@MISC{Jardine10diagonalmodel, author = {J. F. Jardine}, title = {Diagonal model structures}, year = {2010} }

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Abstract

The original purpose of this note was to display a model structure for the category s 2 Set of bisimplicial sets whose cofibrations are the monomorphisms and whose weak equivalences are the diagonal weak equivalences, and then show that it is cofibrantly generated in a very precise way. The project grew to include

... fibrations is technically interesting, and is the subject of the second section of this paper, leading to Theorem 24. This result mirrors well known results for simplicial sets and cubical sets [1], =-=[5]-=-. Roughly speaking, the overall idea of the proof is to produce the minimal fibrewise model structure for the category of bisimplicial set maps X → ∆p,q over the bisimplex ∆p,q in which the anodyne ex...

...ts on the site C. Say that a map f : X → Y of bisimplicial presheaves is a diagonal weak equivalence if the induced simplicial presheaf map d(X) → d(Y ) is a local weak equivalence in the usual sense =-=[3]-=-, [4]. A monomorphism of bisimplicial presheaves is a cofibration. An injective fibration of bisimplicial presheaves is a morphism which has the right lifting property with respect to trivial cofibrat...

... : d between the standard model structure on simplicial sets and the diagonal model structure on bisimplicial sets is an immediate consequence. The Moerdijk model structure for bisimplicial sets [7], =-=[2]-=- is induced from the standard model structure for simplicial sets by the diagonal functor — this was the first published example of a model structure for bisimplicial sets whose 1weak equivalences ar...

...e Kan fibrations is technically interesting, and is the subject of the second section of this paper, leading to Theorem 24. This result mirrors well known results for simplicial sets and cubical sets =-=[1]-=-, [5]. Roughly speaking, the overall idea of the proof is to produce the minimal fibrewise model structure for the category of bisimplicial set maps X → ∆p,q over the bisimplex ∆p,q in which the anody...

... the existence of a diagonal model structure for bisimplicial sets. This model structure on bisimplicial sets, and the localization argument leading to it, has already been displayed by other authors =-=[2]-=-, [11]. It is also easy to show that the diagonal functor and its left adjoint d ∗ define a Quillen equivalence d ∗ : s Pre(C) ⇆ s 2 Pre(C) : d between the injective model structure on simplicial pres...

...existence of a diagonal model structure for bisimplicial sets. This model structure on bisimplicial sets, and the localization argument leading to it, has already been displayed by other authors [2], =-=[11]-=-. It is also easy to show that the diagonal functor and its left adjoint d ∗ define a Quillen equivalence d ∗ : s Pre(C) ⇆ s 2 Pre(C) : d between the injective model structure on simplicial presheaves...

...2 Set : d between the standard model structure on simplicial sets and the diagonal model structure on bisimplicial sets is an immediate consequence. The Moerdijk model structure for bisimplicial sets =-=[7]-=-, [2] is induced from the standard model structure for simplicial sets by the diagonal functor — this was the first published example of a model structure for bisimplicial sets whose 1weak equivalenc...

... the site C. Say that a map f : X → Y of bisimplicial presheaves is a diagonal weak equivalence if the induced simplicial presheaf map d(X) → d(Y ) is a local weak equivalence in the usual sense [3], =-=[4]-=-. A monomorphism of bisimplicial presheaves is a cofibration. An injective fibration of bisimplicial presheaves is a morphism which has the right lifting property with respect to trivial cofibrations....

... the proof given in this paper is again direct, and does not use Cisinski’s localization techniques. Theorem 9 is a translation of the intermediate model structures story for simplicial presheaves of =-=[8]-=-. The results of this paper have been collected together here in anticipation of concrete applications. In particular, they are used in the analysis of homotopy types of diagrams and dynamical systems...

...e results have been collected together here in anticipation of concrete applications. In particular, they are used in the analysis of homotopy types of diagrams and dynamical systems which appears in =-=[8]-=-. 1 Bisimplicial presheaves Recall that a bisimplicial set X is a functor X : ∆ op × ∆ op → Set, and a morphism of bisimplicial sets is a natural transformation of such functors. Write Xp,q = X(p, q) ...