the homotopy category of infinity-stacks

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next: principal infinity-bundles

So far we have

Now, recalling our motivation for sheaves, cohomology and higher stacks, we want to generalize this procedue from presheaves with values in Set to those with values in (sufficiently nice) topological spaces or equivalently infinity-groupoids modeled by simplicial sets.

Brown category of simplicial sheaves

We now describe

We are intersted in finding the corresponding homotopy Ho(SSh(C))Ho(SSh(C)). That shall be our homotopy category of \infty-stacks and provide us with a general notion of cohomology.

To obtain a tractable formula that describes this homotopy category, we follow K. Brown's work and consider

Remark Nowadays there are various model category structures on simplicial (pre)sheaves that provide yet more additional structure, but also more constraints (see descent). The remarkable fact is however that the axioms of a category of fibrant objects are very light-weight and tractable, and certainly sufficient for our purposes here. Establishing a series of lemmas and propositions that exhibit the structure implied by a category of fibrant objects is our goal here.

This is described at


In the previous section we had finally obtained the homotopy category Ho W(SSh(C))Ho_W(SSh(C)) of locally Kan simplicial sheaves, whose

But more. We have actually obtained a concrete formula for the hom-sets in this category. An element in the hom-set Ho(SSh(C))(X,A)Ho(SSh(C))(X,A)

  • is a (homotopy class of a) “cover” YWXY \stackrel{\in W}{\to} X of XX: an object weakly equivalent to XX;

  • equipped with a (homotopy class of a morphism g:YAg : Y \to A to AA out of this cover.

And two such pairs (Y,g)(Y,g), (Y,g)(Y',g') are identified, if there is a finer cover YWYWYY \stackrel{\in W}{\leftarrow} Y''\stackrel{\in W}{\to} Y' pulled back to which gg and gg' coincide.

It turns out that this captures an important phenonon. To emphasize this, we now pass to the following terminology:

  • we write H:=Ho W(SSh(C))H := Ho_W(SSh(C))

  • a representative (Y,g)(Y,g) in H(X,A)H(X,A) is called a cocycle

  • its class in H(X,A)H(X,A) is called its cohomology class

  • H(X,A)H(X,A) itself is the cohomology of XX with coefficients in AA.

A large number of concepts is subsumed by this notion of cohomology given by hom-sets in homotopy categories of \infty-stacks. We discuss various of them at

After considering the general definition, we look at very concrete descriptions of cocycles by considering

This leads then over seamlessly to the discussion of next: principal infinity-bundles in the next section

Last revised on July 8, 2009 at 17:17:01. See the history of this page for a list of all contributions to it.