When is the diagonal functor Frobenius?

TL;DR

This paper characterizes when the diagonal functor is Frobenius in complete, cocomplete categories, providing necessary and sufficient conditions in specific cases like Set and module categories.

Contribution

It offers a detailed analysis of Frobenius diagonal functors, including necessary conditions and explicit characterizations for Set and module categories.

Findings

Necessary conditions on index categories for Frobenius property

Characterization of Frobenius diagonal functors in Set

Characterization in module categories over rings

Abstract

Given a complete, cocomplete category $\mathcal C$, we investigate the problem of describing those small categories $I$ such that the diagonal functor $\Delta:\mathcal C\to {\rm Functors}(I,\mathcal C)$ is a Frobenius functor. This condition can be rephrased by saying that the limits and the colimits of functors $I\to\mathcal C$ are naturally isomorphic. We find necessary conditions on $I$ for a certain class of categories $\mathcal C$, and, as an application, we give both necessary and sufficient conditions in the two special cases $\mathcal C={\bf Set}$ or $_R\mathcal M$, the category of left modules over a ring $R$.