Convenient categories of smooth spaces
HTML articles powered by AMS MathViewer
- by John C. Baez and Alexander E. Hoffnung PDF
- Trans. Amer. Math. Soc. 363 (2011), 5789-5825
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.References
- Philippe Antoine, Étude élémentaire des catégories d’ensembles structurés, Bull. Soc. Math. Belg. 18 (1966), 142–164 (French). MR 200321
- J. Baez, Quantum Gravity Seminar notes, U. C. Riverside, $\langle$http://math.ucr.edu/home/ baez/qg-spring2007/index.html#quantization$\rangle$, Spring 2007.
- J. Baez and U. Schreiber, Higher gauge theory II: 2-connections. Available at arXiv:hep-th/0412325.
- John C. Baez and Urs Schreiber, Higher gauge theory, Categories in algebra, geometry and mathematical physics, Contemp. Math., vol. 431, Amer. Math. Soc., Providence, RI, 2007, pp. 7–30. MR 2342821, DOI 10.1090/conm/431/08264
- Tobias Keith Bartels, Higher gauge theory: 2-bundles, ProQuest LLC, Ann Arbor, MI, 2006. Thesis (Ph.D.)–University of California, Riverside. MR 2709030
- R. Brown, Ten topologies for $X\,\times \,Y$, Quart. J. Math. Oxford Ser. (2) 14 (1963), 303–319. MR 159299, DOI 10.1093/qmath/14.1.303
- Kuo-tsai Chen, Iterated integrals of differential forms and loop space homology, Ann. of Math. (2) 97 (1973), 217–246. MR 380859, DOI 10.2307/1970846
- Kuo Tsai Chen, Iterated integrals, fundamental groups and covering spaces, Trans. Amer. Math. Soc. 206 (1975), 83–98. MR 377960, DOI 10.1090/S0002-9947-1975-0377960-0
- Kuo Tsai Chen, Iterated path integrals, Bull. Amer. Math. Soc. 83 (1977), no. 5, 831–879. MR 454968, DOI 10.1090/S0002-9904-1977-14320-6
- Kuo Tsai Chen, On differentiable spaces, Categories in continuum physics (Buffalo, N.Y., 1982) Lecture Notes in Math., vol. 1174, Springer, Berlin, 1986, pp. 38–42. MR 842915, DOI 10.1007/BFb0076932
- Eduardo J. Dubuc, Concrete quasitopoi, Applications of sheaves (Proc. Res. Sympos. Appl. Sheaf Theory to Logic, Algebra and Anal., Univ. Durham, Durham, 1977) Lecture Notes in Math., vol. 753, Springer, Berlin, 1979, pp. 239–254. MR 555548
- E. J. Dubuc and L. Español, Topological functors as familiarly-fibrations. Available at arXiv:math/0611701.
- E. J. Dubuc and L. Español, Quasitopoi over a base category. Available at arXiv: math/0612727.
- Charles Ehresmann, Catégories topologiques et catégories différentiables, Colloque Géom. Diff. Globale (Bruxelles, 1958) Centre Belge Rech. Math., Louvain, 1959, pp. 137–150 (French). MR 0116360
- Alfred Frölicher, Smooth structures, Category theory (Gummersbach, 1981) Lecture Notes in Math., vol. 962, Springer, Berlin-New York, 1982, pp. 69–81. MR 682945
- Robert Goldblatt, Topoi, 2nd ed., Studies in Logic and the Foundations of Mathematics, vol. 98, North-Holland Publishing Co., Amsterdam, 1984. The categorial analysis of logic. MR 766560
- Marco Grandis, Finite sets and symmetric simplicial sets, Theory Appl. Categ. 8 (2001), 244–252. MR 1825431
- P. Iglesias-Zemmour, Diffeology. Draft available at $\langle$http://math.huji.ac.il/$\sim$piz/Site/ The%20Book/The%20Book.html$\rangle$.
- Klaus Jänich, On the classification of $O(n)$-manifolds, Math. Ann. 176 (1968), 53–76. MR 226674, DOI 10.1007/BF02052956
- Peter T. Johnstone, Sketches of an elephant: a topos theory compendium. Vol. 2, Oxford Logic Guides, vol. 44, The Clarendon Press, Oxford University Press, Oxford, 2002. MR 2063092
- Anders Kock, Synthetic differential geometry, 2nd ed., London Mathematical Society Lecture Note Series, vol. 333, Cambridge University Press, Cambridge, 2006. MR 2244115, DOI 10.1017/CBO9780511550812
- M. Kreck, Differential Algebraic Topology, draft available at $\langle$http://www.him.uni-bonn.de/kreck-stratifolds$\rangle$.
