Abstract
For a small category \(\mathbf{A}\), we prove that the homotopy colimit functor from the category of simplicial diagrams on \(\mathbf{A}\) to the category of simplicial sets over the nerve of \(\mathbf{A}\) establishes a left Quillen equivalence between the projective (or Reedy) model structure on the former category and the covariant model structure on the latter. We compare this equivalence to a Quillen equivalence in the opposite direction previously established by Lurie. From our results we deduce that a categorical equivalence of simplicial sets induces a Quillen equivalence on the corresponding over-categories, equipped with the covariant model structures. Also, we show that a version of Quillen’s Theorem A for \(\infty \)-categories easily follows.
Keywords
Weak Equivalence Lift Property Simplicial Diagram Adjoint Pair Covariant Model StructureReferences
- 1.Berger, C., Moerdijk, I.: On an extension of the notion of Reedy category. Math. Z. 269, 977–1004 (2011)Google Scholar
- 2.Cegarra, A.M., Remedios, J.: The relationship between the diagonal and the bar constructions on a bisimplicial set. Topol. Appl. 153(1), 21–51 (2005)CrossRefzbMATHMathSciNetGoogle Scholar
- 3.Dror, E., Dwyer, W.G., Kan, D.M.: Equivariant maps which are self homotopy equivalences. Proc. Am. Math. Soc. 80(4), 670–672 (1980)CrossRefzbMATHMathSciNetGoogle Scholar
- 4.Dugger, D.: Replacing model categories with simplicial ones. Trans. Am. Math. Soc. 353(12), 5003–5027 (2001)Google Scholar
- 5.Grothendieck, A.: Séminaire de Géométrie Algébrique du Bois-Marie, année 1960–61 (SGA 1), Revêtements étales et groupe fondamental. Institut des Hautes Etudes Scientifiques, Paris (1964)Google Scholar
- 6.Hirschhorn, P.S.: Model Categories and Their Localizations, vol. 99. AMS Bookstore (2009)Google Scholar
- 7.Hollander, S.: Diagrams indexed by Grothendieck constructions. Homol. Homotopy. Appl. 10(3), 193–221 (2008)CrossRefzbMATHMathSciNetGoogle Scholar
- 8.Illusie, L.: Complexe Cotangent et Deformations II. Springer, Berlin (1972)zbMATHGoogle Scholar
- 9.Jardine, J.F.: Fibred sites and stack cohomology. Math. Z. 254(4), 811–836 (2006)CrossRefzbMATHMathSciNetGoogle Scholar
- 10.Joyal, A.: The theory of quasi-categories I. preprint (2008)Google Scholar
- 11.Lurie, J.: Higher Topos Theory, 170th edn. Princeton University Press, Princeton (2009)CrossRefzbMATHGoogle Scholar