2013/08/03 by Gijs Heuts, Heuts, Gijs, Ieke Moerdijk +1
Mathematics · #Advanced Topics in Algebra #Algebraic Topology (math.AT) #Algebraic structures and combinatorial models #Category Theory (math.CT) #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #math.AT #math.CT
paper · pdf · doi:10.48550/arxiv.1308.0704
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
openalex publication_date 2013/08/03 · arxiv created 2016/02/03 · arxiv updated 2016/02/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.