Behavior of Quillen (co)homology with respect to adjunctions

TL;DR

This paper investigates how Quillen (co)homology behaves under adjunctions between categories, providing conditions for comparison diagrams and clarifying categorical assumptions involved.

Contribution

It establishes necessary and sufficient conditions for comparison diagrams of Quillen (co)homology under adjunctions and clarifies the categorical framework needed.

Findings

Identifies conditions for comparison diagrams to arise

Provides explicit comparison maps between Quillen (co)homologies

Clarifies categorical assumptions for Quillen (co)homology

Abstract

This paper aims to answer the following question: Given an adjunction between two categories, how is Quillen (co)homology in one category related to that in the other? We identify the induced comparison diagram, giving necessary and sufficient conditions for it to arise, and describe the various comparison maps. Examples are given. Along the way, we clarify some categorical assumptions underlying Quillen (co)homology: cocomplete categories with a set of small projective generators provide a convenient setup.