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

The category of internal coalgebras in a cocomplete category with respect to a variety is equivalent to the category of left adjoint functors from to . This can be seen best when considering such coalgebras as finite coproduct preserving functors from o p , the dual of the Lawvere theory of , into : coalgebras are restrictions of left adjoints and any such left adjoint is the left Kan extension of a coalgebra along the embedding of o p into A l g . Since S Mod -coalgebras in the variety R Mod for rings R and S are nothing but left S -, right R -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 is locally presentable and comonadic over 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 is a commutative variety, are coreflectors from the category C o a l g (,) into . [ABSTRACT FROM AUTHOR]