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Two-track categories

Published online by Cambridge University Press:  05 May 2010

David Blanc  and
Simona Paoli
David Blanc
Affiliation:
Department of Mathematics, University of Haifa, 31905 Haifa, Israel, blanc@math.haifa.ac.il
Simona Paoli
Affiliation:
Department of Mathematics, Penn State Altoona, Altoona, PA 16601, USA, sup24@psu.edu
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Abstract

We describe a 2-dimensional analogue of track categories, called two-track categories, and show that it can be used to model categories enriched in 2-type mapping spaces. We also define a Baues-Wirsching type cohomology theory for track categories, and explain how it can be used to classify two-track extensions of a track category by a module over .

Keywords

Track categoriesdouble groupoids2-typessimplicially enriched categoriesBaues-Wirsching cohomology

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Type
Research Article
Information
Journal of K-Theory , Volume 8 , Issue 1 , August 2011 , pp. 59 - 106
Copyright
Copyright © ISOPP 2010

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References

1.Artin, M. & Mazur, B., “On the van Kampen theorem”, Topology 5 (1966), 178–189.CrossRefGoogle Scholar
2.Baues, H.-J., The algebra of secondary cohomology operations, Birkhäuser Prog. in Math. 247, Basel, 2006.Google Scholar
3.Baues, H.-J., “Higher order track categories and the algebra of higher order cohomology operations”, preprint, 2009, arXiv:0903.2876.CrossRefGoogle Scholar
4.Baues, H.-J. & Blanc, D. “Comparing cohomology obstructions”, preprint, 2010.Google Scholar
5.Baues, H.-J. & Blanc, D. “Stems and spectral sequences”, preprint, 2010.CrossRefGoogle Scholar
6.Baues, H.-J. & Dreckman, W., “The cohomology of homotopy categories and the general linear group”, K-Theory 3 (1989), 307–338.CrossRefGoogle Scholar
7.Baues, H.-J. & Jibladze, M.A., “Computation of the E 3-term of the Adams spectral sequence”, preprint, 2004.Google Scholar
8.Baues, H.-J. & Jibladze, M.A., “Secondary derived functors and the Adams spectral sequence”, Topology 45 (2006), 295–324.CrossRefGoogle Scholar
9.Baues, H.-J., Jibladze, M.A., & Tonks, A.P., “Cohomology of monoids in monoidal categories”, in Loday, J.-L., Stasheff, J.D., & Voronov, A.A., eds., Operads: Proceedings of Renaissance Conference (Hartford,CT/Luminy, 1995) Contemp. Math. 202, AMS, Providence, RI 1997, 137–165.Google Scholar
10.Baues, H.-J. & Wirsching, G., “The cohomology of small categories”, J. Pure & Appl. Alg. 38 (1985), 187–211.CrossRefGoogle Scholar
11.Berger, C., “A cellular nerve for higher categories”, Adv. in Math. 169 (2002), 118–175.CrossRefGoogle Scholar
12.Blanc, D., “Generalized André-Quillen Cohomology”, J. Homotopy & Rel. Structures 3 (2008), 161–191.Google Scholar
13.Blanc, D., Dywer, W.G., & Goerss, P.G., “The realization space of a Π-algebra: a moduli problem in algebraic topology”, Topology 43 (2004), 857–892.CrossRefGoogle Scholar
14.Blanc, D., Johnson, M.W., & Turner, J.M., “On Realizing Diagrams of Π-algebras”, Alg. & Geom. Top. 6 (2006), 763–807.CrossRefGoogle Scholar
