The tensor functor from the category of $A_\infty$-algebras into the category of differential modules with $\infty$-simplicial faces
S.V. Lapin
TL;DR
This paper constructs a tensor functor from the category of $A_ abla$-algebras to differential modules with $ abla$-simplicial faces, preserving homotopy equivalences, advancing the understanding of algebraic structures in homotopy theory.
Contribution
It introduces a new tensor functor linking $A_ abla$-algebras and differential modules with $ abla$-simplicial faces, preserving homotopy equivalences.
Findings
The tensor functor is explicitly constructed.
Homotopy equivalences are preserved under the functor.
The work advances the algebraic understanding of $ abla$-simplicial structures.
Abstract
The tensor functor from the category of -algebras into the category of differential modules with -simplicial faces is constructed. Further, it is showed that this functor sends homotopy equivalent -algebras into homotopy equivalent differential modules with -simplicial faces.
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Taxonomy
TopicsHomotopy and Cohomology in Algebraic Topology · Algebraic structures and combinatorial models · Advanced Topics in Algebra
The tensor functor from the category of -algebras into the category of differential modules
with -simplicial faces.
S.V. Lapin
Abstract
In the present paper the tensor functor from the category of -algebras into the category of differential modules with -simplicial faces is constructed. Further, it is showed that this functor sends homotopy equivalent -algebras into homotopy equivalent differential modules with -simplicial faces.
In [1]-[7] the homotopy technique of differential modules with -simplicial faces was developed. This homotopy technique is closely related to the homotopy technique of -differential modules that developed in [8]-[16]. On the other hand, in [4] by any -algebra the tensor differential module with -simplicial faces was constructed. By using the technique of these tensor differential modules with -simplicial faces that defined by -algebras the concepts of cyclic homology, dihedral homology and reflexive homology of -algebras were developed in [17]-[19]. The fact [4] that by each -algebra corresponds a tensor differential module with -simplicial faces give rise the very important and interesting problem of extension of this correspondence until a tensor functor from the category of -algebras into the category of differential modules with -simplicial faces.
The present paper is devoted to solving the specified above problem. The paper consists of three paragraphs. In the first paragraph, we recall the necessary information from [1]-[7] related to the notion of a differential module with -simplicial faces. In the second paragraph, we recall the necessary information from [20]-[22] related to the notion of an -algebra. Moreover, in this paragraph, we pay special attention to the problem of correctly writing signs in the structural relations of -algebras because it is very important in the further considerations. In the third paragraph, we construct a tensor functor from the category of -algebras into the category of differential modules with -simplicial faces that continues the correspondence from [4] between -algebras and tensor differential modules with -simplicial faces. Further, we show that this tensor functor sends homotopy equivalent -algebras into homotopy equivalent differential modules with -simplicial faces.
We proceed to precise definitions and statements. All modules and maps of modules considered in this paper are assumed to be -modules and -linear maps of modules, respectively, where is an arbitrary commutative ring with unity.
§ 1. Necessary information about
differential modules with -simplicial faces.
In what follows, by a bigraded module we mean any bigraded module , , , and by a differential bigraded module, or, briefly, a differential module , we mean any bigraded module endowed with a differential of bidegree .
Recall that a differential module with simplicial faces is defined as a differential module together with a family of module maps , , which are maps of differential modules and satisfy the simplicial commutation relations , . The maps are called the simplicial face operators or, more briefly, the simplicial faces of the differential module .
Now, we recall the notion of a differential module with -simplicial faces [1] (see also [3]-[7]), which is a homotopy invariant analogue of the notion of a differential module with simplicial faces.
Let be the symmetric group of permutations on a -element set. Given an arbitrary permutation and any -tuple of nonnegative integers , where , we consider the -tuple , where acts on the -tuple in the standard way, i.e., permutes its components. For the -tuple , we define a -tuple by the following formulae
[TABLE]
where each is the number of those elements of on the right of that are smaller than .
A differential module with -simplicial faces is defined as a differential module together with a family of module maps
[TABLE]
[TABLE]
which satisfy the relations
[TABLE]
where is the set of all partitions of the -tuple into two tuples and , , such that the conditions and holds. The maps are called the -simplicial faces of the differential module with -simplicial faces .
