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.
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[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.
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