  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.
