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 $\mathcal{ V}$.
Categories: math.CT