nLab
cohesive (infinity,1)-topos -- structures II

Context

Cohesive \infty-Toposes

cohesive topos

cohesive (∞,1)-topos

cohesive homotopy type theory

Backround

Definition

Presentation over a site

Structures in a cohesive (,1)(\infty,1)-topos

structures in a cohesive (∞,1)-topos

Structures with infinitesimal cohesion

infinitesimal cohesion?

Models

This is a sub-section of the entry cohesive (∞,1)-topos . See there for background and context


Contents

Structures in a cohesive (,1)(\infty,1)-topos

This continues the list of structures whose first part is at cohesive (infinity,1)-topos -- structures .

Concordance

Since H\mathbf{H} is an (∞,1)-topos it carries canonically the structure of a cartesian closed (∞,1)-category. For
X,YHX, Y \in \mathbf{H}, write Y XHY^X \in \mathbf{H} for the corresponding internal hom.

Since Π:H\Pi : \mathbf{H} \to ∞Grpd preserves products, we have for all X,Y,ZHX,Y, Z \in \mathbf{H} canonically induced a morphism

Π(Y X)×Π(Z Y)Π(Y X×Z Y)Π(comp X,Y,Z)Π(Z X). \Pi(Y^X) \times \Pi(Z^Y) \stackrel{\simeq}{\to} \Pi(Y^X \times Z^Y) \stackrel{\Pi(comp_{X,Y,Z})}{\to} \Pi(Z^X) \,.

This should yield an (∞,1)-category H˜\tilde \mathbf{H} with the same objects as H\mathbf{H} and hom-\infty-groupoids defined by

H˜(X,Y):=Π(Y X). \tilde \mathbf{H}(X,Y) := \Pi(Y^X) \,.

We have that

H˜(X,BG)=Π(BG X) \tilde \mathbf{H}(X,\mathbf{B}G) = \Pi(\mathbf{B}G^X)

is the \infty-groupoid whose objects are GG-principal ∞-bundles on XX and whose morphisms have the interpretaton of GG-principal bundles on the cylinder X×IX \times I. These are concordances of \infty-bundles.

Geometric homotopy and Galois theory

We discuss canonical internal realizations of the notions of homotopy group, local system and Galois theory in H\mathbf{H}.

Definition

For H\mathbf{H} a locally ∞-connected (∞,1)-topos and XHX \in \mathbf{H} an object, we call ΠX\Pi X \in ∞Grpd the fundamental ∞-groupoid in a locally ∞-connected (∞,1)-topos of XX.

The (categorical) homotopy groups of Π(X)\Pi(X) we call the geometric homotopy groups of XX

π geom(X):=π (Π(X)). \pi_\bullet^{geom}(X) := \pi_\bullet(\Pi (X)) \,.
Definition

For ||:\vert - \vert : ∞Grpd \stackrel{\simeq}{\to} Top the homotopy hypothesis-equivalence we write

|X|:=|ΠX|Top \vert X \vert := \vert \Pi X \vert \in Top

and call this the topological geometric realization of XX, or just the geometric realization for short.

Note

In presentations of H\mathbf{H} by a model structure on simplicial presheaves as in prop. \ref{SimplicialPresheavesOverInfinityCohesviveSite} this abstract notion reproduces the notion of geometric realization of ∞-stacks in (Simpson). See remark 2.22 in (SimpsonTeleman).

Definition

We say a geometric homotopy between two morphism f,g:XYf,g : X \to Y in H\mathbf{H} is a diagram

X (Id,i) f X×I η Y (Id,o) g X \array{ X \\ \downarrow^{\mathrlap{(Id,i)}} & \searrow^{\mathrlap{f}} \\ X \times I &\stackrel{\eta}{\to}& Y \\ \uparrow^{\mathrlap{(Id,o)}} & \nearrow_{\mathrlap{g}} \\ X }

such that II is geometrically connected, π 0 geom(I)=*\pi_0^{geom}(I) = *.

Proposition

If f,g:XYf,g : X\to Y are geometrically homotopic in H\mathbf{H}, then their images Π(f),Π(g)\Pi(f), \Pi(g) are equivalent in Grpd\infty Grpd.

