$k$-linear Morita theory

TL;DR

This paper establishes a comprehensive comparison between dg-categories and stable $f$-categories over a commutative ring $k$, using $f$-categorical methods to unify Morita theory.

Contribution

It reinterprets and extends Morita theory for dg-categories within the $f$-categorical framework, providing a new $f$-categorical perspective.

Findings

Proves the comparison between dg-categories and stable $f$-categories over $k$

Reinterprets Morita theory for dg-categories in the $f$-categorical setting

Sets the foundation for Morita theory in $k$-linear stable $f$-categories

Abstract

In this paper, we prove the standard comparison used by mathematicians between the idempotent complete pretriangulated dg-categories, over a unitary and commutative ring $k$, and the idempotent complete $k$-linear stable $\infty$-categories. Our approach is completely included in the $\infty$-categorical theory. To achieve the target we will reinterpret the Morita theory for dg-categories and we set the Morita theory for $k$-linear stable $\infty$-category.