A note on deriving unbounded functors of exact categories, with applications to Ind- and Pro- functors

TL;DR

This paper demonstrates how to derive unbounded functors between exact categories under mild conditions and explores applications to Ind- and Pro- categories, expanding the toolkit for working with complex categorical structures.

Contribution

It introduces conditions under which functors between exact categories can be derived at the unbounded complex level and applies these results to Ind- and Pro- categories.

Findings

Derived functors exist under mild conditions.

Applications to Ind- and Pro- categories are established.

Provides a framework for unbounded functor derivation.

Abstract

In this short note we show that under very mild conditions on a functor between exact categories $F:\mathcal{D}\rightarrow\mathcal{E}$ it is possible to derive $F$ at the level of unbounded complexes. We also give applications to deriving functors between $Pro$- and $Ind$- categories.