Convenient categories of smooth spaces

Baez, John; Hoffnung, Alexander

doi:10.1090/S0002-9947-2011-05107-X

Convenient categories of smooth spaces
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by John C. Baez and Alexander E. Hoffnung PDF

Trans. Amer. Math. Soc. 363 (2011), 5789-5825

Abstract:

A ‘Chen space’ is a set $X$ equipped with a collection of ‘plots’, i.e., maps from convex sets to $X$, satisfying three simple axioms. While an individual Chen space can be much worse than a smooth manifold, the category of all Chen spaces is much better behaved than the category of smooth manifolds. For example, any subspace or quotient space of a Chen space is a Chen space, and the space of smooth maps between Chen spaces is again a Chen space. Souriau’s ‘diffeological spaces’ share these convenient properties. Here we give a unified treatment of both formalisms. Following ideas of Penon and Dubuc, we show that Chen spaces, diffeological spaces, and even simplicial complexes are examples of ‘concrete sheaves on a concrete site’. As a result, the categories of such spaces are locally Cartesian closed, with all limits, all colimits, and a weak subobject classifier. For the benefit of differential geometers, our treatment explains most of the category theory we use.

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