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Abstract
Given a complete, cocomplete category 𝒞, we investigate the problem of describing those small categories I such that the diagonal functor Δ: 𝒞 → Functors(I, 𝒞) is a Frobenius functor. This condition can be rephrased by saying that the limits and the colimits of functors I → 𝒞 are naturally isomorphic. We find necessary conditions on I for a certain class of categories 𝒞, and, as an application, we give both necessary and sufficient conditions in the two special cases 𝒞 =Set or R ℳ, the category of left modules over a ring R.
ACKNOWLEDGMENTS
The author wishes to thank Professor Gigel Militaru, who posed the problem and suggested this line of inquiry, for the insight gained through countless discussions on the topic, as well as the referee for valuable suggestions on how to revise an initial version of this article.
Notes
Communicated by T. Albu.