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cohomology with constant coefficients

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cohomology

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Definition

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

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

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

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

H(X,𝒜):=π 0H(X,LConst𝒜). H(X,\mathcal{A}) := \pi_0 \mathbf{H}(X, LConst \mathcal{A}) \,.

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

Remark

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

Properties

Observation

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

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

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

A cocycle

(˜:XLConst𝒜)(:Π(X)𝒜) (\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 XX with coefficients in 𝒜\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.

Created on November 8, 2010 17:22:37 by Urs Schreiber (131.211.232.76)