@MISC{Jardine10cosimplicialspaces, author = {J. F. Jardine}, title = {Cosimplicial spaces and cocycles}, year = {2010} }

Bookmark

OpenURL

Abstract

This paper is a retelling of the basic homotopy theory of cosimplicial spaces, from a point of view that is informed by sheaf theoretic homotopy theory. The overall plan is to interpolate ideas associated with the injective model structure for cosimplicial spaces with classical results of Bousfield and Kan. The

...ions are defined by a right lifting property with respect to trivial cofibrations. The weak equivalences coincide with the weak equivalences of the BousfieldKan model structure on cosimplicial spaces =-=[1]-=-. A Bousfield-Kan cofibration is a sectionwise cofibration which induces an isomorphism in maximal augmentations. Explcitly, the maximal augmentation X −1 of a cosimplicial set is the simplicial set w...

...d for a Bousfield-Kan fibrant cosimplicial space Y by Tot(Y ) = hom(∆, Y ), where ∆ is the cosimplicial space n ↦→ ∆ n and hom(∆, Y ) is the usual diagram theoretic function complex. Recall also (see =-=[6]-=-, for example) that the homotopy inverse limit ←−−− holim n Xn of a cosimplicial space X is defined by taking an injective fibrant model j : X → Z (a weak equivalence with Z injective fibrant), and th...

...r of fibrations hom(∗, GP1X) q∗ ←− hom(∗, GP2X) q∗ ←− hom(∗, GP3X) ← . . . is a special case of the descent spectral sequence for a simplicial presheaf, with E1-terms given by sheaf cohomology groups =-=[4]-=-. Thomason’s reindexing trick [10, 5.54] converts these E1-terms to E2-terms of the form appearing in the Bousfield-Kan spectral sequence for the tower of fibrations {Tots(GX)} — see also [5]. 22�� �...

...valences (respectively injective fibrations) of cosimplicial spaces. This is a special case of general results about sheaves and/or presheaves of groupoids, for which the usual references are [9] and =-=[3]-=-. Following the same point of view, we say that a cosimplicial groupoid G is a stack if and only if the cosimplicial space BG is injective fibrant. A weak equivalence G → H of cosimplicial groupoids s...

...eak equivalences (respectively injective fibrations) of cosimplicial spaces. This is a special case of general results about sheaves and/or presheaves of groupoids, for which the usual references are =-=[9]-=- and [3]. Following the same point of view, we say that a cosimplicial groupoid G is a stack if and only if the cosimplicial space BG is injective fibrant. A weak equivalence G → H of cosimplicial gro...

...ck site C, then the simplicial sheaf K(F, 0) is fibrant for the injective model structure for simplicial presheaves on C which is defined by the topology — this appears, for example, as Lemma 6.10 in =-=[5]-=-, with the same proof. In diagram categories every presheaf is a sheaf, and so every diagram of simplicial sets which is simplicially constant is injective fibrant. Example 5. There are cosimplicial s...

...g is a weak equivalence. A morphism of cocycles is a diagram V � ≃ ��� ������� �� X Y V ′ ������� � ��� ≃ �� These are the objects and morphisms of the cocycle category h(X, Y ). It is a basic result =-=[8]-=- for injective structures on diagram categories that the assignment which sends a cocycle (g, f) to the morphism fg −1 in the homotopy category defines a bijection π0Bh(X, Y ) ∼ = [X, Y ]. We also hav...

..., n+1) represents an element of the stack cohomology group [pb(Pn−1X), K(Hn(ZnX, n + 1] where stack cohomology is interpreted to mean abelian group cohomology for diagrams over the category ∫ H — see =-=[7]-=-. ∆ 25The ideas displayed in this section admit substantial generalization. One could, for example, start with an I-diagram X of Kan complexes, and observe that its Postnikov tower is defined over th...