Internal Coalgebras in Cocomplete Categories: Generalizing the Eilenberg-Watts-Theorem

Abstract

The category of internal coalgebras in a cocomplete category $\mathcal{C}$ with respect to a variety $\mathcal{V}$ is equivalent to the category of left adjoint functors from $\mathcal{V}$ into $\mathcal{C}$. This can be seen best when considering such coalgebras as finite coproduct preserving functors from $\mathcal{T}_\mathcal{V}^\mathsf{op}$, the dual of the Lawvere theory of $\mathcal{V}$, into $\mathcal{C}$: coalgebras are restrictions of left adjoints and any such left adjoint is the left Kan extension of a coalgebra along the embedding of $\mathcal{T}_\mathcal{V}^\mathsf{op}$ into $\mathsf{Alg}\mathcal{T}$. Since ${_S\mathit{Mod}}$-coalgebras in the variety ${_R\mathit{Mod}}$ 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. 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 $\mathcal{C}$ is locally presentable and comonadic over $\mathcal{C}$ 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 $\mathcal{V}$ is a commutative variety, are coreflectors from the category $\mathsf{Coalg}(\mathcal{T},\mathcal{V})$ into $\mathcal{V}$.