Tensor product of A${}_{\infty }$-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${}_{\infty }$Cat${}^u$/${}_{\approx }$ a closed symmetric monoidal category. Moreover, we define the derived tensor product making Ho(A${}_{\infty }$Cat), the homotopy category of the A${}_{\infty }$-categories, a closed symmetric monoidal category. We provide also an explicit description of the internal homs in terms of A${}_{\infty }$- functors.

Preliminary version, 31 pages. Comments are welcome