Left fibrations and homotopy colimits
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.
Cite
@article{arxiv.1308.0704,
title = {Left fibrations and homotopy colimits},
author = {Gijs Heuts and Ieke Moerdijk},
journal= {arXiv preprint arXiv:1308.0704},
year = {2016}
}
Comments
Version 4: Added Quillen's Theorem B for infinity-categories. Version 3: We thank Joost Nuiten for pointing out an oversight in the proof of Lemma 7.2. We have fixed this and sharpened the statement and proof of Lemma 7.3. Version 2: Added a section on homotopy invariance of the covariant model structure and a section on Quillen's Theorem A for infinity-categories