A-infinity algebras, modules and functor categories (original) (raw)
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European Journal of Pure and Applied Mathematics, 2021
In this paper we study an elementary use of E-infinity modules and E-infinity algebras as together they have a use in terms of describing triangulated categories. Also, we show an interpretation of E-infinity algebras where the modules are fibrant objects within the categories of differential graded co-algebras and co-modules.
On representation categories of AinftyA_\inftyAinfty-algebras and AinftyA_\inftyAinfty-coalgebras
2021
In this paper, we use the language of monads, comonads and Eilenberg-Moore categories to describe a categorical framework for A∞-algebras and A∞-coalgebras, as well as A∞-modules and A∞-comodules over them respectively. The resulting formalism leads us to investigate relations between representation categories of A∞algebras and A∞-coalgebras. In particular, we relate A∞-comodules and A∞-modules by considering a rational pairing between an A∞-coalgebra C and an A∞-algebra A. The categorical framework also motivates us to introduce A∞-contramodules over an A∞-coalgebra C. MSC(2020) Subject Classification: 18C15, 18C20, 18G70
Curved A-infinity-categories : Adjunction and Homotopy
We develop a theory of curved A ∞-categories around equivalences of their module categories. This allows for a uniform treatment of curved and uncurved A ∞-categories which generalizes the classical theory of uncurved A ∞-algebras. Furthermore, the theory is sufficiently general to treat both Fukaya categories and categories of matrix factorizations, as well as to provide a context in which unitification and categorification of pre-categories can be carried out. Our theory is built around two functors: the adjoint algebra functor U e and the functor Q *. The bulk of the paper is dedicated to proving crucial adjunction and homotopy theorems about these functors. In addition, we explore the non-vanishing of the module categories and give a precise statement and proof the folk result known as "Positselski-Kontsevich vanishing".
An Algebraic Definition of (∞, N)-Categories
2015
In this paper we define a sequence of monads T(∞,n)(n ∈ N) on the category ∞-Gr of ∞-graphs. We conjecture that algebras for T(∞,0), which are defined in a purely algebraic setting, are models of∞-groupoids. More generally, we conjecture that T(∞,n)-algebras are models for (∞, n)-categories. We prove that our (∞, 0)-categories are bigroupoids when truncated at level 2. Introduction The notion of weak (∞, n)-category can be made precise in many ways depending on our approach to higher categories. Intuitively this is a weak∞-category such that all its cells of dimension greater than n are equivalences. Models of weak (∞, 1)-categories (case n = 1) are diverse: for example there are the quasicategories studied by Joyal and Tierney (see [24]), but also there are other models which have been studied like the Segal categories, the complete Segal spaces, the simplicial categories, the topological categories, the relative categories, and there are known to be equivalent (a survey of models ...
A∞-Algebras and the Cyclic Bar Complex
Illinois J. Math, 1990
This paper arose from our use of Chen's theory of iterated integrals as a tool in the study of the complex of S 1-equivariant differential forms on the free loop space LX of a manifold X (see [2]). In trying to understand the behaviour of the iterated integral map with respect to products, we were led to a natural product on the space of S 1-equivariant differential forms Ω(Y)[u] of a manifold Y with circle action, where u is a variable of degree 2. This product is not associative but is homotopy associative in a precise way; indeed there is whole infinite family of "higher homotopies". It turns out that this product structure is an example of Stasheff's A ∞-algebras, which are a generalization of differential graded algebras (dgas). Using the iterated integral map, it is a straightforward matter to translate this product structure on the space of S 1-equivariant differential forms on LX into formulas on the cyclic bar complex of Ω(X). Our main goal in this paper is to show that in general, the cyclic bar complex of a commutative dga A has a natural A ∞-structure and we give explicit formulas for this structure. In particular, this shows that the cyclic homology of A has a natural associative product, but it is a much stronger result, since it holds at the chain level. Thus, it considerably strengthens the results of Hood and Jones [3]. We also show how to construct the cyclic bar complex of an A ∞-algebra, and in particular define its cyclic homology. As hinted at in [2], this construction may have applications to the problem of giving models for the S 1 × S 1-equivariant cohomology of double loop spaces LL(X) of a manifold and, since the space of equivariant differential forms on a smooth S 1-manifold Y is an A ∞-algebra, to the problem of finding models for the space of S 1 × S 1-equivariant differential forms on LY. Although the methods that we use were developed independently, they bear a strong resemblance with those of Quillen [6]. Finally, we discuss in our general context the Chen normalization of the cyclic bar complex of an A ∞-algebra. This is a quotient of the cyclic bar complex by a complex of degenerate chains which is acyclic if A is connected, and which was shown by Chen to coincide with the kernel of the iterated integral map in the case A = Ω(X). This normalization is an important tool, since it allows us to remove a large contractible sub-complex of the cyclic bar complex. The first two sections of this paper are devoted to generalities concerning coalgebras and A ∞algebras; a good reference for further background on coalgebras is the book of McCleary [5]. The cyclic bar complex of an A ∞-algebra is constructed in Section 3, the A ∞-structure on the cyclic bar complex of a commutative dga in Section 4, and we discuss Chen normalization in Section 5. All our algebra will be carried out over a fixed coefficient ring K; in fact nothing will be lost by thinking of the case where K is the integers Z. In particular, all tensor products are taken over K unless explicitly stated otherwise. We will make use of the sign-convention in the category of Z 2-graded K-modules, which may be phrased as follows: the canonical map S 21 from V 1 ⊗ V 2 to V 2 ⊗ V 1 is defined by S 21 (v 1 ⊗ v 2) = (−1) |v1||v2| v 2 ⊗ v 1. *In the preprint of [2], the maps m andm are exchanged, for which we beg the reader's forgiveness.
Differential Calculus of Hochschild Pairs for Infinity-Categories
Symmetry, Integrability and Geometry: Methods and Applications
In this paper, we provide a conceptual new construction of the algebraic structure on the pair of the Hochschild cohomology spectrum (cochain complex) and Hochschild homology spectrum, which is analogous to the structure of calculus on a manifold. This algebraic structure is encoded by a two-colored operad introduced by Kontsevich and Soibelman. We prove that for a stable idempotent-complete infinity-category, the pair of its Hochschild cohomology and homology spectra naturally admits the structure of algebra over the operad. Moreover, we prove a generalization to the equivariant context.