Talk Abstract: We introduce the language of locally small spaces and smops that gives a topological background for gluing together (usually infinite) families of definable sets in structures with topologies. It replaces the language of some Grothendieck sites that were used in o-minimal homotopy theory due to a concrete isomorphism of constructs. Some analogues of the o-minimal spectrum (or the real spectrum) are given by new variants of Stone duality for Kolmogorov locally small spaces that see those spaces as certain patch-dense subsets of spectral or up-spectral spaces. By these means, we rebuild the theory of locally definable and nice weakly definable spaces over structures with topologies.

Bio: Artur Piękosz is a Polish mathematician. He earned his doctorate from the Jagiellonian University and is employed by Cracow University of Technology.

His scientific interests include o-minimality and generalized topology.