15.Blanc, D., Johnson, M.W., & Turner, J.M., “Higher homotopy operations and cohomology”, J. K-Theory 5 (2010), 167–200.CrossRefGoogle Scholar
16.Blanco, V., Lorenzo, M. Bullejos, & Faro, E., A Full and faithful Nerve for 2-categories”, Appl. Cat. Str. 13 (2005), 223–233.Google Scholar
17.Borceux, F., Handbook of Categorical Algebra, Vol. 1: Basic Category Theory, Encyc. Math. & its Appl. 50, Cambridge U. Press, Cambridge, UK, 1994.Google Scholar
18.Borceux, F., Handbook of Categorical Algebra, Vol. 2: Categories and Structures, Encyc. Math. & its Appl. 51, Cambridge U. Press, Cambridge, UK, 1994.Google Scholar
19.Bousfield, A.K & Friedlander, E.M, “Homotopy theory of Γ-spaces, spectra, and bisimplicial sets”, in Barratt, M.G. and Mahowald, M.E., eds., Geometric Applications of Homotopy Theory, II, Lec. Notes Math. 658, Springer, Berlin-New York, 1978, 80–130.Google Scholar
20.Bousfield, A.K. & Kan, D.M., Homotopy limits, Completions, and Localizations, Lec.Notes Math. 304, Springer, Berlin-New York, 1972.Google Scholar
21.Brown, R., Hardie, K.A., Kamps, K.H., & Porter, T., “A homotopy double groupoid of a Hausdorff space”, Theory Appl. Categ. 10 (2002), 71–93.Google Scholar
22.Brown, R. & Spencer, C.B., “Double groupoids and crossed modules”, Cahiers Top. Géom. Diff. Cat. 17 (1976), 343–362.Google Scholar
23.Bullejos, M., Cegarra, A.M., & Duskin, J.W., “On catn-groups and homotopy types”, J. Pure & Appl. Alg. 86 (1993), 135–154.CrossRefGoogle Scholar
24.Cegarra, A.M. & Remedios, J., “The relationship between the diagonal and the bar constructions on a bisimplicial set”, Top. & Appl. 153 (2005), 21–51.CrossRefGoogle Scholar
25.Chachólski, W. & Scherer, J., Homotopy theory of diagrams, Memoirs AMS 736, American Mathematical Society, Providence, RI, 2002.Google Scholar
26.Cisinski, D.-C., “Les préfaisceaux comme modèles des types d'homotopie”, Astérisque 308, Soc. Math. France, Paris, 2006.Google Scholar
27.Cordier, J.-M., “Sur la notion de diagramme homotopiquement cohérent”, Cahiers Top. Géom. Cat. 23 (1982), 93–112.Google Scholar
28.Duskin, J.W., Simplicial methods and the interpretation of “triple” cohomology, Memoirs AMS 3 (No. 163), AMS, Providence, RI, 1975.Google Scholar
29.Dwyer, W.G. & Kan, D.M., “Function complexes in homotopical algebra”, Topology 19 (1980), 427–440.CrossRefGoogle Scholar
30.Dwyer, W.G. & Kan, D.M., “Simplicial localizations of categories”, J. Pure & Appl. Alg. 17 (1980), No. 3, 267–284.CrossRefGoogle Scholar
31.Dwyer, W.G. & Kan, D.M., “An obstruction theory for diagrams of simplicial sets”, Proc. Kon. Ned. Akad. Wet. - Ind. Math. 46 (1984) 139–146.Google Scholar
32.Dwyer, W.G., Kan, D.M., & Smith, J.H., “An obstruction theory for simplicial categories”, Proc. Kon. Ned. Akad. Wet. - Ind. Math. 89 (1986), 153–161.Google Scholar
33.Ehresmann, C., “Catégories doubles et catégories structurées”, Comptes Rend. Acad. Sci., Paris 256 (1963), 1198–1201.Google Scholar
34.Ellis, G.J., “Homology of 2-types”, J. Lond. Math. Soc. (2) 46 (1992), 1–27.CrossRefGoogle Scholar