Proof

By the condition that Π\Pi preserves products in a cohesive (,1)(\infty,1)-topos we have that the image of the geometric homotopy in Grpd\infty Grpd is a diagram of the form

Π(X) (Id,Π(i)) Π(f) Π(X)×Π(I) Π(η) Π(Y) (Id,Π(o)) Π(g) Π(X). \array{ \Pi(X) \\ \downarrow^{\mathrlap{(Id,\Pi(i))}} & \searrow^{\mathrlap{\Pi(f)}} \\ \Pi(X) \times \Pi(I) &\stackrel{\Pi(\eta)}{\to}& \Pi(Y) \\ \uparrow^{\mathrlap{(Id,\Pi(o))}} & \nearrow_{\mathrlap{\Pi(g)}} \\ \Pi(X) } \,.

Now since Π(I)\Pi(I) is connected by assumption, there is a diagram

* Id Π(i) * Π(I) Id Π(o) * \array{ && * \\ & {}^{\mathllap{Id}}\nearrow & \downarrow^{\mathrlap{\Pi(i)}} \\ * &\to& \Pi(I) \\ & {}_{\mathllap{Id}}\searrow & \uparrow^{\mathrlap{\Pi(o)}} \\ && * }

in ∞Grpd.

Taking the product of this diagram with Π(X)\Pi(X) and pasting the result to the above image Π(η)\Pi(\eta) of the geometric homotopy constructs the equivalence Π(f)Π(g)\Pi(f) \Rightarrow \Pi(g) in Grpd\infty Grpd.

Proposition

For H\mathbf{H} a locally ∞-connected (∞,1)-topos, also all its objects XHX \in \mathbf{H} are locally \infty-connected, in the sense their petit over-(∞,1)-toposes H/X\mathbf{H}/X are locally \infty-connected.

The two notions of fundamental \infty-groupoids of XX induced this way do agree, in that there is a natural equivalence

Π X(XH/X)Π(XH). \Pi_X(X \in \mathbf{H}/X) \simeq \Pi(X \in \mathbf{H}) \,.
Proof

By the general facts recalled at étale geometric morphism we have a composite essential geometric morphism

(Π XΔ XΓ X):H /XX *X *X !HΓΔΠGrpd (\Pi_X \dashv \Delta_X \dashv \Gamma_X) : \mathbf{H}_{/X} \stackrel{\overset{X_!}{\to}}{\stackrel{\overset{X^*}{\leftarrow}}{\underset{\X_*}{\to}}} \mathbf{H} \stackrel{\overset{\Pi}{\to}}{\stackrel{\overset{\Delta}{\leftarrow}}{\underset{\Gamma}{\to}}} \infty Grpd

and X !X_! is given by sending (YX)H/X(Y \to X) \in \mathbf{H}/X to YHY \in \mathbf{H}.

Definition

For κ\kappa a regular cardinal write

CoreGrpd κGrpd Core \infty Grpd_\kappa \in \infty Grpd

for the ∞-groupoid of κ\kappa-small ∞-groupoids: the core of the full sub-(∞,1)-category of ∞Grpd on the κ\kappa-small ones.

Remark

We have

CoreGrpd κ iBAut(F i), Core \infty Grpd_\kappa \simeq \coprod_i \mathbf{B} Aut(F_i) \,,

where the coproduct ranges over all κ\kappa-small homotopy types [F i][F_i] and Aut(F i)Aut(F_i) is the automorphism ∞-group of any representative F iF_i of [F i][F_i].

Definition

For XHX \in \mathbf{H} write

LConst(X):=H(X,DiscCoreGrpd κ). LConst(X) := \mathbf{H}(X, Disc Core \infty Grpd_\kappa) \,.

We call this the \infty-groupoid of locally constant ∞-stacks on XX.

Observation

Since DiscDisc is left adjoint and right adjoint it commutes with coproducts and with delooping and therefore

DiscCoreGrpd κ iBDiscAut(F i). Disc Core \infty Grpd_\kappa \simeq \coprod_i \mathbf{B} Disc Aut(F_i) \,.

