Skip to main content Accessibility help

On the structure of simplicial categories associated to quasi-categories

  • EMILY RIEHL (a1)

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.

Hide All
[1]Bergner, J.A model category structure on the category of simplicial categories. Trans. Amer. Math. Soc. 359 (2007), 20432058.
[2]Bergner, J. A survey of (∞, 1)-categories. Towards higher categories. IMA Vol. Math. Appl. 152 (Springer, 2010), pp. 6983.
[3]Cordier, J.-M.Sur la notion de diagramme homotopiquement cohérent. Cah. Top. Géom. Difféc. 1 XXIII (1982), 93112.
[4]Cordier, J.-M. and Porter, T.Vogts theorem on categories of homotopy coherent diagrams. Math. Proc. Cambr. Phil. Soc. 100 (1986), 6590.
[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.
[6]Dugger, D. and Spivak, D.Rigidification of quasi–categories. Algebr. Geom. Topol. 11 (2011), 225261.
[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.
[8]Duskin, J.Simplicial methods and the interpretation of “triple” cohomology. Mem. Amer. Math. Soc. 3 (2), no. 16 (1975).
[9]Dwyer, W. and Kan, D.Simplicial localizations of categories. J. Pure Appl. Algebra 17 (1980), 267284.
[10]Gabriel, P. and Zisman, M.Calculus of fractions and homotopy theory. Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 35 (Springer-Verlag, 1967).
[11]Goerss, P. and Jardine, J. Simplicial homotopy theory. Progr. Math. 174 (Birkhauser Verlag, 1999).
[12]Hovey, M. Model categories. Math. Surv. Monogr. vol. 63 (American Mathematical Society, 1999).
[13]Joyal, A.Quasi-categories and Kan complexes. J. Pure Appl. Algebra 175 (2002), 207222.
[14]Joyal, A. The theory of quasi-categories I. In progress (2008).
[15]Lurie, J.Higher topos theory. Ann. of Math. Stud. 170 (2009).
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Mathematical Proceedings of the Cambridge Philosophical Society
  • ISSN: 0305-0041
  • EISSN: 1469-8064
  • URL: /core/journals/mathematical-proceedings-of-the-cambridge-philosophical-society
Please enter your name
Please enter a valid email address
Who would you like to send this to? *


Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed