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Description: |
Following the paper of M.M. Clementino and D. Hofmann [2], and having as starting point Barr's description of topological spaces as lax algebras for the ultrafilter monad [1], I am going to present the categories of lax algebras. They represent a suitable generalization of the relational algebras which allow to describe topological structures as well (it turns out that they have indeed a topological nature). The category of reflexive lax algebras has actually a "better behaviour" than the category of reflexive and transitive lax algebras; in fact, it is a quasi-topos [3]. In this talk I show that, contrarily to what happens in particular in the category of topological spaces, in categories of reflexive lax algebras regular epimorphisms are pullback-stable. Also some examples are given.
[1] M. Barr, Relational Algebras, Springer Lecture Notes in Math. 137 (1970), 39-55.
[2] M.M. Clementino and D. Hofmann, Topological features of lax algebras, Applied Categorical Structures 11 (2003), 267-286.
[3] M.M. Clementino, D. Hofmann and W. Tholen, Exponentiability in categories of lax algebras, Theory and Applications of Categories 11 (2003), 337-352.
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