Criteria for preserving the category cohomology for the inverse image

TL;DR

This paper explores conditions under which functors between small categories preserve various types of category cohomology when taking inverse images, providing criteria and counterexamples for invariance.

Contribution

It establishes necessary and sufficient conditions for preserving Baues-Wirsching, Hochschild-Mitchell, and Thomason cohomology under inverse image functors, and generalizes these cohomologies.

Findings

Counterexamples show left adjoint functors may not preserve certain cohomologies.

Provides criteria for invariance of cohomology under inverse image functors.

Generalizes cohomology theories for small categories.

Abstract

The article investigates the question of under what conditions a functor between small categories preserves cohomology groups when passing to the inverse image. For example, it is known that the left adjoint functor preserves the category cohomology with local coefficients or the category cohomology constructed as derived of the limit functor. We give counterexamples showing that the left adjoint functor may not preserve the Baues-Wirsching, Hochschild-Mitchell, and Thomason cohomology. To solve the arising problems, we propose necessary and sufficient conditions for the invariance of these types of cohomology under the transition to the inverse image of a functor between small categories. Moreover, we generalize these cohomology of small categories and find similar criteria for the obtained generalization.