### Nonabelian Cohomology in Three (∞,1)-Toposes

#### Posted by Urs Schreiber

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

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

where $|A|$ is the geometric realization of $A$ and $Sing X$ the fundamental ∞-groupoid of $X$.

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

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

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

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

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

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

Write

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

for the global sections terminal geometric morphism.

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

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

$H'(X,A) := \pi_0 Sh_{(\infty,1)}(X)(X, LConst A) \,.$

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

$\cdots \simeq \pi_0 \Gamma LConst A \,.$

**Theorem** (Jacob Lurie)

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

$H(X,A) := \pi_0 \infty Grpd(Sing X,A) \simeq Sh_{(\infty,1)}(X)(X,LConst A) \,.$

This is theorem 7.1.0.1 in Higher Topos Theory.

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

Another alternative is to regard the space $X$ as an object in the gros (∞,1)-sheaf topos $Sh_{(\infty,1)}(CartSp)$ 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

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

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

**Theorem** (my claim)

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

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

and hence in particular an isomorphism on cohomology

$H(X,A) \simeq \pi_0 Sh_{(\infty,1)}(CartSp)(X, LConst A)$

**Proof**

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

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

**Remark**

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

## 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 $X$ is a locally compact CW complex. In particular, this implies that it is

hereditarily m-cofibrant, i.e. every open subset of $X$ 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 $X$ 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$Top/X \to [Op(X)^{op},sSet]$

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 $X$ embeds in the homotopy theory of $(\infty,1)$-sheaves on $X$. One can also identify its image as consisting of the locally constant $(\infty,1)$-sheaves. This is a homotopical version of the identification of covering spaces with locally constant 1-sheaves.

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