For example, at the relations take, respectively, the following view
[TABLE]
[TABLE]
[TABLE]
Simplest examples of differential modules with -simplicial faces are differential modules with simplicial faces. Indeed, given any differential module with simplicial faces , we can define by setting , , and , , thus obtaining the differential module with -simplicial faces .
It is worth mentioning that the notion of an differential module with -simplicial faces specified above is a part of the general notion of a differential -simplicial module introduced in [3] by using the homotopy technique of differential Lie modules over curved colored coalgebras.
Now, we recall that a map of differential modules with simplicial faces is defined as a map of differential modules that satisfies the relations , .
Let us consider the notion of a morphism of differential modules with -simplicial faces [1], which homotopically generalizes the notion of a map differential modules with simplicial faces.
A morphism of differential modules with -simplicial faces is defined as a family of module maps
[TABLE]
[TABLE]
(at we will use the denotation ), which satisfy the relations
[TABLE]
[TABLE]
[TABLE]
where is the same as in . The maps are called the components of the morphism .
For example, at the relations take, respectively, the following view
[TABLE]
[TABLE]
[TABLE]
[TABLE]
[TABLE]
Under a composition of morphisms of differential modules with -simplicial faces and we mean [1] a morphism differential modules with -simplicial faces whose components are defined by
[TABLE]
where is the set of all partitions of the -tuple into two tuples and , , such that the conditions and holds.
For example, at the formulae take, respectively, the following form
[TABLE]
[TABLE]
[TABLE]
[TABLE]
Given any differential module with -simplicial faces , there is an identity morphism
[TABLE]
where is the identity map of the module and is the zero map of modules for each . Thus, the class of all differential modules with -simplicial faces and their morphisms is a category.
Now, we recall that a homotopy between morphisms of differential modules with simplicial faces is defines as a homotopy between morphisms of differential modules satisfies the relations
[TABLE]
Let us consider the notion of a homotopy between morphisms of differential modules with -simplicial faces [1], which homotopically generalizes the notion of a homotopy between morphisms of differential modules with simplicial faces.
A homotopy between morphisms of differential modules with -simplicial faces is defined as a family of module maps
[TABLE]
[TABLE]
(at we will use the denotation ), which satisfy the relations
[TABLE]
[TABLE]
[TABLE]
where is the same as in . The maps are called the components of the homotopy .
For example, at the relations take, respectively, the following view
[TABLE]
[TABLE]
[TABLE]
[TABLE]
[TABLE]
By using the notion of a homotopy between morphisms of differential modules with -simplicial faces the notion homotopy equivalent differential modules with -simplicial faces is introduced in the usual way.
In conclusion of this paragraph it is worth mentioning that the homotopy invariance of the structure of a differential module with -simplicial faces under homotopy equivalences of the type of an SDR-data of differential modules was established in [1].
§ 2. Necessary information about -algebras.
Recall, following [20] (see also [21]), that an -algebra is any differential module with , , , , equipped with a family of maps , , satisfying the following relations for any integer :
[TABLE]
where .
For example, at the relations take the forms
[TABLE]
[TABLE]
For greater clarity and assurance that in the sign is written correctly, let us describe the procedure of finding of this sign.
Given any differential module with , , , , consider the suspension over , where and the differential at any element is defined by . It is well known (see, for example, [22]) that introduction of the structure of an -algebra on the differential module is equivalent to consideration of a family of module maps
[TABLE]
satisfying the following relations for any integer :
[TABLE]
where . Indeed, if we consider the maps of differential modules and defined by and , then the specified above maps and define each other by . By using equalities , , , , , and also by applying the Koszul rule of computing of signs, we obtain
[TABLE]
[TABLE]
[TABLE]
[TABLE]
[TABLE]
where . It follows that the relations holds because the map is a isomorphism of degree of graded modules and the congruence is true.
Now, recall [20] that a morphism of -algebras is defined as a family of module maps , which, for all integers , satisfy the relations
[TABLE]
[TABLE]
where and
[TABLE]
[TABLE]
For example, at the relations take, respectively, the following view
[TABLE]
[TABLE]
For greater clarity and assurance that in the signs and are written correctly, let us describe the procedure of finding of these signs.
