Hostname: page-component-586b7cd67f-2plfb Total loading time: 0 Render date: 2024-11-20T16:25:28.988Z Has data issue: false hasContentIssue false
  • English
  • Français

Two-Categorical Bundles and their Classifying Spaces

Published online by Cambridge University Press:  23 February 2012

Nils A. Baas ,
Marcel Bökstedt  and
Tore August Kro
Nils A. Baas
Affiliation:
Department of Mathematical Sciences, NTNU, NO-7034 Trondheim, [email protected]
Marcel Bökstedt
Affiliation:
Department of Mathematical Sciences, University of Aarhus, Ny Munkegade, DK-8000 Aarhus C, [email protected]
Tore August Kro
Affiliation:
Østfold University College, NO-1757 Halden, [email protected]
Get access

Abstract

For a 2-category 2C we associate a notion of a principal 2C-bundle. For the 2-category of 2-vector spaces, in the sense of M.M. Kapranov and V.A. Voevodsky, this gives the 2-vector bundles of N.A. Baas, B.I. Dundas and J. Rognes. Our main result says that the geometric nerve of a good 2-category is a classifying space for the associated principal 2-bundles. In the process of proving this we develop powerful machinery which may be useful in further studies of 2-categorical topology. As a corollary we get a new proof of the classification of principal bundles. Another 2-category of 2-vector spaces has been proposed by J.C. Baez and A.S. Crans. A calculation using our main theorem shows that in this case the theory of principal 2-bundles splits, up to concordance, as two copies of ordinary vector bundle theory. When 2C is a cobordism type 2-category we get a new notion of cobordism-bundles which turns out to be classified by the Madsen–Weiss spaces.

Keywords

2-bundlesclassifying spacesgeometric nerve
Type
Research Article
Information
Journal of K-Theory , Volume 10 , Issue 2 , October 2012 , pp. 299 - 369
Copyright
Copyright © ISOPP 2012

