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.
Categories: math.AG math.CT