As above, given any differential module with , , , , consider the suspension over and the maps of differential modules and . It is well known (see, for example, [22]) that consideration of the morphism of -algebras is equivalent to consideration of the family of module maps
[TABLE]
satisfying the following relations for any integer :
[TABLE]
[TABLE]
where is the same as in and . The maps , , were defined above. The specified above maps and define each other by , . Since and for all , we have
[TABLE]
By using equalities , , , , , and also by applying the Koszul rule of computing of signs, we obtain
[TABLE]
[TABLE]
[TABLE]
[TABLE]
[TABLE]
[TABLE]
By using equalities , , , , , and also by applying the Koszul rule of computing of signs, we have
[TABLE]
[TABLE]
[TABLE]
[TABLE]
[TABLE]
[TABLE]
[TABLE]
[TABLE]
where
[TABLE]
The obtained above equalities follows that the relations holds because the map is a isomorphism of degree of graded modules.
Under a composition of morphisms of -algebras and we mean [20] a morphism of -algebras
[TABLE]
defined by
[TABLE]
where and
[TABLE]
For example, at the formulae take, respectively, the following view
[TABLE]
[TABLE]
For greater clarity and assurance that in the sign is written correctly, let us describe the procedure of finding of this sign.
As above, given any differential module with , , , , consider the suspension over and the map of differential modules . It is well known (see, for example, [22]) that consideration of the composition of morphisms of -algebras and is equivalent to consideration of the composition of families of maps and corresponding to the morphisms of -algebras and . The composition of families of maps and is defined by
[TABLE]
where is the same as in . By using equalities
[TABLE]
[TABLE]
and also by applying the Koszul rule of computing of signs, we obtain
[TABLE]
[TABLE]
[TABLE]
[TABLE]
[TABLE]
[TABLE]
[TABLE]
[TABLE]
where
[TABLE]
The obtained above equalities follows that the relations holds because the map is a isomorphism of degree of graded modules.
Given any -algebra , there is an identity morphism
[TABLE]
where is the identity map of the module and is the zero map of modules for each . Thus, the class of all -algebras and their morphisms is a category.
Now, recall (see, for example [22]) that a homotopy between morphisms of -algebras is defined as a family of module maps , which, for all integers , satisfy the relations
[TABLE]
[TABLE]
where and
[TABLE]
[TABLE]
For example, at the relations take, respectively, the following view
[TABLE]
[TABLE]
[TABLE]
For greater clarity and assurance that in the signs and are written correctly, let us describe the procedure of finding of these signs.
As above, given any differential module with , , , , consider the suspension over and the maps of differential modules and . It is well known (see, for example, [22]) that consideration of the homotopy between morphisms of -algebras is equivalent to consideration of the family of module maps
[TABLE]
satisfying the following relations for any integer :
[TABLE]
[TABLE]
[TABLE]
where is the same as in and . The maps , , , , were considered above. The specified above maps and define each other by , . Since and for all , we have
[TABLE]
It is obvious that the equality
[TABLE]
holds. By using , , , , , and also by applying the Koszul rule of computing of signs, we obtain
[TABLE]
[TABLE]
[TABLE]
[TABLE]
[TABLE]
[TABLE]
By using equalities
[TABLE]
[TABLE]
[TABLE]
and also by applying the Koszul rule of computing of signs, we have
[TABLE]
[TABLE]
[TABLE]
[TABLE]
[TABLE]
[TABLE]
[TABLE]
[TABLE]
[TABLE]
[TABLE]
where
[TABLE]
[TABLE]
[TABLE]
[TABLE]
[TABLE]
[TABLE]
[TABLE]
The obtained above equalities follows that the relations holds because the map is a isomorphism of degree of graded modules.
By using the notion of a homotopy between morphisms of -algebras the notion homotopy equivalent -algebras is introduced in the usual way.
In conclusion of this paragraph we mention that the homotopy invariance of the structure of an -algebra under homotopy equivalences of the type of an SDR-data of differential modules was established in [20].
§ 3. The tensor functor from the category of -algebras
into the category of differential modules with -simplicial faces.
In [4] (see also [1]) it was shown that each -algebra defines the tensor differential module with -simplicial faces . Let us consider the construction of this tensor differential module with -simplicial faces.
Given any -algebra , consider the tensor differential module defined by
[TABLE]
and is the usual differential in a tensor product. Consider also the family of maps
[TABLE]
[TABLE]
defined by
[TABLE]
In [4] it was proved the following assertion, which establish the connection between of -algebras and differential modules with -simplicial faces.
Theorem 3.1. Given any -algebra , the above triple is a differential module with -simplicial faces.
Let us show that each morphism of -algebra defines the morphism of differential modules with -simplicial faces.