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1.Ausoni, C., ‘On the algebraic K-theory of the complex K-theory spectrum’, Invent. Math. 180 (2010), no.3, 611–668.CrossRefGoogle Scholar
2.Baas, N. A., Dundas, B. I., Richter, B.Rognes, J., ‘Stable bundles over rig categories’, J. Topology 4 (2011) 623–640.Google Scholar
3.Baas, N. A., Dundas, B. I., Richter, B.Rognes, J., ‘Ring completion of rig categories’, preprint, 2010; arXiv:math/0706.0531v4.Google Scholar
4.Baas, N. A., Dundas, B. I.Rognes, J., ‘Two-vector bundles and forms of elliptic cohomology’, Topology, geometry and quantum field theory (ed Tillmann, U.), London Math. Soc. Lecture Note Ser. 308 (Cambridge Univ. Press, Cambridge, 2004), pp. 18–45.CrossRefGoogle Scholar
5.Baez, J. C.Crans, A. S., ‘Higher-dimensional algebra. VI. Lie 2-algebras’, Theory Appl. Categ. 12 (2004) 492–538 (electronic).Google Scholar
6.Beke, T., ‘Higher Čech theory’, K-Theory 32 (2004) 293–322.Google Scholar
7.Bénabou, J., ‘Introduction to bicategories’, Reports of the Midwest Category Seminar, Lecture Notes in Mathematics 47, (Springer, Berlin, 1967), pp. 1–77.CrossRefGoogle Scholar
8.Brown, E. H. Jr., Szczarba, R. H., ‘Real and rational homotopy theory’, Handbook of algebraic topology (ed James, I. M.), (North-Holland, Amsterdam, 1995), pp. 867–915.Google Scholar
9.Bullejos, M.Cegarra, A. M., ‘On the geometry of 2-categories and their classifying spaces’, K-Theory 29 (2003) 211–229.CrossRefGoogle Scholar
10.Dugger, D.Isaksen, D. C., ‘Topological hypercovers and -realizations’, Math. Z. 246 (2004) 667–689.CrossRefGoogle Scholar
11.Duskin, J. W., ‘Simplicial matrices and the nerves of weak n-categories. I. Nerves of bicategories’, Theory Appl. Categ. 9 (2001/02), 198–308 (electronic), CT2000 Conference (Como).Google Scholar
12.Dwyer, W. G.Spaliński, J., ‘Homotopy theories and model categories’, Handbook of algebraic topology (ed James, I. M.), (North-Holland, Amsterdam, 1995), pp. 73–126.Google Scholar
13.Dyer, E.Eilenberg, S., ‘An adjunction theorem for locally equiconnected spaces’, Pacific J. Math. 41 (1972) 669–685.Google Scholar
14.Galatius, S., ‘Stable homology of automorphism groups of free groups’, Ann. of Math (2) 173 (2011) 705–768.Google Scholar
15.Galatius, S., Madsen, I., Tillmann, U.Weiss, M., ‘The homotopy type of the cobordism category’, Acta Math. 202 (2009), no.2, 195–239.Google Scholar
16.Heath, P. R., ‘A pullback theorem for locally-equiconnected spaces’, Manuscripta Math. 55 (1986) 233–237.Google Scholar
17.Hirschhorn, P. S., Model categories and their localizations, Mathematical Surveys and Monographs 99, (American Mathematical Society, Providence, RI, 2003).Google Scholar
18.Jardine, J. F., ‘Simplicial approximation’, Theory Appl. Categ. 12 (2004) 34–72 (electronic).Google Scholar
19.Kapranov, M. M.Voevodsky, V. A., ‘2-categories and Zamolodchikov tetrahedra equations’, Algebraic groups and their generalizations: quantum and infinite-dimensional methods, University Park, PA, 1991, (eds Haboush, W. J. and Parshall, B. J.), Proc. Sympos. Pure Math. 56, (Amer. Math. Soc., Providence, RI, 1994), pp. 177–259.Google Scholar
20.Lurie, J., ‘What is … an ∞-category?’, Notices Amer. Math. Soc. (8) 55 (2008) 949–950.Google Scholar
21.Mac Lane, S., Categories for the working mathematician, second ed., Graduate Texts in Mathematics 5, (Springer-Verlag, New York, 1998).Google Scholar
22.Madsen, I., ‘Algebraic K-theory and traces’, Current developments in mathematics, 1995, Cambridge, MA, (eds Bott, R., Hopkins, M., Jaffe, A., Singer, I., Stroock, D. and Yau, S.-T.), (Internat. Press, Cambridge, MA, 1994), pp. 191–321.Google Scholar
23.Madsen, I.Weiss, M., ‘The stable moduli space of Riemann surfaces: Mumford's conjecture’, Ann. of Math. (2) 165 (2007) 843–941.Google Scholar
24.Moerdijk, I., Classifying spaces and classifying topoi, Lecture Notes in Mathematics 1616, (Springer-Verlag, Berlin, 1995).Google Scholar
25.Murray, M. K.Stevenson, D., ‘Bundle gerbes: stable isomorphism and local theory’, J. London Math. Soc. (2) 62 (2000) 925–937.Google Scholar
26.Quillen, D., ‘Higher algebraic K-theory. I’, Algebraic K-theory, I: Higher K-theories, Proc. Conf., Battelle Memorial Inst., Seattle, Wash., 1972, (ed Bass, H.), Lecture Notes in Math. 341, (Springer, Berlin, 1973), pp. 85–147.Google Scholar
27.Rezk, C., Schwede, S.Shipley, B., ‘Simplicial structures on model categories and functors’, Amer. J. Math. 123 (2001) 551–575.Google Scholar
28.Segal, G., ‘Classifying spaces and spectral sequences’, Inst. Hautes Études Sci. Publ. Math. 34 (1968) 105–112.Google Scholar
29.Segal, G., ‘Categories and cohomology theories’, Topology 13 (1974) 293–312.Google Scholar
30.Spanier, E. H., Algebraic topology, (Springer-Verlag, New York, 1991); Corrected reprint of the 1966 original.Google Scholar
31.Street, R., ‘The algebra of oriented simplexes’, J. Pure Appl. Algebra 49 (1987) 283–335.Google Scholar
32.Street, R., ‘Categorical structures’, Handbook of algebra 1, (North-Holland, Amsterdam, 1996), pp. 529–577.Google Scholar
33.Strøm, A., ‘The homotopy category is a homotopy category’, Arch. Math. (Basel) 23 (1972) 435–441.Google Scholar
34.Tillmann, U., ‘On the homotopy of the stable mapping class group’, Invent. Math. 130 (1997) 257–275.Google Scholar
35.Weiss, M., ‘What does the classifying space of a category classify ?’, Homology, Homotopy and Applications 7 (2005) 185–195; http://intlpress.com/HHA.Google Scholar
36.Whitney, H., ‘Analytic extensions of differentiable functions defined in closed sets’, Trans. Amer. Math. Soc. 36 (1934) 63–89.Google Scholar
37.Wirth, J.Stasheff, J., ‘Homotopy transition cocycles’, J. Homotopy Relat. Struct. 1 (2006), no.1, 273–283 (electronic).Google Scholar