Homotopy limits for 2-categories

Abstract

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.

Weighted limits in simplicial homotopy theory

“…On the other hand, our approach differs also from the one taken in the literature on weighted limits in homotopy theory [1,10,11], which does not consider model structures. Here, as in [7], the combination of ideas of enriched category theory and of homotopical algebra plays instead an essential role. This is in a spirit similar to that of [22], which relates the formulas for homotopy limits involving the bar construction [18] with the abstract approach of homotopical categories [5].…”

Shadows are Bicategorical Traces

“…It is important to note that the colimit of Diagram (3.3) is evaluated in the (2, 1)category Cat, which differs from the 1-categorical colimit in the category of categories, but rather corresponds to the pseudocolimit [Kel89]. Computing general pseudocolimits can be quite challenging, although there certain helpful methods via weighted colimits of enriched categories [Kel05] and homotopy colimits of model categories [Gam08]. The example below illustrates the challenges that arise even in simple examples.…”

A Colimit Decomposition for Homotopy Algebras in Cat

“…commutes. This is just the colimit variant of a result on limits of [8]. To see where the isomorphism comes from recall that the cotensor A X (or power) of an object A ∈ C by a category X is the limit defined by a natural isomorphism C(B, A)) Given a weight W : J → Cat and diagram D : J op → C this isomorphism extends in a pointwise manner to a natural isomorphism as on the left below…”

“…By the adjunction Q ⊣ ι we have a natural isomorphism [J, Cat](QW, C(D−, A)) ∼ = P s(J, Cat)(W, C(D−, A)) so that the pseudocolimit is nothing but the weighted colimit QW ⋆ D. That pseudocolimits are flexible is easy to see: the adjunction Q ⊣ ι generates a comonad (Q, q, ∆) on [J, Cat] with counit the same q : Q → 1 as before; in particular each QW admits a (co-free) coalgebra structure, and so is certainly a flexible weight. Since the pseudocolimit W ⋆ p D is the genuine colimit of D weighted by a cofibrant replacement of W , pseudocolimits are closely related to homotopy colimits-this relationship was studied in [8].…”