The n-Category Café

Petit $(\mathrm{\infty },1)$-sheaf $(\mathrm{\infty },1)$-topos

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

$$H:={\mathrm{Sh}}_{(\mathrm{\infty },1)}(X):={\mathrm{Sh}}_{(\mathrm{\infty },1)}(\mathrm{Op}(X))$$

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

Write

$$(\mathrm{LConst}⊣\Gamma ):{\mathrm{Sh}}_{(\mathrm{\infty },1)}(X)\stackrel{\stackrel{\mathrm{LConst}}{\leftarrow }}{\underset{\Gamma }{\to }}\mathrm{\infty }\mathrm{Grpd}$$

for the global sections terminal geometric morphism.

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

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

$$H\prime (X,A):={\pi }_0{\mathrm{Sh}}_{(\mathrm{\infty },1)}(X)(X,\mathrm{LConst}A).$$

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

$$\cdots \simeq {\pi }_0\Gamma \mathrm{LConst}A.$$

Theorem (Jacob Lurie)

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

$$H(X,A):={\pi }_0\mathrm{\infty }\mathrm{Grpd}(\mathrm{Sing}X,A)\simeq {\mathrm{Sh}}_{(\mathrm{\infty },1)}(X)(X,\mathrm{LConst}A).$$

This is theorem 7.1.0.1 in Higher Topos Theory.

Gros $(\mathrm{\infty },1)$-sheaf $(\mathrm{\infty },1)$-topos

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

$$(\Pi ⊣\mathrm{LConst}⊣\Gamma ):{\mathrm{Sh}}_{(\mathrm{\infty },1)}(\mathrm{CartSp})\stackrel{\stackrel{\Pi }{\to }}{\stackrel{\stackrel{\mathrm{LConst}}{\leftarrow }}{\underset{\Gamma }{\to }}}\mathrm{\infty }\mathrm{Grpd},$$

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

Theorem (my claim)

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

$${\mathrm{Sh}}_{(\mathrm{\infty },1)}(\mathrm{CartSp})(X,\mathrm{LConst}A)\simeq \mathrm{Top}(X,\mid A\mid )$$

and hence in particular an isomorphism on cohomology

$$H(X,A)\simeq {\pi }_0{\mathrm{Sh}}_{(\mathrm{\infty },1)}(\mathrm{CartSp})(X,\mathrm{LConst}A)$$

Proof

The key point is that for a paracompact manifold $X$, the nerve theorem asserts that $\Pi (X)$ is weakly homotopy equivalent to $\mathrm{Sing}X$, 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 $(\Pi ⊣\mathrm{LConst})$, that is discussed in detail at Unstructured homotopy ∞-groupoid.

Remark

In fact, for $X$ locally contractible, it should be true that also ${\mathrm{Sh}}_{(\mathrm{\infty },1)}(X)$ 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 ${\mathrm{Sh}}_{(\mathrm{\infty },1)}(X)$ we have under the left adjoint $\Pi$ of $\mathrm{LConst}$ that $\Pi (X)\in \mathrm{\infty }\mathrm{Grpd}$ is equivalent to $\mathrm{Sing}X$. This is the old Artin-Mazur theorem in slight disguise.