Internal coalgebras in cocomplete categories: Generalizing the Eilenberg–Watts theorem

Abstract

The category of internal coalgebras in a cocomplete category [Formula: see text] with respect to a variety [Formula: see text] is equivalent to the category of left adjoint functors from [Formula: see text] to [Formula: see text]. This can be seen best when considering such coalgebras as finite coproduct preserving functors from [Formula: see text], the dual of the Lawvere theory of [Formula: see text], into [Formula: see text]: coalgebras are restrictions of left adjoints and any such left adjoint is the left Kan extension of a coalgebra along the embedding of [Formula: see text] into [Formula: see text]. Since [Formula: see text]-coalgebras in the variety [Formula: see text] for rings [Formula: see text] and [Formula: see text] are nothing but left [Formula: see text]-, right [Formula: see text]-bimodules, the equivalence above generalizes the Eilenberg–Watts theorem and all its previous generalizations. By generalizing and strengthening Bergman’s completeness result for categories of internal coalgebras in varieties, we also prove that the category of coalgebras in a locally presentable category [Formula: see text] is locally presentable and comonadic over [Formula: see text] and, hence, complete in particular. We show, moreover, that Freyd’s canonical constructions of internal coalgebras in a variety define left adjoint functors. Special instances of the respective right adjoints appear in various algebraic contexts and, in the case where [Formula: see text] is a commutative variety, are coreflectors from the category [Formula: see text] into [Formula: see text].