Left fibrations and homotopy colimits

TL;DR

This paper establishes a Quillen equivalence between homotopy colimits of simplicial diagrams and simplicial sets over a nerve, connecting classical and infinity-category theories with practical implications.

Contribution

It proves a new Quillen equivalence linking homotopy colimits and over-categories, extending Lurie's results and simplifying the proof of Quillen's Theorems A and B for infinity-categories.

Findings

Homotopy colimit functor induces a Quillen equivalence.

Categorical equivalences induce Quillen equivalences on over-categories.

Simplified proofs of Quillen's Theorems A and B for infinity-categories.

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 infinity-categories easily follow.