2020/03/18 by Laurent Poinsot, Poinsot, Laurent, Hans-E. Porst +1
Computer Science · Mathematics · #Advanced Topics in Algebra #Algebraic structures and combinatorial models #Category Theory (math.CT) #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Logic, programming, and type systems #Primary 18D35 #Secondary 08C5
paper · pdf · doi:10.48550/arxiv.2003.08113
openalex publication_date 2020/03/18 · openalex created_date 2022/07/25 · openalex updated_date 2026/07/28
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 TVop, the dual of the Lawvere theory 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 TVop into AlgT. Since SMod-coalgebras in the variety RMod 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.