# On the twisted tensor product of small dg categories

## TL;DR

This paper introduces a new twisted tensor product for small dg categories, establishes its adjunction properties, and compares it to the standard tensor product in the homotopy category, providing new insights into dg category theory.

## Contribution

It defines the non-symmetric twisted tensor product of small dg categories and proves its adjunction with the dg functor category, relating it to the standard tensor product in the homotopy category.

## Key findings

- The twisted tensor product is left adjoint to the dg functor category.
- For cofibrant categories, the twisted tensor product coincides with the standard tensor product in the homotopy category.
- The construction generalizes the classical tensor product within the dg category framework.

## Abstract

Given two small dg categories $C,D$, defined over a field, we introduce their (non-symmetric) twisted tensor product $C\overset{\sim}{\otimes} D$. We show that $-\overset{\sim}{\otimes} D$ is left adjoint to the functor $Coh(D,-)$, where $Coh(D,E)$ is the dg category of dg functors $D\to E$ and their coherent natural transformations. This adjunction holds in the category of small dg categories (not in the homotopy category of dg categories $\mathrm{Hot}$). We show that for $C,D$ cofibrant, the adjunction descends to the corresponding adjunction in the homotopy category. Then comparison with a result of To\"{e}n shows that, for $C,D$ cofibtant, $C\overset{\sim}{\otimes} D$ is isomorphic to $C\otimes D$, as an object of the homotopy category $\mathrm{Hot}$.