A Quillen model structure for Gray-categories

By Stephen Lack. 2002

91 citations recorded by the source corpus.

A Quillen model structure on the category Gray-Cat of Gray-categories is described, for which the weak equivalences are the triequivalences. It is shown to restrict to the full subcategory Gray-Gpd of Gray-groupoids. This is used to provide a functorial and model-theoretic proof of the unpublished theorem of Joyal and Tierney that Gray-groupoids model homotopy 3-types. The model structure on Gray-Cat is conjectured to be Quillen equivalent to a model structure on the category Tricat of tricategories and strict homomorphisms of tricategories.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/1e4258b308126bba9a3dd51df5ede1e52091111d

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Algebra and Number Theory

1. Ieke Moerdijk
2. C. Berger
3. Stephen Lack
4. Simona Paoli
5. B. Noohi
6. Michael Shulman
7. Benoit Fresse
8. Emily Riehl

In Geometry and Topology

1. Ieke Moerdijk
2. C. Berger
3. Stephen Lack
4. Simona Paoli
5. B. Noohi
6. Michael Shulman
7. Michael Batanin
8. Carlos Simpson

In Mathematical Physics

1. Ieke Moerdijk
2. C. Berger
3. Stephen Lack
4. Simona Paoli
5. B. Noohi
6. Michael Shulman
7. Michael Batanin
8. Chenchang Zhu

Citations in the corpus

• 2-nerves for bicategories (2006, methodology): “In fact, in the case where one uses the weak equivalences, the resulting categories are also equivalent to the homotopy categories of the Quillen model categories 2-Cat and Bicats of [ 11 , 12]; here 2-Cat is the category of 2-categories and 2-functors, and Bicats the category of bicategories and st”
• Homotopy-theoretic aspects of 2-monads (2006, background): “The definitions of weak equivalence and fibration for 2-functors coming from the model structure on 2-Cat of [12] make perfectly good sense for large 2-categories, and it is only size issues which prevent this from making 2-CAT/K into a model category, and sem into a right Quillen functor.”
• NOTES ON 2-GROUPOIDS, 2-GROUPS AND CROSSED MODULES (2005, background): “For a nice description of Gray tensor product see [ La02 ], last paragraph of Section 2.”
• On the homotopy theory of enriched categories (2012, methodology): “The rest of this section is devoted to the identification of the weak equivalences of the canonical model structure with the so-called Dwyer–Kan equivalences, often used in the literature; see [5, 9, 13, 14, 23, 24].”