Given any morphism of -algebras , consider the family of module maps
[TABLE]
[TABLE]
defined by the following rules:
1). If , then
[TABLE]
2). If or , then
[TABLE]
3). If , , ,
[TABLE]
[TABLE]
then
[TABLE]
[TABLE]
where
[TABLE]
For example, given the map , the formula take the following view
[TABLE]
It is worth mentioning that an arbitrary collection of numbers , where , always can be represented in the view, which specified in the point 3).
The following assertion establish the connection between morphisms of -algebras and morphisms of differential modules with -simplicial faces.
Theorem 3.2. Given any morphism of -algebras , the family of maps defined by – is the morphism of differential modules with -simplicial faces.
Proof. For the family of maps defined by – , we need to check that the relations holds. It is clearly that at we have because . It is easy to see that, for any collection of numbers , where , and any permutation , we have that each partition satisfies the condition or the condition . It follows the required relation . Similarly, for any collection of numbers , where , and any permutation , we have that each partition satisfies the condition or the condition . It follows the required relation . Now, we check that the maps
[TABLE]
satisfy the relations . First, we rewrite the relations in the form
[TABLE]
[TABLE]
[TABLE]
where
[TABLE]
[TABLE]
[TABLE]
Given any fixed collections and , consider the partition of the collection of numbers into the following blocks
[TABLE]
[TABLE]
[TABLE]
[TABLE]
Any specified above partition defines the permutation , which act on the collection of numbers by the following rule:
[TABLE]
[TABLE]
It is easy to check that the equality of collections
[TABLE]
is true. The equalities – and imply that the relations in the considered case can be rewritten in the form
[TABLE]
[TABLE]
[TABLE]
where by we denote the permutation such that and . Now, we calculate , . Denote by the number of elements in the block , , and by the number of elements in the block , . Since is the permutation acting on the collection by partitioning this collection into blocks as and permuting this blocks by the formula , the number of inversions of the permutation is equal
[TABLE]
[TABLE]
Taking into account the congruence , and by using the equalities
[TABLE]
we obtain the congruence
[TABLE]
[TABLE]
In particular, at , , we have the following congruence
[TABLE]
Now, we show that the relations are equivalence to the relations . Indeed, by using the equalities , and we rewrite the relations in the form
[TABLE]
[TABLE]
[TABLE]
where
[TABLE]
[TABLE]
[TABLE]
[TABLE]
For the exponent , we have
[TABLE]
[TABLE]
[TABLE]
For the exponent , by using the equality we have
[TABLE]
[TABLE]
[TABLE]
[TABLE]
[TABLE]
[TABLE]
[TABLE]
[TABLE]
Thus the relations are equivalent to the relations and, consequently, the maps , , satisfy the relations . In the same way, as it was done above, it is checked that all maps
[TABLE]
[TABLE]
[TABLE]
satisfy the relations . By using this and also by using the obvious equality
[TABLE]
[TABLE]
[TABLE]
we easily obtain that all maps defined by satisfy the relations .
The following assertion establish the connection between the composition of morphisms of -algebras and the composition of morphisms of differential modules with -simplicial faces. This assertion is proved in the similar way as Theorem 2.
Theorem 3.3. Given any morphisms of -algebras and , there is the equality
[TABLE]
of morphisms of differential modules with -simplicial faces.
Theorems – implies that there is the functor of taking of a tensor module or, more briefly, the tensor functor
[TABLE]
[TABLE]
where by denoted the category of -algebras over the ground ring and by denoted the category of differential modules with -simplicial faces over the ground ring .
Let us show that the tensor functor sends homotopies between morphisms into homotopies between morphisms. More precisely, let us show that each homotopy between morphism of -algebras defines the homotopy between morphisms of differential modules with -simplicial faces.
Given any homotopy between morphisms of -algebras , consider the family of module maps
[TABLE]
[TABLE]
defined by the following rules:
1). If , then
[TABLE]
2). If or , then
[TABLE]
3). If , , ,
[TABLE]
[TABLE]
then
[TABLE]
[TABLE]
[TABLE]
[TABLE]
[TABLE]
[TABLE]
where
[TABLE]
[TABLE]
For example, given the map , the formula take the following view
[TABLE]
[TABLE]
[TABLE]
[TABLE]
[TABLE]
As above we must note that an arbitrary collection of numbers , where , always can be represented in the view, which specified in the point 3).
