Geodesic
Classifying Spaces for Monoidal Categories Through Geometric Nerves
Canadian mathematical bulletin, Tome 47 (2004) no. 3, pp. 321-331

Voir la notice de l'article provenant de la source Cambridge University Press

The usual constructions of classifying spaces for monoidal categories produce $\text{CW}$ -complexes with many cells that,moreover, do not have any proper geometric meaning. However, geometric nerves of monoidal categories are very handy simplicial sets whose simplices have a pleasing geometric description: they are diagrams with the shape of the 2-skeleton of oriented standard simplices. The purpose of this paper is to prove that geometric realizations of geometric nerves are classifying spaces for monoidal categories.
DOI : 10.4153/CMB-2004-031-8
Mots-clés : 18D10, 18G30, 55P15, 55P35, 55U40, monoidal category, pseudo-simplicial category, simplicial set, classifying space, homotopy type 
Bullejos, M.; Cegarra, A. M. Classifying Spaces for Monoidal Categories Through Geometric Nerves. Canadian mathematical bulletin, Tome 47 (2004) no. 3, pp. 321-331. doi: 10.4153/CMB-2004-031-8
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[1] [1] Bénabou, J., Introduction to bicategories. Reports of the Midwest Category Seminar, Springer, Berlin, 1967. Google Scholar

[2] [2] Bousfield, A. K. and Kan, D. M., Homotopy limits, completions and localizations. Lecture Notes in Mathematics, 304, Springer, Berlin, 1972. Google Scholar

[3] [3] Duskin, J., Simplicial matrices and nerves of weak n-categories, I: Nerves of bicategories. Theory Appl. Categ. 9 (2002), 198–308. Google Scholar

[4] [4] Carrasco, P. and Cegarra, A. M., Braided Tensor structures on homotopy groupoids and nerves of (braided) categorical groups. Comm. Algebra 24 (1996), 3995–4058. Google Scholar

[5] [5] Giraud, J., Cohomologie non abélienne. Grundlehren Math.Wiss. 179(1971). Google Scholar

[6] [6] Gray, J. W., Fibred and Cofibred Categories. Proc. Conf. Categorical Algebra, Springer, New York, 1966 pp. 21–83. Google Scholar

[7] [7] Grothendieck, A., Catégories fibrées et descente, S.G.A. I, Exposé VI. Lecture Notes in Mathematics, 224, Springer-Verlag, 1971). Google Scholar

[8] [8] Hinich, V. A. and Schechtman, V. V., Geometry of a category of complexes and algebraic K-theory. Duke Math. J. 52 (1985), 339–430. Google Scholar

[9] [9] Jardine, J. F., Supercoherence. J. Pure Appl. Algebra 75 (1991), 103–194. Google Scholar

[10] [10] Joyal, A. and Street, R., Braided tensor categories. Adv. Math. 102 (1993), 20–78. Google Scholar

[11] [11] Kan, D. M., On homotopy theory and c.s.s. groups. Ann. Math. 68 (1958), 38–53. Google Scholar

[12] [12] Mac Lane, S., Categories for the working mathematician. Graduate Texts in Mathematics, 5, 2nd Edition, Springer, Berlin, 1998. Google Scholar

[13] [13] May, J. P., Pairing of categories and spectra. J. Pure Appl. Algebra 19 (1980), 299–346. Google Scholar

[14] [14] Quillen, D., Higher algebraic K-theory: I. In: Algebraic K-theory I, Lecture Notes in Mathematics, 341, Springer, Berlin, 1973, pp. 85–147. Google Scholar

[15] [15] Thomason, R. W., Homotopy colimits in the category of small categories. Math. Proc. Camb. Phil. Soc. 85 (1979), 91–109. Google Scholar

[16] [16] Segal, G. B., Classifying spaces and spectral sequences. Inst. Hautes ´Etudes Sci. Publ. Math. 34 (1968), 105–112. Google Scholar

[17] [17] Segal, G. B., Categories and cohomology theories. Topology 13 (1974), 293–312. Google Scholar

[18] [18] Street, R., Two constructions on lax functors. Cahiers Topologie Géometrie Différentielle 13 (1972), 217–264. Google Scholar

[19] [19] Street, R., Categorical structures. In: Handbook of Algebra, Vol. I, North-Holland, Amsterdam, 1996, pp. 529–574. Google Scholar

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