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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

   

 

Convenient categories of smooth spaces


Authors: John C. Baez and Alexander E. Hoffnung
Journal: Trans. Amer. Math. Soc. 363 (2011), 5789-5825
MSC (2000): Primary 58A40; Secondary 18F10, 18F20
Published electronically: June 6, 2011
MathSciNet review: 2817410
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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.


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Additional Information

John C. Baez
Affiliation: Department of Mathematics, University of California, Riverside, California 92521
Email: baez@math.ucr.edu

Alexander E. Hoffnung
Affiliation: Department of Mathematics and Statistics, University of Ottawa, 585 King Edward, Ottawa, Ontario, Canada K1N 6N5
Email: hoffnung@uottawa.ca

DOI: http://dx.doi.org/10.1090/S0002-9947-2011-05107-X
PII: S 0002-9947(2011)05107-X
Received by editor(s): September 13, 2008
Received by editor(s) in revised form: October 13, 2009
Published electronically: June 6, 2011
Article copyright: © Copyright 2011 John C. Baez and Alexander E. Hoffnung