Left fibrations and homotopy colimits

Abstract

Abstract. For a small category A, we prove that the homotopy colimit functor from the category of simplicial diagrams on A to the category of simplicial sets over the nerve of 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 versions of Quillen's Theorems A and B for ∞-categories easily follow.

Fibrations in ∞-Category Theory

“…We know four proofs: the original one due to Joyal, a modification thereof due to Lurie[7], a simplification due to Dugger and Spivak[3], and a recent extreme simplification due to Heuts and Moerdijk[4, 5].…”

Quasi-categories vs. Segal spaces: Cartesian edition

“…where n ≥ 0, 0 < i ≤ n. Unlike Cartesian fibrations we can in fact endow the category of simplicial sets over a fixed simplicial set with a model structure, the contravariant model structure, such that the fibrant objects are precisely the right fibrations. It was first proven by Lurie [31] (and later many other authors [23,24,44]), that this model structure is Quillen equivalent to presheaves valued in spaces. Moreover, we can give an analogous definition of right fibrations and contravariant model structure in the context of simplicial spaces in a way that is Quillen equivalent to the contravariant model structure for simplicial sets [12,37].…”

“…However, fibrant objects are of the form (W , W hoequiv ), where W is a complete Segal space. By direct computation, 23). Moreover, fibrations between fibrant objects are just unmarked CSS fibrations (Theorem 2.25), which are, again by Theorem 1.23, taken to Joyal fibrations.…”

“…We can use a similar equivalence between the contravariant model structure and a presheaf category to similarly deduce that right fibrations are invariant under categorical equivalences. However, in this case there is also an alternative proof, which uses the combinatorics of the contravariant model structure [23]. Using the complete Segal object approach to Cartesian fibrations we can generalize that proof immediately from right fibrations to Cartesian fibrations without having to translate to presheaf categories.…”

“…-Grothendieck Construction over 1-Categories: One particular instance of the previous point is the study of Cartesian fibrations over ordinary categories. In [23], the authors give a Grothendieck construction for right fibrations very much along the lines of the classical Grothendieck construction for categories (also known as the category of elements). They then prove that this gives us a Quillen equivalence.…”