Internal Coalgebras in Cocomplete Categories: Generalizing the Eilenberg-Watts Theorem
Résumé
The category of internal coalgebras in a cocomplete category C with respect to a variety V is equivalent to the category of left adjoint functors from V into C. This can be seen best when considering such coalgebras as finite coproduct preserving functors from the the dual of the Lawvere theory T_V of V, into C: coalgebras are restrictions of left adjoints and any such left adjoint is the left Kan extension of a coalgebra along the embedding of the dual of T_V into AlgT. 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. 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 C is locally presentable and comonadic over 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 V is a commutative variety, are coreflectors from the category Coalg(T, V) into V. MSC 2010: Primary 18D35, Secondary 08C5
Origine : Fichiers produits par l'(les) auteur(s)
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