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On the structure of simplicial categories associated to quasi-categories

Published online by Cambridge University Press:  11 March 2011

Department of Mathematics, University of Chicago, 5734 S. University Ave., Chicago, IL 60637, USA e-mail:


The homotopy coherent nerve from simplicial categories to simplicial sets and its left adjoint are important to the study of (∞, 1)-categories because they provide a means for comparing two models of their respective homotopy theories, giving a Quillen equivalence between the model structures for quasi-categories and simplicial categories. The functor also gives a cofibrant replacement for ordinary categories, regarded as trivial simplicial categories. However, the hom-spaces of the simplicial category X arising from a quasi-category X are not well understood. We show that when X is a quasi-category, all Λ21 horns in the hom-spaces of its simplicial category can be filled. We prove, unexpectedly, that for any simplicial set X, the hom-spaces of X are 3-coskeletal. We characterize the quasi-categories whose simplicial categories are locally quasi, finding explicit examples of 3-dimensional horns that cannot be filled in all other cases. Finally, we show that when X is the nerve of an ordinary category, X is isomorphic to the simplicial category obtained from the standard free simplicial resolution, showing that the two known cofibrant “simplicial thickenings” of ordinary categories coincide, and furthermore its hom-spaces are 2-coskeletal.

Research Article
Copyright © Cambridge Philosophical Society 2011

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[1]Bergner, J.A model category structure on the category of simplicial categories. Trans. Amer. Math. Soc. 359 (2007), 20432058.CrossRefGoogle Scholar
[2]Bergner, J. A survey of (∞, 1)-categories. Towards higher categories. IMA Vol. Math. Appl. 152 (Springer, 2010), pp. 6983.CrossRefGoogle Scholar
[3]Cordier, J.-M.Sur la notion de diagramme homotopiquement cohérent. Cah. Top. Géom. Difféc. 1 XXIII (1982), 93112.Google Scholar
[4]Cordier, J.-M. and Porter, T.Vogts theorem on categories of homotopy coherent diagrams. Math. Proc. Cambr. Phil. Soc. 100 (1986), 6590.CrossRefGoogle Scholar
[5]Day, B. On closed categories of functors. Reports of the Midwest Category Seminar IV. Lecture Notes in Math. vol. 137 (Springer 1970), pp. 138.Google Scholar
[6]Dugger, D. and Spivak, D.Rigidification of quasi–categories. Algebr. Geom. Topol. 11 (2011), 225261.CrossRefGoogle Scholar
[7]Duskin, J.Simplicial matrices and the nerves of weak n-categories I: Nerves of bicategories. CT2000 Conference (Como). Theory Appl. Categ. (10) 9 (2001/2002), 198308.Google Scholar
[8]Duskin, J.Simplicial methods and the interpretation of “triple” cohomology. Mem. Amer. Math. Soc. 3 (2), no. 16 (1975).Google Scholar
[9]Dwyer, W. and Kan, D.Simplicial localizations of categories. J. Pure Appl. Algebra 17 (1980), 267284.CrossRefGoogle Scholar
[10]Gabriel, P. and Zisman, M.Calculus of fractions and homotopy theory. Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 35 (Springer-Verlag, 1967).CrossRefGoogle Scholar
[11]Goerss, P. and Jardine, J. Simplicial homotopy theory. Progr. Math. 174 (Birkhauser Verlag, 1999).Google Scholar
[12]Hovey, M. Model categories. Math. Surv. Monogr. vol. 63 (American Mathematical Society, 1999).Google Scholar
[13]Joyal, A.Quasi-categories and Kan complexes. J. Pure Appl. Algebra 175 (2002), 207222.CrossRefGoogle Scholar
[14]Joyal, A. The theory of quasi-categories I. In progress (2008).Google Scholar
[15]Lurie, J.Higher topos theory. Ann. of Math. Stud. 170 (2009).Google Scholar