On representation categories of $A_\infty$-algebras and $A_\infty$-coalgebras
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| Format: | Preprint |
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2021
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| _version_ | 1866910443079991296 |
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| author | Banerjee, Abhishek Naolekar, Anita |
| author_facet | Banerjee, Abhishek Naolekar, Anita |
| contents | In this paper, we use the language of monads, comonads and Eilenberg-Moore categories to describe a categorical framework for $A_\infty$-algebras and $A_\infty$-coalgebras, as well as $A_\infty$-modules and $A_\infty$-comodules over them respectively. The resulting formalism leads us to investigate relations between representation categories of $A_\infty$-algebras and $A_\infty$-coalgebras. In particular, we relate $A_\infty$-comodules and $A_\infty$-modules by considering a rational pairing between an $A_\infty$-coalgebra $C$ and an $A_\infty$-algebra $A$. The categorical framework also motivates us to introduce $A_\infty$-contramodules over an $A_\infty$-coalgebra $C$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2112_02707 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | On representation categories of $A_\infty$-algebras and $A_\infty$-coalgebras Banerjee, Abhishek Naolekar, Anita Category Theory Algebraic Topology 18C15, 18C20, 18G70 In this paper, we use the language of monads, comonads and Eilenberg-Moore categories to describe a categorical framework for $A_\infty$-algebras and $A_\infty$-coalgebras, as well as $A_\infty$-modules and $A_\infty$-comodules over them respectively. The resulting formalism leads us to investigate relations between representation categories of $A_\infty$-algebras and $A_\infty$-coalgebras. In particular, we relate $A_\infty$-comodules and $A_\infty$-modules by considering a rational pairing between an $A_\infty$-coalgebra $C$ and an $A_\infty$-algebra $A$. The categorical framework also motivates us to introduce $A_\infty$-contramodules over an $A_\infty$-coalgebra $C$. |
| title | On representation categories of $A_\infty$-algebras and $A_\infty$-coalgebras |
| topic | Category Theory Algebraic Topology 18C15, 18C20, 18G70 |
| url | https://arxiv.org/abs/2112.02707 |