Tensor product of A∞-categories

Abstract

In this paper we define the tensor product of two $A_{\infty }$-categories and two $A_{\infty }$-functors. This tensor product makes the category of $A_{\infty }$-categories symmetric monoidal (up to homotopy), and the category $A_{\infty }{\mathrm{Cat}}^u/\approx$ a closed symmetric monoidal category. Moreover, we define the derived tensor product making $\text{Ho}\left(A_{\infty }\mathrm{Cat}\right)$, the homotopy category of the $A_{\infty }$-categories, a closed symmetric monoidal category. We also provide an explicit description of the internal homs in terms of $A_{\infty }$-functors.