Therefore a cocycle PLConst(X)P \in LConst(X) may be identified on each geometric connected component of XX as a DiscAut(F i)Disc Aut(F_i)-principal ∞-bundle PXP \to X over XX for the ∞-group object DiscAut(F i)HDisc Aut(F_i) \in \mathbf{H}. We may think of this as an object PH/XP \in \mathbf{H}/X in the little topos over XX. This way the objects of LConst(X)LConst(X) are indeed identified \infty-stacks over XX.

The following proposition says that the central statements of Galois theory hold for these canonical notions of geometric homotopy groups and locally constant \infty-stacks.

Proposition

For H\mathbf{H} locally ∞-connected and ∞-connected, we have

  • a natural equivalence

    LConst(X)Grpd(Π(X),Grpd κ) LConst(X) \simeq \infty \mathrm{Grpd}(\Pi(X), \infty Grpd_\kappa)

    of locally constant \infty-stacks on XX with \infty-permutation representations of the fundamental ∞-groupoid of XX (local systems on XX);

  • for every point x:*Xx : * \to X a natural equivalence of the endomorphisms of the fiber functor x *x^* and the loop space of Π(X)\Pi(X) at xx

    End(x *:LConst(X)Grpd)Ω xΠ(X). End( x^* : LConst(X) \to \infty Grpd ) \simeq \Omega_x \Pi(X) \,.
Proof

The first statement is just the adjunction (ΠDisc)(\Pi \dashv Disc).

LConst(X) :=H(X,DiscCoreGrpd κ) Grpd(Π(X),CoreGrpd κ) Grpd(Π(X),Grpd κ). \begin{aligned} LConst(X) & := \mathbf{H}(X, Disc Core \infty Grpd_\kappa) \\ & \simeq \infty Grpd(\Pi(X), Core \infty Grpd_\kappa) \\ & \simeq \infty Grpd(\Pi(X), \infty Grpd_\kappa) \end{aligned} \,.

Using this and that Π\Pi preserves the terminal object, so that the adjunct of (*XDiscCoreGrpd κ)(* \to X \to Disc Core \infty Grpd_\kappa) is (*Π(X)Grpd κ)(* \to \Pi(X) \to \infty Grpd_\kappa) the second statement follows with an iterated application of the (∞,1)-Yoneda lemma (this is pure Tannaka duality as discussed there):

The fiber functor x *:Func(Π(X),Grpd)Grpdx^* : Func(\Pi(X), \infty Grpd) \to \infty Grpd evaluates an (,1)(\infty,1)-presheaf on Π(X) op\Pi(X)^{op} at xΠ(X)x \in \Pi(X). By the (∞,1)-Yoneda lemma this is the same as homming out of j(x)j(x), where j:Π(X) opFunc(Π(X),Grpd)j : \Pi(X)^{op} \to Func(\Pi(X), \infty Grpd) is the (∞,1)-Yoneda embedding:

x *Hom PSh(Π(X) op)(j(x),). x^* \simeq Hom_{PSh(\Pi(X)^{op})}(j(x), -) \,.

This means that x *x^* itself is a representable object in PSh(PSh(Π(X) op) op)PSh(PSh(\Pi(X)^{op})^{op}). If we denote by j˜:PSh(Π(X) op) opPSh(PSh(Π(X) op) op)\tilde j : PSh(\Pi(X)^{op})^{op} \to PSh(PSh(\Pi(X)^{op})^{op}) the corresponding Yoneda embedding, then

x *j˜(j(x)). x^* \simeq \tilde j (j (x)) \,.

With this, we compute the endomorphisms of x *x^* by applying the (∞,1)-Yoneda lemma two more times:

Endx * End PSh(PSh(Π(X) op) op)(j˜(j(x))) End(PSh(Π(X)) op)(j(x)) End Π(X) op(x,x) Aut xΠ(X) =:Ω xΠ(X). \begin{aligned} End x^* & \simeq End_{PSh(PSh(\Pi(X)^{op})^{op})} (\tilde j(j (x))) \\ & \simeq End(PSh(\Pi(X))^{op}) (j(x)) \\ & \simeq End_{\Pi(X)^{op}}(x,x) \\ & \simeq Aut_x \Pi(X) \\ & =: \Omega_x \Pi(X) \end{aligned} \,.

van Kampen theorem

A higher van Kampen theorem asserts that passing to fundamental ∞-groupoids preserves certain colimits.

