Browsing by Subject "Algebras"

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    Characterization of coextensive varieties of universal algebras
    (2025) Broodryk, David Neal; Janelidze, George; Janelidze-Gray, Tamar
    A coextensive category can be defined as a category C with finite products such that for each pair X, Y of objects in C, the canonical functor × : X/C × Y /C / / (X × Y )/C is an equivalence. In this thesis we give a syntactic characterization of coextensive varieties of universal algebras. We first show that any such variety must have what we call a diagonalizing term. The existence of such a term is a Mal'tsev condition which is interesting in its own right, and we show that it is sufficient to prove many useful subconditions of coextensivity. We also introduce the notion of a category with upward closed subproducts as a categorical generalization of varieties with diagonalizing terms, which we study in the more general context of Barr-exact categories.