nLab cohomology with constant coefficients

Definition

The cohomology of an object $X$ in an (∞,1)-topos $\mathbf{H}$ with coefficients in another object $A$ is the set of connected components of the hom-space from $X$ to $A$.

Notice that for every (∞,1)-sheaf (∞,1)-topos there is the terminal global section (∞,1)-geometric morphism

$$(LConst \dashv \Gamma) : \mathbf{H} \stackrel{\overset{LConst}{\leftarrow}}{\underset{\Gamma}{\to}} \infty Grpd \,.$$

Definition

In the case that $A = LConst \mathcal{A}$ for $\mathcal{A} \in$ ∞Grpd we say that

$$H(X,\mathcal{A}) := \pi_0 \mathbf{H}(X, LConst \mathcal{A}) \,.$$

the cohomology of $X$ with constant coefficients, constant on $\mathcal{A}$

Remark

For $\mathbf{H}$ the (∞,1)-sheaf (∞,1)-topos over an (∞,1)-site $C$, we have that $LConst \mathcal{A}$ is the constant ∞-stack over $C$. Notice that this is the ∞-stackification of the (∞,1)-presheaf that is literally constant (as an (∞,1)-functor) on $\mathcal{A}$. So unless over $C$ constant presheaves already satisfy descent (as for instance over an (∞,1)-cohesive site) the object $LConst \mathcal{A}$ is not itself given by a constant functor on $C^{op}$.

Properties

Observation

If $\mathbf{H}$ is a locally ∞-connected (∞,1)-topos in that we have a further left adjoint (∞,1)-functor $\Pi$ to $LConst$

$$(\Pi \dashv LConst \dashv \Gamma) : \mathbf{H} \to \infty Grpd$$

then by the adjunction hom-equivalence $\mathbf{H}(X, LConst \mathcal{A}) \simeq \infty Grpd(\Pi(X), \mathcal{A})$ we have that cohomology with constant coefficients in $\mathbf{H}$ is equivalently the cohomology of the fundamental ∞-groupoid in a locally ∞-connected (∞,1)-topos $\Pi(X)$ in ∞Grpd with coefficients in $\mathcal{A}$.

A cocycle

$$(\tilde \nabla : X \to LConst \mathcal{A}) \sim (\nabla : \Pi(X) \to \mathcal{A})$$

in this cohomology may then be identified with what is called a local system on $X$ with coefficients in $\mathcal{A}$. So in this case we have

• Cohomology with constant coefficients classifies local systems.

Examples

The relation between cohomology with local coefficients cohomology in ∞Grpd $\simeq$ Top is discussed at nonabelian cohomology in the section nonabelian sheaf cohomology.