- Andreas Kriegl, A Cartesian closed extension of the category of smooth Banach manifolds, Categorical topology (Toledo, Ohio, 1983) Sigma Ser. Pure Math., vol. 5, Heldermann, Berlin, 1984, pp. 323–336. MR 785022
- A. Kriegl, Remarks on germs in infinite dimensions, Acta Math. Univ. Comenian. (N.S.) 66 (1997), no. 1, 117–134. MR 1474553
- Andreas Kriegl and Peter W. Michor, The convenient setting of global analysis, Mathematical Surveys and Monographs, vol. 53, American Mathematical Society, Providence, RI, 1997. MR 1471480, DOI 10.1090/surv/053
- M. Laubinger, A Lie algebra for Frölicher groups, available at arXiv:0906.4486.
- Gerd Laures, On cobordism of manifolds with corners, Trans. Amer. Math. Soc. 352 (2000), no. 12, 5667–5688. MR 1781277, DOI 10.1090/S0002-9947-00-02676-3
- Kirill C. H. Mackenzie, General theory of Lie groupoids and Lie algebroids, London Mathematical Society Lecture Note Series, vol. 213, Cambridge University Press, Cambridge, 2005. MR 2157566, DOI 10.1017/CBO9781107325883
- Saunders Mac Lane, Categories for the working mathematician, 2nd ed., Graduate Texts in Mathematics, vol. 5, Springer-Verlag, New York, 1998. MR 1712872
- J. P. May, A concise course in algebraic topology, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 1999. MR 1702278
- Peter Michor, A convenient setting for differential geometry and global analysis, Cahiers Topologie Géom. Différentielle 25 (1984), no. 1, 63–109. MR 764972
- Mark A. Mostow, The differentiable space structures of Milnor classifying spaces, simplicial complexes, and geometric realizations, J. Differential Geometry 14 (1979), no. 2, 255–293. MR 587553
- Saunders Mac Lane and Ieke Moerdijk, Sheaves in geometry and logic, Universitext, Springer-Verlag, New York, 1994. A first introduction to topos theory; Corrected reprint of the 1992 edition. MR 1300636
- Jacques Penon, Quasi-topos, C. R. Acad. Sci. Paris Sér. A-B 276 (1973), A237–A240 (French). MR 332921
- Jacques Penon, Sur les quasi-topos, Cahiers Topologie Géom. Différentielle 18 (1977), no. 2, 181–218 (French). MR 480693
- Urs Schreiber and Konrad Waldorf, Parallel transport and functors, J. Homotopy Relat. Struct. 4 (2009), no. 1, 187–244. MR 2520993
- Roman Sikorski, Differential modules, Colloq. Math. 24 (1971/72), 45–79. MR 482794, DOI 10.4064/cm-24-1-45-79
- J. Wolfgang Smith, The de Rham theorem for general spaces, Tohoku Math. J. (2) 18 (1966), 115–137. MR 202154, DOI 10.2748/tmj/1178243443
- J.-M. Souriau, Groupes différentiels, Differential geometrical methods in mathematical physics (Proc. Conf., Aix-en-Provence/Salamanca, 1979) Lecture Notes in Math., vol. 836, Springer, Berlin-New York, 1980, pp. 91–128 (French). MR 607688
- A. Stacey, Comparative smootheology, available at arXiv:0802.2225.
- N. E. Steenrod, A convenient category of topological spaces, Michigan Math. J. 14 (1967), 133–152. MR 210075
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
- Received by editor(s): September 13, 2008
- Received by editor(s) in revised form: October 13, 2009
- Published electronically: June 6, 2011
- © Copyright 2011 John C. Baez and Alexander E. Hoffnung
- Journal: Trans. Amer. Math. Soc. 363 (2011), 5789-5825
- MSC (2000): Primary 58A40; Secondary 18F10, 18F20
- DOI: https://doi.org/10.1090/S0002-9947-2011-05107-X
- MathSciNet review: 2817410