What do abelian categories form?

TL;DR

This paper provides an overview of constructing an abelian category of continuous functors between finitely presentable abelian categories, revealing connections to Hochschild and Mac Lane Cohomology.

Contribution

It introduces an explicit, absolute construction of an abelian category of functors with enhanced 2-categorical properties, linking derived categories and cohomology theories.

Findings

Constructs an abelian category of continuous functors with good 2-categorical behavior

Provides an explicit model for the stable category of stable functors between derived categories

Connects the construction to Hochschild and Mac Lane Cohomology

Abstract

This is mostly an overview. Given finitely presentable abelian categories $A$ and $B$, we sketch the construction of an abelian category of continuous functors from $A$ to $B$ that has nice $2$-categorical behaviour and gives an explicit model for the stable category of stable functors between the derived categories of $A$ and $B$. The construction is absolute, thus allows one to recover not only Hochschild but also Mac Lane Cohomology.