By a closure space we mean a set equipped with a closed under intersections set of its subsets. The purpose of this talk is to briefly recall our recent results on descent theory of closure spaces, and consider their E-versions, where E consists either of closed or of open maps, comparing them with their topological counterparts. In particular, we explain why, in contrast to the topological case, closed maps are pullback stable while non-surjective open ones are not. This simple observation is crucial in proving that 'effective global descent implies effective E-descent'.
This is joint work with G. Janelidze (University of Cape Town).
[1] M. M. Clementino and G. Janelidze, Another note on effective descent morphisms of topological spaces and relational algebras, Topology and its Applications 273, 2020, 106961, 8 pp. [2] G. Janelidze and M. Sobral, Finite preorders and topological descent I, Journal of Pure and Applied Algebra 175(1-3), 2002, 187-205. [3] G. Janelidze and M. Sobral, Strict monadic topology I: First separation axioms and reflections. Topology and its Applications 273 (2020), 106963, 10 pp. [4] G. Janelidze and M. Sobral, Strict monadic topology II: Descent for closure spaces. CMUC Preprint (2023), 23-31. [5] G. Janelidze, M. Sobral, and W. Tholen, Beyond Barr exactness: effective descent morphisms, Categorical Foundations: Special Topics in Order, Topology, Algebra, and Sheaf Theory, Cambridge University Press, 2004, 359-405. [6] G. Janelidze and W. Tholen, Facets of Descent I, Applied Categorical Structures 2, 1994, 245-281. [7] Gr. Mirhosseinkhani, On some classes of quotient maps in closure spaces, International Mathematical Forum 6 (2011), no. 21-24, 1155-1161. [8] M. Sobral and W. Tholen, Effective descent morphisms and effective equivalence relations, Category theory 1991 (Montreal, PQ, 1991), 421-433, CMS Conference Proceedings 13, American Mathematical Society, Providence, RI, 1992.
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