# Quotients of unital $A_\infty$-categories

## Volodymyr Lyubashenko and Oleksandr Manzyuk

Assuming that $B$ is a full $A_\infty$-subcategory of a unital $A_\infty$-category $\cc$ we construct the quotient unital $A_\infty$-category $\cd=$`$\cc/\cb$'. It represents the $A_\infty^u$-2-functor $A \mapsto A_\infty^u(C,A)_{mod B}$, which associates with a given unital $A_\infty$-category $A$ the $A_\infty$-category of unital $A_\infty$-functors $C \to A$, whose restriction to $B$ is contractible. Namely, there is a unital $A_\infty$-functor $e: C \to D$ such that the composition $B \hookrightarrow C \to^e D$ is contractible, and for an arbitrary unital $A_\infty$-category $A$ the restriction $A_\infty$-functor $(e\boxtimes 1)M : A_\infty^u(D,A)\to A_\infty^u(C,A)_{mod B}$ is an equivalence.

Let $C_k$ be the differential graded category of differential graded $k$-modules. We prove that the Yoneda $A_\infty$-functor $Y: A \to A_\infty^u(A^{op}, C_k)$ is a full embedding for an arbitrary unital $A_\infty$-category $A$. In particular, such $A$ is $A_\infty$-equivalent to a differential graded category with the same set of objects.

Keywords: $A_\infty$-categories, $A_\infty$-categories, $A_\infty$-functors, $A_\infty$-transformations, 2-categories, 2-functors

2000 MSC: 18D05, 18D20, 18G55, 55U15

Theory and Applications of Categories, Vol. 20, 2008, No. 13, pp 405-495.

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