Resolving by a free action linear category and applications to Hochschild-Mitchell (co)homology

TL;DR

This paper develops a method to resolve categories with group actions into free actions, enabling new insights into Hochschild-Mitchell (co)homology computations and their relation to skew categories and quotients.

Contribution

It introduces a resolution technique for categories with group actions, facilitating the analysis of Hochschild-Mitchell (co)homology via spectral sequences and decomposition formulas.

Findings

Categories can be resolved to have free group actions on objects.

Hochschild-Mitchell (co)homology invariants relate to skew categories and quotients.

Spectral sequences and decompositions provide tools for computing (co)homology in this setting.

Abstract

Let $G$ be a group acting on a small category $\mathcal C$ over a field $k$, that is $\mathcal C$ is a $G$-$k$-category. We first obtain that $\mathcal C$ is resolvable by a category which is $G$-$k$-equivalent to it, on which $G$ acts freely on objects. This resolvent category enables to show that if the coinvariants and the invariants functors are exact, then the coinvariants and invariants of the Hochschild-Mitchell (co)homology of $\mathcal C$ are isomorphic to the trivial component of the Hochschild-Mitchell (co)ho\-mo\-logy of the skew category $\mathcal C[G]$. Otherwise the corresponding spectral sequence can be settled. If the action of $G$ is free on objects, there is a canonical decomposition of the Hochschild-Mitchell (co)homology of the quotient category $\mathcal C/G$ along the conjugacy classes of $G$. This way we provide a general frame for monomorphisms which have been described previously in low degrees.