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A Model Structure on the Category of Small Categories for Coverings
, 2009
"... We define a new model structure on the category of small categories, which is intimately related to the notion of coverings and fundamental groups of small categories. Fibrant objects in the model structure coincide with groupoids, and the fibrant replacement is the groupoidification. ..."
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We define a new model structure on the category of small categories, which is intimately related to the notion of coverings and fundamental groups of small categories. Fibrant objects in the model structure coincide with groupoids, and the fibrant replacement is the groupoidification.
HOMOTOPYTHEORETIC ASPECTS OF 2MONADS STEPHEN LACK
"... We study 2monads and their algebras using a Catenriched version of Quillen model categories, emphasizing the parallels between the homotopical and 2categorical points of view. Every 2category with finite limits and colimits has a canonical model structure in which the weak equivalences are the e ..."
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We study 2monads and their algebras using a Catenriched version of Quillen model categories, emphasizing the parallels between the homotopical and 2categorical points of view. Every 2category with finite limits and colimits has a canonical model structure in which the weak equivalences are the equivalences; we use these to construct more interesting model structures on 2categories, including a model structure on the 2category of algebras for a 2monad T, and a model structure on a 2category of 2monads on a fixed 2category K. 1.
HOMOTOPY LIMITS FOR 2CATEGORIES
"... Abstract. We study homotopy limits for 2categories using the theory of Quillen model categories. In order to do so, we establish the existence of projective and injective model structures on diagram 2categories. Using this result, we describe the homotopical behaviour not only of conical limits bu ..."
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Abstract. We study homotopy limits for 2categories using the theory of Quillen model categories. In order to do so, we establish the existence of projective and injective model structures on diagram 2categories. Using this result, we describe the homotopical behaviour not only of conical limits but also of weighted limits for 2categories. Finally, homotopy limits are related to pseudolimits. 1. Quillen model structures in 2category theory The 2category of groupoids, functors, and natural transformations admits a model structure in which the weak equivalences are the equivalence of categories and the fibrations are the Grothendieck fibrations [1, 5, 13]. Similarly, the 2category of small categories, functors, and natural transformations admits a model structure in which the weak equivalences are the equivalence of categories and the fibrations are the isofibrations, which are functors satisfying a restricted version of the lifting condition for Grothendieck fibrations which involves only isomorphisms [13, 19]. Steve Lack has vastly
ON THE HOMOTOPY THEORY OF ENRICHED CATEGORIES
"... Abstract. We give sufficient conditions for the existence of a Quillen model structure on small categories enriched in a given monoidal model category. This yields a unified treatment for the known model structures on simplicial, topological, dg and spectral categories. Our proof is mainly based on ..."
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Abstract. We give sufficient conditions for the existence of a Quillen model structure on small categories enriched in a given monoidal model category. This yields a unified treatment for the known model structures on simplicial, topological, dg and spectral categories. Our proof is mainly based on a fundamental property of cofibrant enriched categories on two objects, stated below as the Interval Cofibrancy Theorem. 1. Introduction and
Lecture 08 (January 6, 2011)
"... 18 Torsors for groups Suppose that G is a sheaf of groups. A Gtorsor is traditionally defined to be a sheaf X with a free Gaction such that X/G ∼ = ∗ in the sheaf category. The requirement that the action G × X → X is free means that the isotropy subgroups of G for the action are trivial in all ..."
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18 Torsors for groups Suppose that G is a sheaf of groups. A Gtorsor is traditionally defined to be a sheaf X with a free Gaction such that X/G ∼ = ∗ in the sheaf category. The requirement that the action G × X → X is free means that the isotropy subgroups of G for the action are trivial in all sections, which is equivalent to requiring that all sheaves of fundamental groups for the Borel construction EG ×G X are trivial. There is an isomorphism of sheaves ˜π0(EG ×G X) ∼ = X/G. Also the simplicial sheaf EG ×G X is the nerve of a sheaf of groupoids, which is given in each section by the translation category for the action of G(U) on X(U); this means, in particular, that all sheaves of higher homotopy groups for EG ×G X vanish. It follows that a Gsheaf X is a Gtorsor if and only if the map EG ×G X → ∗ is a local weak equivalence. 1 Example 18.1. The Borel construction EH ×H H = EH for a group H is the nerve of the translation category for the action H × H → H which is given by the multiplication of H. There is a unique map e h − → h for all h ∈ H, so that EH ×H H is a contractible simplicial set. If G is a sheaf of groups, then EG ×G G is contractible in each section, so that the map EG ×G G → ∗ is a local weak equivalence, and G is a Gtorsor. This object is often called the trivial Gtorsor. Example 18.2. Suppose that L/k is a finite Galois extension with Galois group G. Then the étale covering Sp(L) → Sp(k) has Čech resolution C(L) and there is an isomorphism of simplicial schemes C(L) ∼ = EG ×G Sp(L). The simplicial presheaf map