The n-Category Café

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.