On a cohesive (,1)(\infty,1)-topos H\mathbf{H} the fundamental \infty-groupoid functor Π:HGrpd\Pi : \mathbf{H} \to \infty Grpd is a left adjoint (∞,1)-functor and hence preserves all (∞,1)-colimits.

More interesting is the question which (,1)(\infty,1)-colimits of concrete spaces in

Conc(H)injconcH Conc(\mathbf{H}) \stackrel{\overset{conc}{\leftarrow}}{\underset{inj}{\hookrightarrow}} \mathbf{H}

are preserved by Πinj:Conc(H)Grpd\Pi \circ inj : Conc(\mathbf{H}) \to \infty Grpd. These colimits are computed by first computing them in H\mathbf{H} and then applying the concretization functor. So we have

Observation

Let U :KConc(H)U_\bullet : K \to Conc(\mathbf{H}) be a diagram such that the (∞,1)-colimit lim injU \lim_\to inj \circ U_\bullet is concrete, inj(X)\cdots \simeq inj(X).

Then the fundamental ∞-groupoid of XX is computed as the (,1)(\infty,1)-colimit

Π(X)lim Π(U ). \Pi(X) \simeq {\lim_\to} \Pi(U_\bullet) \,.

In the Examples we discuss the cohesive (,1)(\infty,1)-topos H=(,1)Sh(TopBall)\mathbf{H} = (\infty,1)Sh(TopBall) of topological ∞-groupoids For that case we recover the ordinary higher van Kampen theorem:

Proposition

Let XX be a paracompact or locally contractible topological spaces and U 1XU_1 \hookrightarrow X, U 2XU_2 \hookrightarrow X a covering by two open subsets.

Then under the singular simplicial complex functor Sing:TopSing : Top \to sSet we have a homotopy pushout

Sing(U 1)Sing(U 2) Sing(U 2) Sing(U 1) Sing(X). \array{ Sing(U_1) \cap Sing(U_2) &\to& Sing(U_2) \\ \downarrow && \downarrow \\ Sing(U_1) &\to& Sing(X) } \,.
Proof

We inject the topological space via the external Yoneda embedding

TopSh(TopBalls)H:=(,1)Sh(OpenBalls) Top \hookrightarrow Sh(TopBalls) \hookrightarrow \mathbf{H} := (\infty,1)Sh(OpenBalls)

as a 0-truncated topological ∞-groupoid in the cohesive (,1)(\infty,1)-topos H\mathbf{H}. Being an (∞,1)-category of (∞,1)-sheaves this is presented by the left Bousfield localization Sh(TopBalls,sSet) inj,locSh(TopBalls, sSet)_{inj,loc} of the injective model structure on simplicial sheaves on TopBallsTopBalls (as described at models for ∞-stack (∞,1)-toposes).

Notice that the injection TopSh(TopBalls)Top \hookrightarrow Sh(TopBalls) of topological spaces as concrete sheaves on the site of open balls preserves the pushout X=U 1 U 1U 2U 2X = U_1 \coprod_{U_1 \cap U_2} U_2. (This is effectively the statement that XX as a representable on Diff is a sheaf.) Accordingly so does the further inclusion into Sh(TopBall,sSet)Sh(TopBalls) Δ opSh(TopBall,sSet) \simeq Sh(TopBalls)^{\Delta^{op}} as simplicially constant simplicial sheaves.