35.Goerss, P.G. & Jardine, J.F., Simplicial Homotopy Theory, Progress in Mathematics 179, Birkhäuser, Basel-Boston, 1999.Google Scholar
36.Gordon, R., Power, A.J., & Street, R.H., Coherence of tricategories, Memoirs AMS 117, AMS, Providence, RI, 1995.Google Scholar
37.Hardie, K.A., Kamps, K.H., & Kieboom, R.W., “A homotopy bigroupoid of a topological space”, Appl. Cat. Struct. 9 (2001), 311–327.CrossRefGoogle Scholar
38.Higgins, P.J., Notes on Categories and Groupoids, Van Nostrand Reinhold Mathematical Studies 32, Van Nostrand Reinhold Co., London, 1971.Google Scholar
39.Hirschhorn, P.S., Model Categories and their Localizations, Math. Surveys & Monographs 99, AMS, Providence, RI, 2002.Google Scholar
40.Illusie, L., Complexe cotangent et déformations. II, Lec. Notes Math. 283, Springer, Berlin-New York, 1972.Google Scholar
41.Jardine, J.F., “Cubical homotopy theory: a beginning”, preprint, 2002.Google Scholar
42.Jardine, J.F., “Categorical homotopy theory”, Homology, Homotopy & Applic. 8 (2006), 71–144.CrossRefGoogle Scholar
43.Johnson, M. & Walters, R.F.C., “On the nerve of an n-category”, Cahiers Top. Géom. Diff. Cat. 28 (1987), 257–282.Google Scholar
44.Joyal, A. & Street, R., “Pullbacks equivalent to pseudo pullbacks”, Cahiers de Topologie et Géom. Diff. Catég. 34 (1993), 153–156.Google Scholar
45.Kan, D.M., “Is an ss complex a css complex?”, Adv. in Math. 4 (1970), 170–171.CrossRefGoogle Scholar
46.Lack, S. & Paoli, S., “2-Nerves for bicategories”, K-Theory 38 (2008), 153–175.CrossRefGoogle Scholar
47.Mac Lane, S. & Whitehead, J.H.C., “On the 3-type of a complex”, Proc. Nat. Acad. Sci. USA 36 (1950), 41–48.CrossRefGoogle Scholar
48.Moerdijk, I. & Svensson, J.-A., “Algebraic classification of equivariant homotopy 2-types, I”, J. Pure Appl. Alg. 89 (1993), 187–216.CrossRefGoogle Scholar
49.Paoli, S., “(Co)homology of crossed modules with coefficients in a π1-module”, Homol. Homotopy Applic. 5 (2003), 261–296.CrossRefGoogle Scholar
50.Paoli, S., “Weakly globular catn-groups and Tamsamani's model”, Adv. in Math. 222 (2009), 621–727.CrossRefGoogle Scholar
51.Quillen, D.G., Homotopical Algebra, Springer-Verlag Lec. Notes Math. 20, Berlin-New York, 1963.Google Scholar
52.Quillen, D.G., “On the (co-)homology of commutative rings”, Applications of Categorical Algebra, Proc. Symp. Pure Math. 17, AMS, Providence, RI, 1970, 65–87.CrossRefGoogle Scholar
53.Segal, G.B., “Classifying spaces and spectral sequences”, Pub. Math. Inst. Hautes Et. Sci. 34 (1968), 105–112.CrossRefGoogle Scholar
54.Segal, G.B., “Categories and cohomology theories”, Topology 13 (1974), 293–312.CrossRefGoogle Scholar
55.Street, R.H., “The algebra of oriented simplices”, J. Pure Appl. Alg. 49 (1987), 283–335.CrossRefGoogle Scholar
56.Tamsamani, Z., “Sur des notions de n-catégorie et n-groupoide non-strictes via des ensembles multi-simpliciaux”, K-theory 16 (1999), 51–99.CrossRefGoogle Scholar
57.Verity, D., Complicial sets characterising the simplicial nerves of strictω-categories, Memoirs AMS 193, American Mathematical Society, Providence, RI, 2008.Google Scholar