Schreiber the homotopy category of infinity-stacks

previous: infinity-groupoids

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So far we have

• described the fundamental concept of a category of presheaves with values in Set;

• characterized geometric embeddings into presheaf categories as categories of sheaves;

  • and saw that the category of sheaves is equivalently described as the homotopy category of the category of presheaves relative to the local isomorphisms.

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.

• In order to do so, we shall proceed in complete analogy to what we had previously:

  • we describe now the category Infinity-Grpd of infinity-groupoids generalizing the category Set of sets, modeled on simplicial sets that are Kan complexes;

  • then characterize weak equivalences of infinity-groupoids;

  • and then describe the homotopy category $H$ of the category of presheaves with values in $\infty$-groupoids as the homotopy category with respect to weak equivalences given by stalkwise weak equivalences of $\infty$-groupoids.

• This gives us a notion of cohomology of one sheaf of $\infty$-groupoids $X$ with coefficients in another such sheaf $A$ simply as the hom-set

  $$H(X,A) \,.$$
  • This is sometimes called nonabelian cohomology. It generalizes abelian sheaf cohomology, that we discuss in more detail later. It describes the homotopy classes of morphisms between the infinity-stackifications of $X$ and $A$,
    $$H(X,A) = \pi_0 \mathbf{H}(\bar X, \bar A) \,.$$

    We speak of the Brown model for infinity-stacks, often also called the Brown-Joyal-Jardine model, since Joyal and Jardine refined the tools here to a model structure on simplicial presheaves.

• In the sequel we then discuss how the category $Ch_+(Ab)$ of non-negatively chain complexes of abelian groups provides a particularly tractable subcategory of Infinity-Grpd,

  $$DoldKan : Ch_+(Ab) \hookrightarrow \infty Grpd \,.$$

  This is the famous Dold-Kan correspondence.

  • This will allow to understand abelian sheaf cohomology of a sheaf $F$ of complexes of abelian groups simply as the restriction of the above situation to coefficient sheaves $A_F$ in the image of the Dold-Kan inclusion.

Brown category of simplicial sheaves

We now describe

• the structure of a homotopica category on $SSh(C)$ simplicial sheaves on some site $C$;

We are intersted in finding the corresponding homotopy $Ho(SSh(C))$. 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

• an extension of that homotoptical structure to that of a category of fibrant objects.

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

• category of fibrant objects.

Cohomology

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

• objects we may think of as infinity-stacks

• morphisms are equivalence classes of morphisms between infinity-stacks.

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

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

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

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

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

• we write $H := Ho_W(SSh(C))$

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

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

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

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

• cohomology.

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

• Cech cohomology .

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