Schreiber
de Rham cohomology in an (∞,1)-topos

Contents

In a locally contractible (∞,1)-topos H\mathbf{H} a cocycle in (nonabelian) de Rham cohomology is a cocycle Π(X)A\Pi(X) \to A in flat differential cohomology whose underlying cocycle XΠ(X)AX \hookrightarrow \Pi(X) \to A in (nonabelian) cohomology is trivial: it encodes a trivial principal ∞-bundle with possibly nontrivial but flat connection.

If H\mathbf{H} is a smooth (∞,1)-topos, then nonabelian deRham cocycles are represented by flat? ∞-Lie algebroid valued differential forms ω\omega:

Π inf(X) ω 𝔞 Π(X) A. \array{ \mathbf{\Pi}_{inf}(X) &\stackrel{\omega}{\to}& \mathfrak{a} \\ \downarrow && \downarrow \\ \mathbf{\Pi}(X) &\to& A } \,.

If A=B nR/ZA = \mathbf{B}^n R/Z then ω\omega is an ordinary closed n-form.

For more see differential cohomology in an (∞,1)-topos.

Idea

A differential 1-form AΩ 1(X)A \in \Omega^1(X) on a smooth manifold XX may be thought of as a connection on the trivial U(1)U(1)- or \mathbb{R}-principal bundle on XX.

Similarly a differential 2-form BΩ 2(X)B \in \Omega^2(X) on a manifold XX may be thought of as a connection on the trivial U(1)U(1)-bundle gerbe on XX; or on the trivial B(1)\mathbf{B}(1)-principal 2-bundle.

This pattern continues: a differential nn-form is the same as a connection on a trivial B nU(1)\mathbf{B}^n U(1)-principal ∞-bundle.

Moreover this pattern generalizes to GG-principal bundles for nonabelian groups GG:

for 𝔤\mathfrak{g} the Lie algebra of a Lie group GG – possibly nonabelian – a Lie-algebra valued 1-form AΩ 1(X,g)A \in \Omega^1(X,g) may be thought of as a connection on the trivial GG-principal bundle on XX.

While it may seem that the notion of differential form is more fundamental than that of a connection, in the context of differential nonabelian cohomology in an arbitrary path-structured (∞,1)-topos? the most fundamental notion of a differential cocycle is that of a flat connection on a principal ∞-bundle : on an space XX this is simply given by a morphism Π(X)A\Pi(X) \to A from the path ∞-groupoid to the given coefficient object AA.

The underlying principal ∞-bundle is that characterized by the cocycle that is given by the composite morphism XΠ(X)AX \to \Pi(X) \to A.

We may therefore characterize flat connections on trivial AA-principal ∞-bundles as those morphisms Π(X)A\Pi(X) \to A for which the composite XΠ(X)AX \to \Pi(X) \to A trivializes. This way we characterize AA-valued deRham cohomology in the (∞,1)-topos H\mathbf{H}.

Definition

Fix a model for the (∞,1)-topos H\mathbf{H} in terms of the local model structure on simplicial presheaves SPSh(C) locSPSh(C)^{loc} as described at path ∞-groupoid.

Definition (deRham differential refinement)

For ASPSh(C)A \in SPSh(C) a pointed object with point pt A:*Apt_A : {*} \to A define A dRSPSh(C)A_{dR} \in SPSh(C) by

A dR:U[I,SPSh(C)](U Π(U),* pt A A). A_{dR} : U \mapsto [I,SPSh(C)] \left( \array{ U \\ \downarrow \\ \Pi(U) } \,,\; \array{ {*} \\ \downarrow^{\mathrlap{pt_A}} \\ A } \right) \,.

This we call the de Rham differential refinement of AA.

The cohomology with coefficients in A dRA_{dR}

H dR(X,A):=π 0H(X,A dR) H_{dR}(X,A) := \pi_0 \mathbf{H}(X,A_{dR})

we call AA-valued de Rham cohomology

Remark

(de Rham cohomology in terms of differential forms)

The definition does not actually presuppose that the ambient (∞,1)-topos is a smooth (∞,1)-topos in which a concrete notion of ∞-Lie algebroid valued differential forms exists. It defines a notion of “de Rham cohomology” even in the absence of an ordinary notion of differential forms.

But if H\mathbf{H} does happen to be a smooth (∞,1)-topos then both notions are compatible.

Revised on May 5, 2010 22:29:17 by Urs Schreiber (87.212.203.135)