Since cofibrations in that model structure are objectwise and degreewise injective maps, it follows that the ordinary pushout diagram

U 1U 2 U 2 U 1 X \array{ U_1 \cap U_2 &\to& U_2 \\ \downarrow && \downarrow \\ U_1 &\to& X }

in Sh(TopBalls,sSet) inj,locSh(TopBalls, sSet)_{inj,loc} has all objects cofibrant and is the pushout along a cofibration, hence is a homotopy pushout (as described there). By the general theorem at (∞,1)-colimit homotopy pushouts model (,1)(\infty,1)-pushouts, so that indeed XX is the (,1)(\infty,1)-pushout

XU 1 U 1U 2U 2H. X \simeq U_1 \coprod_{U_1 \cap U_2} U_2 \in \mathbf{H} \,.

The proposition now follows with the above observation that Π\Pi preserves all (,1)(\infty,1)-colimits and with the statement (from topological ∞-groupoid) that for a topological space (locally contractible or paracompact) we have ΠXSingX\Pi X \simeq Sing X.

Paths and geometric Postnikov towers

The above construction of the fundamental ∞-groupoid of objects in H\mathbf{H} as an object in ∞Grpd may be reflected back into H\mathbf{H}, where it gives a notion of homotopy path n-groupoids and a geometric notion of Postnikov towers of objects in H\mathbf{H}.

Definition

For H\mathbf{H} a locally ∞-connected (∞,1)-topos define the composite adjoint (∞,1)-functors

(Π):=(DiscΠDiscΓ):HH. (\mathbf{\Pi} \dashv \mathbf{\flat}) := (Disc \Pi \dashv Disc \Gamma) : \mathbf{H} \to \mathbf{H} \,.

We say

Write

(τ ni n):H niτ nH (\tau_n \dashv i_n) : \mathbf{H}_{\leq n} \stackrel{\overset{\tau_{n}}{\leftarrow}}{\underset{i}{\hookrightarrow}} \mathbf{H}

for the reflective sub-(∞,1)-category of n-truncated objects and

τ n:Hτ nH nH \mathbf{\tau}_n : \mathbf{H} \stackrel{\tau_n}{\to} \mathbf{H}_{\leq n} \hookrightarrow \mathbf{H}

for the truncation-localization funtor.

We say

Π n:HΠ nHτ nH \mathbf{\Pi}_n : \mathbf{H} \stackrel{\mathbf{\Pi}_n}{\to} \mathbf{H} \stackrel{\mathbf{\tau}_n}{\to} \mathbf{H}

is the homotopy path n-groupoid functor.

We say that the (truncated) components of the (ΠDisc)(\Pi \dashv Disc)-unit

XΠ(X) X \to \mathbf{\Pi}(X)

are the constant path inclusions. Dually we have canonical morphism

AA. \mathbf{\flat}A \to A \,.
Observation

If H\mathbf{H} is cohesive, then \mathbf{\flat} has a right adjoint Γ\mathbf{\Gamma}

(ΠΓ):=(DiscΠDiscΓcoDiscΓ):HΓΠH. (\mathbf{\Pi} \dashv \mathbf{\flat} \dashv \mathbf{\Gamma}) := (Disc \Pi \dashv Disc \Gamma \dashv coDisc \Gamma) : \mathbf{H} \stackrel{\overset{\mathbf{\Pi}}{\to}}{\stackrel{\overset{\mathbf{\flat}}{\leftarrow}}{\underset{\mathbf{\Gamma}}{\to}}} \mathbf{H} \,.

and this makes H\mathbf{H} be \infty-connected and locally \infty-connected over itself.

Proposition

Let H\mathbf{H} be a locally ∞-connected (∞,1)-topos. If XHX \in \mathbf{H} is small-projective then the over-(∞,1)-topos H/X\mathbf{H}/X is

  1. locally ∞-connected;

  2. local.

Proof

The first statement is proven at locally ∞-connected (∞,1)-topos, the second at local (∞,1)-topos.

Proposition

In a cohesive (,1)(\infty,1)-topos H\mathbf{H}, if XX is small-projective then so is its path ∞-groupoid Π(X)\mathbf{\Pi}(X).

Proof

Because of the adjoint triple of adjoint (∞,1)-functors (ΠΓ)(\mathbf{\Pi} \dashv \mathbf{\flat} \dashv \mathbf{\Gamma}) we have for diagram A:IHA : I \to \mathbf{H} that

H(Π(X),lim iA i) H(X,lim iA i) H(X,lim iA i) lim iH(X,A i), \begin{aligned} \mathbf{H}(\mathbf{\Pi}(X), {\lim_\to}_i A_i) & \simeq \mathbf{H}(X, \mathbf{\flat}{\lim_\to}_i A_i) \\ & \simeq \mathbf{H}(X, {\lim_\to}_i \mathbf{\flat} A_i) \\ & \simeq {\lim_\to}_i \mathbf{H}(X, \mathbf{\flat} A_i) \end{aligned} \,,

where in the last step we used that XX is small-projective by assumption.

Definition

For XHX \in \mathbf{H} we say that the geometric Postnikov tower of XX is the Postnikov tower in an (∞,1)-category of Π(X)\mathbf{\Pi}(X):

Π(X)Π 2(X)Π 1(X)Π 0(X). \mathbf{\Pi}(X) \to \cdots \to \mathbf{\Pi}_2(X) \to \mathbf{\Pi}_1(X) \to \mathbf{\Pi}_0(X) \,.

Universal coverings and geometric Whitehead towers

We discuss an intrinsic notion of Whitehead towers in a locally ∞-connected ∞-connected (∞,1)-topos H\mathbf{H}.

Definition

For XHX \in \mathbf{H} a pointed object, the geometric Whitehead tower of XX is the sequence of objects

X ()X (2)X (1)X (0)X X^{\mathbf{(\infty)}} \to \cdots \to X^{\mathbf{(2)}} \to X^{\mathbf{(1)}} \to X^{\mathbf{(0)}} \simeq X

in H\mathbf{H}, where for each nn \in \mathbb{N} the object X (n+1)X^{(n+1)} is the homotopy fiber of the canonical morphism XΠ n+1XX \to \mathbf{\Pi}_{n+1} X to the path n+1-groupoid of XX.

We call X (n+1)X^{\mathbf{(n+1)}} the (n+1)(n+1)-fold universal covering space of XX.

We write X ()X^{\mathbf{(\infty)}} for the homotopy fiber of the untruncated constant path inclusion.

X ()XΠ(X). X^{\mathbf{(\infty)}} \to X \to \mathbf{\Pi}(X) \,.

Here the morphisms X (n+1)X nX^{\mathbf{(n+1)}} \to X^{\mathbf{n}} are those induced from this pasting diagram of (∞,1)-pullbacks

X (n) * X (n1) B nπ n(X) * X Π n(X) Π (n1)(X), \array{ X^{\mathbf{(n)}} &\to& * \\ \downarrow && \downarrow \\ X^{\mathbf{(n-1)}} & \to & \mathbf{B}^n \mathbf{\pi}_n(X) &\to& * \\ \downarrow && \downarrow && \downarrow \\ X &\to& \mathbf{\Pi}_n(X) &\to& \mathbf{\Pi}_{(n-1)}(X) } \,,

where the object B nπ n(X)\mathbf{B}^n \mathbf{\pi}_n(X) is defined as the homotopy fiber of the bottom right morphism.

Proposition

Every object XHX \in \mathbf{H} is covered by objects of the form X ()X^{\mathbf{(\infty)}} for different choices of base points in XX, in the sense that every XX is the (∞,1)-colimit over a diagram whose vertices are of this form.

Proof

Consider the diagram

lim sΠ(X)(i **) lim sΠ(X)* X i Π(X). \array{ {\lim_\to}_{s \in \Pi(X)} (i^* *) &\to& {\lim_\to}_{s \in \Pi(X)} * \\ {}^{\mathllap{\simeq}}\downarrow && \downarrow^{\mathrlap{\simeq}} \\ X &\stackrel{i}{\to}& \mathbf{\Pi}(X) } \,.

The bottom morphism is the constant path inclusion, the (ΠDisc)(\Pi \dashv Disc)-unit. The right morphism is the equivalence in an (∞,1)-category that is the image under DiscDisc of the decomposition lim S*S{\lim_\to}_S * \stackrel{\simeq}{\to} S of every ∞-groupoid as the (∞,1)-colimit (see there) over itself of the (∞,1)-functor constant on the point.

