Università Cattolica del Sacro Cuore

On the Convenient Category of Diffeological Spaces

Seminario  30 gennaio 2018
Aula: Aula 1 -  Ore: 10.30
Via Musei 41, Brescia

Prof. Mauro SPERA
Università Cattolica del Sacro Cuore

Antonio Michele MITI
Università Cattolica del Sacro Cuore

Diffeological spaces are a class of geometric structures that generalize the notion of smooth manifolds, in this sense, they provide an instance of a "generalized space".
They consist of a set X equipped with a collection of "plots" - maps from open Euclidean subsets to X - satisfying three simple axioms.
While an individual diffeological space can be much worse than a smooth manifold, the category DiffSpaces of all diffeological spaces enjoys many desirable properties not possessed by the category of smooth manifolds.
The aim of this talk is to give a quick review of the notions of diffeological space, concrete site, and concrete sheaf.
We will show that DiffSpaces is indeed a category of "concrete sheaves on a concrete site" - also known as "generalized spaces" - and we will employ the rest of our time to exhibit the properties owned by this class of categories.
The upshot is that any category of concrete sheaves on a concrete site - and thus the category of diffeological spaces - turns out to be a quasitopos with all limits and colimits.

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