Università Cattolica del Sacro Cuore

On the Convenient Category of Diffeological Spaces

 January 30 2018

Room: Aula 1 -  Hour: 10.30
Università Cattolica del Sacro Cuore, Via Musei 41, Brescia, Italy

Flier (pdf)

Prof. Mauro SPERA
Università Cattolica del Sacro Cuore, Italy

Antonio Michele MITI
Università Cattolica del Sacro Cuore, Italy

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.

In cooperation with: