if2), ┌(i)⟹(ii)┐ and (iii).
(i). For any functions s′,t12′,t3′,e12′,e3′,f12′,f3′:S×S×S→Arr(U−Cat), for any a,b∈S, s′(a,b) is a Lan(s(a,b)),
and for any a,b,c∈S,
t12′(a,b,c)is a Lan(t12(a,b,c)) and t3′(a,b,c)is a Lan(t3(a,b,c)) and
e12′(a,b,c) is a Lan(e12) (for some coproduct arrow e12=e1(a,b,c)⊔e2(a,b,c)) and
e3′ is a Lan(e3(a,b,c)) and
[TABLE]
[TABLE]
is the pullback for hˉ, and
f12′(a,b,c) is a Lan(f12) (for some coproduct arrow f12=f1(a,b,c)⊔f2(a,b,c)) and
f3′(a,b,c) is a Lan(f3(a,b,c)) and
[TABLE]
[TABLE]
is the pullback for hˉ, imply that for any a,b,c∈S
[TABLE]
[TABLE]
[TABLE]
[TABLE]
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(ii). The pair F=(idS,s′):Sˉ→Tˉ is an arrow in
WEskˉ(WEAssoc(sk)(A,⊗),×WEAssoc(sk)(A,⊗)), where Sˉ is the constellation formed of the e1,2,e3 data and … .
(iii). For any tuples (s1,t112,t13,s1′,t112′,t13′) and (s2,t212,t23,s2′,t212′,t23′), for any tuple (s3′,t312′,t33′) such that the first two tuples and the tuple (s2⋅s1,t212⋅t112,t23⋅t13,s3′,t313′,t33′) all satisfy condition (i) in the place of (s,t12,t3,s′,t12′,t3′),
For any J∈Ob(A), for any arrow of functors ρ for which (Yoopp(J),ρ):(A,⊗)→(Set,×Set) is an arrow of tensor categories, for any objects a,b∈Ob(S)=Ob(Sˉ),
[TABLE]
i.e. the enriched sets WE(Ob,ρ0)⋅Stell(Bar(λA,ρA)(J))(Sˉ) and WE(Yoopp(J),ρ)(Sˉ) have isomorphic hom objects.
3.3.14. Remark
Any (A,⊗)-enriched set, Sˉ=(S,...) can be reconsidered in several ways as a (WEAssoc(sk)(A,⊗),×WEAssoc(sk)(A,⊗))-enriched set, with such data (s,t3) as above (assignments of functors between composition data) allowing for their comparison. This might be inductively applied to induce an ((n,A)−Cat,×)-enrichment with the same underlying (Set,×)-enrichment as the original enriched set Sˉ.
3.3.15. Example
Trivial. Assign to x,y∈Ob(C) the diagram (x0→y0) . The composition is given by gluing two such diagrams into a diagram
[TABLE]
so that the first arrow is composed of the e1,e2 arrows and the third is the e3 arrow.
3.3.16. Example
“Localization.” Assign to x,y∈Ob(C) the diagram with objects x0,a, and y0, and arrows (x0→a), (a→x0), (a→y0) and (x0→y0), with both arrows from x0 to y0 equal. The composition is given by gluing the individual diagrams at the middle object, and adding an object c, “above both a above x0→y0 and b above y0→z0.”
One might, in applying (3.3.12), send (x00→y00) as in the first example to the arrow (x0→y0).
3.3.17. Example
Fibre functors, their adjoints. Given an admissibility structure ε:E↪Fib on a category T (see [18]), assign to x,y∈Ob(T) the sub-category of diagrams which is the disjoint union of the arrow category over y with itself, with an additional arrow (u,1)→(u,2) for each object u∈Ob(E(y)). The distinguished objects are y0=(idy,1) and x0=(idy,2).
3.3.18. Remark
One obtains (k+1)-arrows from k-arrows by appending a tuple of arrows of C, for the natural transformations.
3.3.19. Theorem (“Lens”)
For any arrows of enriched sets ud,uc:Iˉ=(I,...)→Jˉ=(J,...) with two distinguished objects i1′,i2′∈I, for any sub-enriched set
[TABLE]
for any functions c:L→2L,ld,lc:F⋅Arrˉ(Lˉ)→Arr(WEAssoc(sk)(A,⊗) such that for any k,l,m∈Ob(L), for any ϕ∈Ob(hL(l,m)), ψ∈Ob(hL(k,l)), the following, (i) and (ii), hold,
(i). dom(ld(ϕ))=dom(lc(ψ))=⟨c(m)⟩WEFull(Lˉ)⊆Lˉ, and codom(ld(ϕ))=dom(e2(k,l,m)) and codom(lc(ψ))=dom(e1(k,l,m)).
(ii). skˉ(ϕ⋅ld(ϕ))=skˉ(ψ⋅lc(ψ).
we construct arrows
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[TABLE]
by the following (3.3.19.1), (3.3.19.2), and (3.3.19.3). For any l-constellation Stell(Sˉ,...),
3.3.19.1. For any s∈L⊆S=Ob(Stell(Sˉ,...)), the arrow
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[TABLE]
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[TABLE]
is the inclusion of the sub-enriched set generated by (A,⊗)-functors skˉ factoring through some arrow ϕ∈F∘Arrˉ(Lˉ), which agree on distinguished elements, the (domain and codomain enriched sets).
3.3.19.2. For any s,t∈L, for any ϕ∈Ob(hL(s,t)), let
[TABLE]
[TABLE]
[TABLE]
[TABLE]
be the inclusion of the sub-enriched set generated by (A,⊗)-functors
x:Jˉ→dom(e1(s,t,t)) for which the composition ϕ⋅x for any arrow ϕ∈Ob(hL(s,t)) restricts by both arrows ud,uc:Iˉ→Jˉ to objects of the enriched sets T0(s) and T0(t) respectively.
