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May 26, 2010

Nonabelian Cohomology in Three (∞,1)-Toposes

Posted by Urs Schreiber

For XX a topological space and AA an ∞-groupoid, the standard way to define the nonabelian cohomology of XX with coefficients in AA is to define it as the intrinsic cohomology as seen in ∞Grpd \simeq Top:

H(X,A):=π 0Top(X,|A|)π 0Func(SingX,A), H(X,A) := \pi_0 Top(X, |A|) \simeq \pi_0 \infty Func(Sing X, A) \,,

where |A||A| is the geometric realization of AA and SingXSing X the fundamental ∞-groupoid of XX.

But both XX and AA here naturally can be regarded, in several ways, as objects of (∞,1)-sheaf (∞,1)-toposes H=Sh (,1)(C)\mathbf{H} = Sh_{(\infty,1)}(C) over nontrivial (∞,1)-sites CC. The intrinsic cohomology of such H\mathbf{H} is a nonabelian sheaf cohomology.

The following discusses two such choices for H\mathbf{H} such that the corresponding nonabelian sheaf cohomology coincides with H(X,A)H(X,A) (for paracompact XX).

Petit (,1)(\infty,1)-sheaf (,1)(\infty,1)-topos

For XX a topological space and Op(X)Op(X) its category of open subsets equipped with the canonical structure of an (∞,1)-site, let

H:=Sh (,1)(X):=Sh (,1)(Op(X)) \mathbf{H} := Sh_{(\infty,1)}(X) := Sh_{(\infty,1)}(Op(X))

be the (∞,1)-category of (∞,1)-sheaves on XX. The space XX itself is naturally identified with the terminal object X=*Sh (,1)(X)X = * \in Sh_{(\infty,1)}(X). This is the petit topos incarnation of XX.


(LConstΓ):Sh (,1)(X)ΓLConstGrpd (LConst \dashv \Gamma) : Sh_{(\infty,1)}(X) \stackrel{\overset{LConst}{\leftarrow}}{\underset{\Gamma}{\to}} \infty Grpd

for the global sections terminal geometric morphism.

Under the constant (∞,1)-sheaf functor LConstLConst an ∞-groupoid AGrpdA \in \infty Grpd is regarded as an object LConstASh (,1)(X)LConst A \in Sh_{(\infty,1)}(X).

There is therefore the intrinsic cohomology of the (,1)(\infty,1)-topos Sh (,1)(X)Sh_{(\infty,1)}(X) with coefficients in the constant (∞,1)-sheaf on AA

H(X,A):=π 0Sh (,1)(X)(X,LConstA). H'(X,A) := \pi_0 Sh_{(\infty,1)}(X)(X, LConst A) \,.

Notice that since XX is in fact the terminal object of Sh (,1)(X)Sh_{(\infty,1)}(X) and that Sh (,1)(X)(X,)Sh_{(\infty,1)}(X)(X,-) is the global sections functor, this is equivalently

π 0ΓLConstA. \cdots \simeq \pi_0 \Gamma LConst A \,.

Theorem (Jacob Lurie)

If XX is a paracompact space, then these two definitins of nonabelian cohomology of XX with constant coefficients AGrpdA \in \infty Grpd agree:

H(X,A):=π 0Grpd(SingX,A)Sh (,1)(X)(X,LConstA). H(X,A) := \pi_0 \infty Grpd(Sing X,A) \simeq Sh_{(\infty,1)}(X)(X,LConst A) \,.

This is theorem in Higher Topos Theory.

Gros (,1)(\infty,1)-sheaf (,1)(\infty,1)-topos

Another alternative is to regard the space XX as an object in the gros (∞,1)-sheaf topos Sh (,1)(CartSp)Sh_{(\infty,1)}(CartSp) over the site CartSp, as described at ∞-Lie groupoid. This has the special property that it is a locally ∞-connected (∞,1)-topos, which means that the terminal geometric morphism is an essential geometric morphism

(ΠLConstΓ):Sh (,1)(CartSp)ΓLConstΠGrpd, (\Pi \dashv LConst \dashv \Gamma) : Sh_{(\infty,1)}(CartSp) \stackrel{\overset{\Pi}{\to}}{\stackrel{\overset{LConst}{\leftarrow}}{\underset{\Gamma}{\to}}} \infty Grpd \,,

with the further left adjoint Π\Pi to LConstLConst being the intrinsic path ∞-groupoid functor . The intrinsic nonabelian cohomology in there also coincides with nonabelian cohomology in TopTop; even the full cocycle ∞-groupoids are equivalent:

