Modules in monoidal model categories

Lewis, L; Mandell, Michael A.

doi:10.1016/j.jpaa.2006.10.002

Cited by 17 publications

(33 citation statements)

References 8 publications

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“…Weighted homotopy limits. Proposition 5.2.1 and Proposition 5.2.2 allow us to apply the general theory of model V-categories [7,12] and conclude the existence of the total derived Ho(Cat)-enriched functor of the weighted limit 2-functor. We refer to it as the weighted homotopy limit functor.…”

Section: 23mentioning

confidence: 94%

“…The existence of the projective and injective model structures allows us to apply the general theory of enriched model categories [7,12,20] to study the total derived functors of limit 2-functors. We will consider not only conical limits but also weighted limits [15,16,22].…”

Section: Quillen Model Structures In 2-category Theorymentioning

confidence: 99%

“…We write Ho(Cat) for the homotopy category of Cat, which consists of categories and isomorphism classes of functors, and denote the localization functor as λ : Cat → Ho(Cat). As recalled in [17, §2.2], the cartesian product equips Cat with a symmetric monoidal structure which satisfies the axioms for a monoidal model category [7,12,20].…”

Section: Quillen Model Structures In 2-category Theorymentioning

confidence: 99%

“…Model 2-categories. For any monoidal model category V, there is an associated notion of a model V-enriched category [7,12]. We spell out the general definition in the special case when V is Cat.…”

Section: Quillen Model Structures In 2-category Theorymentioning

confidence: 99%

“…Lemma 2.2.2 is a special case of [7,Proposition 3.4]. It will be useful in the study of homotopy-theoretic aspects of 2-categorical limits.…”

Section: Quillen Model Structures In 2-category Theorymentioning

confidence: 99%

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Homotopy limits for 2-categories

Gambino¹

2008

Math. Proc. Camb. Phil. Soc.

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Abstract. We study homotopy limits for 2-categories using the theory of Quillen model categories. In order to do so, we establish the existence of projective and injective model structures on diagram 2-categories. Using this result, we describe the homotopical behaviour not only of conical limits but also of weighted limits for 2-categories. Finally, homotopy limits are related to pseudo-limits. Quillen model structures in 2-category theoryThe 2-category 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 2-category 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 generalised these results by showing that every 2-category K with finite limits and colimits admits a model structure, called here the natural model structure on K, in which the weak equivalences and the fibrations are the equivalences and the isofibrations in K [17]. The notions of equivalence and isofibration for a map in a 2-category are obtained by suitably generalising the notions of equivalence and of isofibration for a functor. We take Lack's theorem as a starting point to study homotopy limits for 2-categories.Our first step is to show that for every small 2-category A and every 2-category K with finite limits and small colimits, the functor 2-category [A, K] admits a model structure in which the weak equivalences are the pointwise equivalences and the fibrations are the pointwise isofibrations. We refer to this model structure as the projective model structure. When K is assumed to be locally presentable, the existence of the projective model structure follows by a result on the lifting of the natural model structure on a 2-category K to 2-categories of algebras for a 2-monad with rank on K [17, Theorem 4.5]. The special form of the 2-category [A, K], however, allows us to avoid assuming that K is locally presentable, and to give a simple proof of the model category axioms for the projective model structure, which does not Date: April 9th, 2007.

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Section: 23mentioning

confidence: 94%