When is the diagonal functor Frobenius?

Abstract

Given a complete, cocomplete category $\mathcal{C}$, we investigate the problem of describing those small categories $I$ such that the diagonal functor $\Delta :\mathcal{C}\to \mathrm{Functors}(I,\mathcal{C})$ 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 $\mathcal{C}=\mathbf{Set}$ or ${}_R\mathcal{M}$, the category of left modules over a ring $R$.

18 pages; changed definition 2.1+minor modifications; submitted

References in corpus (1)

• When is the Product isomorphic to the Coproduct