Theorem 3.4. Suppose given any homotopy between morphisms of -algebras . Then the family of maps defined by – is the homotopy between morphisms of differential modules with -simplicial faces.
Proof. For the family of maps defined by – , we need to check that the relations holds. Clearly, that at we have because . It is easy to see that, for any collection of numbers , where , and any permutation , we have that each partition satisfies the condition or the condition . It follows the required relation . Similarly, for any collection of numbers , where , and any permutation , we have that each partition satisfies the condition or the condition . It follows the required relation . Now, we check that the maps
[TABLE]
satisfy the relations . First, we rewrite the relations in the form
[TABLE]
[TABLE]
[TABLE]
[TABLE]
[TABLE]
[TABLE]
[TABLE]
[TABLE]
[TABLE]
where
[TABLE]
[TABLE]
[TABLE]
The equalities – , and – imply that the relations in the considered case can be rewritten in the form
[TABLE]
[TABLE]
[TABLE]
where and are the same as in .
Let us show that the relations are equivalent to the relations . By using the relations – and – rewritten the relation in the form
[TABLE]
[TABLE]
[TABLE]
[TABLE]
[TABLE]
[TABLE]
[TABLE]
[TABLE]
[TABLE]
where
[TABLE]
[TABLE]
[TABLE]
[TABLE]
For the exponent , by using we have
[TABLE]
[TABLE]
[TABLE]
For the exponent , by using
[TABLE]
[TABLE]
and , we have
[TABLE]
[TABLE]
[TABLE]
[TABLE]
[TABLE]
[TABLE]
[TABLE]
Thus the relations are equivalent to the relations and, consequently, the maps , , satisfy the relations . In the same way, as it was done above, it is checked that all maps
[TABLE]
[TABLE]
[TABLE]
satisfy the relations . By using this and also by using the usual formula of a differential in tensor products we easily obtain that all maps defined by satisfy the relations .
By using Theorem 3.4 we easily obtain the following assertion.
Corollary 3.1. The tensor functor sends homotopy equivalent -algebras into homotopy equivalent differential modules with -simplicial faces.
The reference list from the paper itself. Each links out to its DOI / PubMed record.
- 1[1] S. V. Lapin, Homotopy simplicial faces and the homology of realizations of simplicial topological spaces (in Russian), Mat. Zametki 94 (5), 661–681 (2013); translation in Math. Notes 94 (5–6), 619–635 (2013).
- 2[2] S. V. Lapin, Homotopy properties of differential Lie modules over curved coalgebras and Koszul duality (in Russian), Mat. Zametki 94 (3), 354–372 (2013); translation in Math. Notes 94 (3–4), 335–350 (2013).
- 3[3] S. V. Lapin, Differential Lie modules over curved colored coalgebras and ∞ \infty -simplicial modules (in Russian), Mat. Zametki 96 (5), 709–731 (2014); translation in Math. Notes 96 (5–6), 698–715 (2014).
- 4[4] S. V. Lapin, Chain realization of differential modules with ∞ \infty -simplicial faces and the B 𝐵 B -construction over \A \A \A -algebras (in Russian), Mat. Zametki 98 (1), 101–124 (2015); translation in Math. Notes 98 (1–2), 111–129 (2015).
- 5[5] S. V. Lapin, Homotopy properties of differential modules with simplicial F ∞ subscript 𝐹 F_{\infty} -faces and D ∞ subscript 𝐷 D_{\infty} -differential modules, Georgian Math. J. 2015; 22(4):543-562.
- 6[6] S. V. Lapin, Homotopy properties of ∞ \infty -simplicial coalgebras and homotopy unital supplemented \A \A \A -algebras (in Russian), Mat. Zametki 99 (1), 55–77 (2016); translation in Math. Notes 99 (1–2), 63–81 (2016).
- 7[7] S. V. Lapin, Differential modules with ∞ \infty -simplicial faces and A ∞ subscript 𝐴 A_{\infty} -algebras, ar Xiv:1809.01853 v 1 [math.AT] 6 Sep 2018, p. 1-26.
- 8[8] S. V. Lapin, Differential perturbations and D ∞ subscript 𝐷 D_{\infty} -differential modules (in Russian), Mat. Sb. 192:11 (2001), 55–76; translation in Sb. Math. 192:11 (2001), 1639–1659.