Theorem (my claim)

For a paracompact manifold XX we have an equivalence of cocycle ∞-groupoids

Sh (,1)(CartSp)(X,LConstA)Top(X,|A|) Sh_{(\infty,1)}(CartSp)(X, LConst A) \simeq Top(X, |A|)

and hence in particular an isomorphism on cohomology

H(X,A)π 0Sh (,1)(CartSp)(X,LConstA) H(X,A) \simeq \pi_0 Sh_{(\infty,1)}(CartSp)(X, LConst A)


The key point is that for a paracompact manifold XX, the nerve theorem asserts that Π(X)\Pi(X) is weakly homotopy equivalent to SingXSing X, the standard fundamental ∞-groupoid of XX. This is discussed in detail in the section geometric realization at path ∞-groupoid.

Using this, the statement follows by the (∞,1)-adjunction (ΠLConst)(\Pi \dashv LConst), that is discussed in detail at Unstructured homotopy ∞-groupoid.


In fact, for XX locally contractible, it should be true that also Sh (,1)(X)Sh_{(\infty,1)}(X) is a locally ∞-connected (∞,1)-topos and one can play the same adjunction game here. Then the above amounts to the statement that for XX regarded as an object of Sh (,1)(X)Sh_{(\infty,1)}(X) we have under the left adjoint Π\Pi of LConstLConst that Π(X)Grpd\Pi(X) \in \infty Grpd is equivalent to SingXSing X. This is the old Artin-Mazur theorem in slight disguise.

Posted at May 26, 2010 1:26 PM UTC

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Re: Nonabelian Cohomology in Three (∞,1)-Toposes

This is a very nice picture.

I can’t resist mentioning another way of looking at the first equivalence. Suppose that XX is a locally compact CW complex. In particular, this implies that it is hereditarily m-cofibrant, i.e. every open subset of XX has the homotopy type of a CW complex. That’s what you need in order to conclude that taking sheaves of sections of spaces over XX is well-behaved homotopically, since only m-cofibrant spaces are good for mapping out of homotopically. You can then prove (see this paper of mine) that the “sheaf of sections” functor

Top/X[Op(X) op,sSet] Top/X \to [Op(X)^{op},sSet]

is the right adjoint in a right Quillen embedding, i.e. a Quillen adjunction whose derived right adjoint is fully faithful. In other words, the homotopy theory of spaces over XX embeds in the homotopy theory of (,1)(\infty,1)-sheaves on XX. One can also identify its image as consisting of the locally constant (,1)(\infty,1)-sheaves. This is a homotopical version of the identification of covering spaces with locally constant 1-sheaves.

Furthermore, if f:XYf\colon X\to Y is a map of such spaces, then the pullback functor f *:Top/YTop/Xf^\ast\colon Top/Y \to Top/X agrees with the inverse image functor f *f^\ast for (,1)(\infty,1)-sheaves. In particular, when YY is a point and AA a space, then the constant (,1)(\infty,1)-sheaf Const(A)Const(A) is identified with (the sheaf of sections of) the space X *A=X×AX^\ast A = X\times A over XX. Therefore, the nonabelian cohomology of XX with coefficients in Const(A)Const(A) is the same as the maps in Top/XTop/X from XX (the terminal object of Top/XTop/X) to X *AX^\ast A. Since the left adjoint of X *:TopTop/XX^\ast:Top \to Top/X just forgets the structure map to XX, this is the same as maps in TopTop from XX to AA. Thereby we recover Lurie’s theorem, in the case when XX is a locally compact CW complex.

Posted by: Mike Shulman on May 26, 2010 4:13 PM | Permalink | Reply to this

Re: Nonabelian Cohomology in Three (∞,1)-Toposes

Thanks, Mike. I should have mentioned this.

For the time being I moved your comment, only slightly edited, into a new section Nonabelian sheaf cohomology with constant coefficients – In terms of covering spaces at nnLab:cohomology.

This should be exanded, but i need to take care of something else for the moment…

Posted by: Urs Schreiber on May 26, 2010 6:01 PM | Permalink | Reply to this


m stands for…???

Posted by: jim stasheff on May 27, 2010 1:41 PM | Permalink | Reply to this

Re: m-cofibrant

the ‘m’ in m-cofibrant stands for the mixed model structure. It’s a mash-up of the standard (Quillen) model structure on TopTop and the Strom model structure.

Posted by: David Roberts on May 28, 2010 1:47 AM | Permalink | Reply to this

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