So, what is a derived functor?

TL;DR

This paper redefines derived functors within infinity categories using correspondences, clarifying the transition from adjoint pairs to derived adjoint pairs and emphasizing the naturality of this process.

Contribution

It introduces a correspondence-based framework for derived functors in infinity categories, simplifying the understanding of derived adjoint pairs and their canonicity.

Findings

Comparison with Deligne's definition (SGA4) included

Diagrams of derived functors discussed in detail

Kan extensions described via correspondences

Abstract

In the context of infinity categories, we rethink the notion of derived functor in terms of correspondences. This is especially convenient for the description of a passage from an adjoint pair (F,G) of functors to a derived adjoint pair (LF,RG). In particular, canonicity of this passage becomes obvious. 2nd version: added comparison to Deligne's definition (SGA4) and a discussion of diagrams of derived functors. Introduction rewritten and references added. 3rd version: description of Kan extensions in terms of correspondences more detailed. 4th version: the final version accepted to HHA.