Convenient Categories of Smooth Spaces

By John C. Baez, Alexander E. Hoffnung. 2008

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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.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/d1fe261a8873298fe8a249be694f73d98edc56cb

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Citations in the corpus

Derived smooth manifolds (2008, background): “The category of Chen spaces is roughly this topos, the difference being that points are given more importance in Chen spaces than in Shv(Conv), in a sense known as concreteness (see [3] for a precise account).”
A convenient category for higher-order probability theory (2017, background): “[3], [10]); but that is not to say that the definition of quasi-Borel space (Def.”
The D-topology for diffeological spaces (2013, result): “ by the inclusion map {A→ X} is called the subset diﬀeology. More generally, we have the following well-known result: Theorem 2.5. The category Diﬀ is both complete and cocomplete. This is proved in [BH], but can be found implicitly in earlier work. We give a brief sketch here. The forgetful functor”
Central extensions of smooth 2–groups and a finite-dimensional string 2–group (2009, methodology): “overcome by embedding abelian Lie groups into a lager category of “smooth” abelian group objects. For example abelian group objects in one of the “convenient categories of smooth spaces” discussed in [BH08] provide such an enhancement. Alternatively one could use the sheaf cohomology of the resultin”