Convenient categories of smooth spaces. (English) Zbl 1237.58006
Authors’ 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.
Reviewer: Eugen Pascu (Montréal)
MSC:
| 58A40 | Differential spaces |
| 18F10 | Grothendieck topologies and Grothendieck topoi |
| 18F20 | Presheaves and sheaves, stacks, descent conditions (category-theoretic aspects) |
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