3 years ago

Internal Coalgebras in Cocomplete Categories: Generalizing the Eilenberg-Watts-Theorem. (arXiv:2003.08113v1 [math.CT])

Laurent Poinsot (LIPN), Hans-E Porst
The category of internal coalgebras in a cocomplete category with respect to a variety is equivalent to the category of left adjoint functors from into . This can be seen best when considering such coalgebras as finite coproduct preserving functors from , 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 into . Since -coalgebras in the variety for rings and are nothing but left -, right -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 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 into .

Publisher URL: http://arxiv.org/abs/2003.08113

DOI: arXiv:2003.08113v1

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