[TABLE]
[TABLE]
be the restrictions. Suppose that
[TABLE]
is a left (right for r-constellations) adjoint of ud∗. Then
[TABLE]
is the element of the set HomWEsk(A,⊗)(T(0)(dom(ϕ)),T(0)(codom(ϕ))) assigned to ϕ.
3.3.19.3. For any s,t∈S, for any ϕ,ψ∈Ob(hL(s,t)) consider the projections
[TABLE]
from the full tuple of hom-objects (a sub-object of which is the object of A specified in the enrichment lemma) to those which appear in the codomain components, i.e. between the images of objects i under the functors T(ϕ)(f) and T(ψ)(f). Composition of this projection with the maps x→∏i∈Ob(dom(ϕ))hS(ϕ(0)(i),ψ(0)(i)) associated to each object in the colimit diagram induces arrows x→∏i∈Ob(dom(f))hS(T10(ϕ)(0)(f)(i),T10(ψ)(0)(f)(i) compatible with the composition requirement, and therefore of the latter colimit diagram, and provided with an arrow into the latter colimit. The colimit map for the enrichment object between ϕ and ψ induces an arrow,
[TABLE]
which similarly induces an arrow, which we define to be
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3.3.19.4. If the condition on arrows τ of (2.3.23) holds, then an arrow c∈HomA(I0,I), where I0 is a unit for the product tensor structure ×A and I is a unit for ⊗ (the arrow c is as that of (2.3.21.3), and is implicitly used thereby, in the below invocation) define an arrow
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by pulling back the product maps,
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by the lemma on the product enriched set, (2.3.23),
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by the lemma on composition functors, (2.3.7.3), with the product unit
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by the restriction uc∗ from the arrow uc:Iˉ→Jˉ, with the product unit,
3.3.19.4.1. For any colimit (Jˉ′,λJ) of the diagram D:({0,1,2},{(0,1),(0,2),id0,...},...)
⟶WEAssoc(sk)(A,⊗) with two non-identity arrows ud,uc:Iˉ→Jˉ and three objects, so that
[TABLE]
[TABLE]
For any arrow of enriched sets d:Jˉ→Jˉ′, using the notation of (3.3.19.2), we temporarily define a sub-enriched set
[TABLE]
[TABLE]
[TABLE]
take the arrow
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to be the restriction, and the arrow
[TABLE]
to be the left (right for r-constellations) Kan extension of udd∗. Applying the arrow of enriched sets of (3.3.19.4) to objects constructed from this, temporarily defining
[TABLE]
and
[TABLE]
we obtain an arrow in A
[TABLE]
where ud′=ud(r,t)′ as for H(r,t) and K is the Lan (Ran) functor used for composition.
3.3.19.4.2. In particular, given an (sk)-unit (I0,λ0,ρ0)∈Ob(A) a pair of arrows (I0→D),(I0→C)∈Arr(A) i.e. “natural transformations”
[TABLE]
and
[TABLE]
determine an arrow
[TABLE]
i.e. a “natural transformation”
[TABLE]
so that, if the former two arrows are (sk)-equivalences, the latter is an (sk)-equivalence.
3.3.19.4.3. If, for every r,s,t∈L, there are such (sk)-equivalences as in (3.3.19.4.2), then (3.3.19.1),(3.3.19.2) and (3.3.19.3) define an arrow T:Lˉ→WEˉ(WEAssoc(sk)(A,⊗)). An arrow To:Lˉopp→WEˉ(WEAssoc(sk)(A,⊗)) would be similarly defined, differing in that the “functors” T10(ϕ) are defined by taking the Kan extensions of functors with “images” in the codomain enriched sets.
Proof.
Articles (3.3.19.1) and (3.3.19.2) are definitions.
The proof of (3.3.19.3) is the argument that the objects p:x→∏i∈dom(e1(s,t,t)hS(ϕ(i),ψi) naturally determine objects in the P-category which determines the object
hhˉ(T0(s),T0(t))(T10(ϕ),T10(ψ))∈Ob(A). The arrows T10(ϕ)(x,y) and T10(ψ)(x,y) factor through ϕ and ψ respectively so that their tensors with the component arrows for h...(ϕ,ψ) are commutative with respect to the composition in Sˉ.
Article (3.3.19.4) is a definition composed of the listed lemmas. Its subsections are corollaries of the existence of the constructed arrow.
∎
3.3.20. Corollary
If the condition (i) of (3.3.8) is satisfied, then any sub-enriched set of (n+1)−Cat generated by the image of an arrow of enriched sets of the form T or Topp of (3.3.19), for (A,⊗)=(n−Cat,×), i.e. whose k-arrows are the images of those of the chosen L, is (skˉ)-associative.
3.3.21. Remark
Roughly speaking, the first arrow of (3.3.19.4.2) expresses a “natural transformation” between the arrow of enriched sets T0(r)→hˉ(Jˉ,codom(e3(r,s,t))) which finds the closest arrow factoring Jˉ directly through dom(e3(r,s,t)), and that which finds the closest arrow factoring Jˉ through dom(e1(r,s,t)) and dom(e2(r,s,t)) together, through a duplication of itself in Jˉ′. It could be thought of as a constellation data in miniature.
The second arrow of (3.3.19.4.2) expresses a “natural transformation” from the composition of the restriction with the Kan extension (that used for the composition) to the identity functor.
3.3.22. Remark
I imagine that one might construct functors End:C→WE(A,⊗), borrowing the enrichment of WE(WE(A,⊗),×) by the arrows L,Lo:C,Copp→WE(A,⊗), and mapping c to hˉ(Lo(c),L(c)) or hˉ(L(c),L(c)).
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