• DOI:10.1090/S0002-9947-2011-05107-X
• Corpus ID: 11125524

Convenient Categories of Smooth Spaces

@article{Baez2008ConvenientCO,
  title={Convenient Categories of Smooth Spaces},
  author={John C. Baez and Alexander E. Hoffnung},
  journal={arXiv: Differential Geometry},
  year={2008}
}

• J. Baez, Alexander E. Hoffnung
• Published 10 July 2008
• Mathematics
• arXiv: Differential Geometry

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.