Left fibrations and homotopy colimits

Gijs Heuts; Ieke Moerdijk

Left fibrations and homotopy colimits

Research output: Contribution to journal › Article › Academic › peer-review

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.

Original language	Undefined/Unknown
Journal	Mathematische Zeitschrift
Publication status	Published - 3 Aug 2013

Bibliographical note

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

Keywords

math.AT
math.CT

Cite this

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title = "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.",

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T1 - Left fibrations and homotopy colimits

AU - Heuts, Gijs

AU - Moerdijk, Ieke

N1 - 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

PY - 2013/8/3

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N2 - 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.

AB - 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.

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KW - math.CT

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SN - 0025-5874

JO - Mathematische Zeitschrift

JF - Mathematische Zeitschrift

ER -