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.
Highlights
Functors having a left adjoint which is a right adjoint were investigated by Morita in [10], where it is shown that given a ring morphism R → S, the restriction of scalars functor has this property if and only if R → S is a Frobenius extension: S is finitely generated and projective in RM, and S ∼= RHom(S, R) as (S, R)-bimodules
In this paper the point of view is the following one: we fix a complete, cocomplete category C, and seek to characterize those small categories I for which the functors CI → C sending a functor in CI to its limit and colimit are naturally isomorphic
The structure of the paper is as follows: In Section 1 we introduce some conventions and prove Lemma 1.4, which allows us later on to break up the main problem into the two cases when I is discrete or connected
Summary
Functors having a left adjoint which is a right adjoint were investigated by Morita in [10], where it is shown that given a ring morphism R → S, the restriction of scalars functor has this property if and only if R → S is a Frobenius extension: S is finitely generated and projective in RM, and S ∼= RHom(S, R) as (S, R)-bimodules. In this paper the point of view is the following one: we fix a complete, cocomplete category C, and seek to characterize those small categories I for which the functors CI → C sending a functor in CI to its limit and colimit are naturally isomorphic Diagonal functor, Frobenius functor, complete, cocomplete, limit, colimit This question is investigated in [6], for discrete small categories I (i.e. sets), and categories C enriched over the category of commutative monoids (referred to as AMon categories), and having a zero object. Since both Set and RM are admissible in the sense of Section 2, the results proven there can be applied to the two particular cases.
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