Tensor product of A$_{\infty}$-categories

TL;DR

This paper introduces a tensor product for A$_{4}$-categories and functors, establishing a symmetric monoidal structure that is compatible with homotopy, and provides explicit descriptions of internal homs.

Contribution

It defines the tensor product of A$_{4}$-categories and functors, proving the resulting category is symmetric monoidal up to homotopy and describing internal homs explicitly.

Findings

The tensor product makes A$_{4}$-categories symmetric monoidal (up to homotopy).

The homotopy category of A$_{4}$-categories is a closed symmetric monoidal category.

Explicit descriptions of internal homs in terms of A$_{4}$-functors are provided.

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.