Convenient Categories of Smooth Spaces

TL;DR

This paper unifies Chen spaces and diffeological spaces within a categorical framework, showing they form well-behaved, locally cartesian closed categories with desirable limits, colimits, and classifiers, enhancing their utility in differential geometry.

Contribution

It provides a unified categorical treatment of Chen and diffeological spaces, demonstrating their structure as concrete sheaves on a site with favorable categorical properties.

Findings

Categories are locally cartesian closed

All limits and colimits exist in these categories

They have a weak subobject classifier

Abstract

A "Chen space" is a set X equipped with a collection of "plots" - 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 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.