The left morphism is the (∞,1)-pullback along ii of this equivalence, hence itself an equivalence. By universal colimits in the (∞,1)-topos H\mathbf{H} the top left object is the (∞,1)-colimit over the single homotopy fibers i ** si^* *_s of the form X ()X^{\mathbf{(\infty)}} as indicated.

Proposition

The inclusion Π(i **)Π(X)\Pi(i^* *) \to \Pi(X) of the fundamental ∞-groupoid Π(i **)\Pi(i^* *) of each of these objects into Π(X)\Pi(X) is homotopic to the point.

Proof

We apply Π()\Pi(-) to the above diagram over a single vertex ss and attach the (ΠDisc)(\Pi \dashv Disc)-counit to get

Π(i **) * ΠX Π(i) ΠDiscΠ(X) Π(X). \array{ \Pi(i^* *) &\to& &\to& * \\ \downarrow && && \downarrow \\ \Pi X &\stackrel{\Pi(i)}{\to}& \Pi Disc \Pi(X) &\to& \Pi(X) } \,.

Then the bottom morphism is an equivalence by the (ΠDisc)(\Pi \dashv Disc)-zig-zag-identity.

Flat \infty-connections and local systems

We describe for a locally ∞-connected (∞,1)-topos H\mathbf{H} a canonical intrinsic notion of flat connections on ∞-bundles, flat higher parallel transport and higher local systems.

Write (Π):=(DiscΠDiscΓ):HH(\mathbf{\Pi} \dashv\mathbf{\flat}) := (Disc \Pi \dashv Disc \Gamma) : \mathbf{H} \to \mathbf{H} for the adjunction given by the path ∞-groupoid. Notice that this comes with the canonical (ΠDisc)(\Pi \dashv Disc)-unit with components

XΠ(X) X \to \mathbf{\Pi}(X)

and the (DiscΓ)(Disc \dashv \Gamma)-counit with components

AA. \mathbf{\flat} A \to A \,.
Definition

For X,AHX, A \in \mathbf{H} we write

H flat(X,A):=H(ΠX,A) \mathbf{H}_{flat}(X,A) := \mathbf{H}(\mathbf{\Pi}X, A)

and call H flat(X,A):=π 0H flat(X,A)H_{flat}(X,A) := \pi_0 \mathbf{H}_{flat}(X,A) the flat (nonabelian) differential cohomology of XX with coefficients in AA.

We say a morphism :Π(X)A\nabla : \mathbf{\Pi}(X) \to A is a flat ∞-connnection on the principal ∞-bundle corresponding to XΠ(X)AX \to \mathbf{\Pi}(X) \stackrel{\nabla}{\to} A, or anAA-local system** on XX.

The induced morphism

H flat(X,A)H(X,A) \mathbf{H}_{flat}(X,A) \to \mathbf{H}(X,A)

we say is the forgetful functor that forgets flat connections.

Remark

The object Π(X)\mathbf{\Pi}(X) has the interpretation of the path ∞-groupoid of XX: it is a cohesive \infty-groupoid whose k-morphisms may be thought of as generated from the kk-morphisms in XX and kk-dimensional cohesive paths in XX.

Accordingly a mophism Pi(X)A\mathbf{Pi}(X) \to A may be thought of as assigning

  • to each point of XX a fiber in AA;

  • to each path in XX an equivalence between these fibers;

  • to each disk in XX a 2-equivalalence between these equivaleces associated to its boundary

  • and so on.

This we think of as encoding a flat higher parallel transport on XX, coming from some flat \infty-connection and defining this flat \infty-connection.

Observation

By the (Π)(\mathbf{\Pi} \dashv \mathbf{\flat})-adjunction we have a natural equivalence

H flat(X,A)H(X,A). \mathbf{H}_{flat}(X,A) \simeq \mathbf{H}(X,\mathbf{\flat}A) \,.

A cocycle g:XAg : X \to A for a principal ∞-bundle on XX is in the image of

H flat(X,A)H(X,A) \mathbf{H}_{flat}(X,A) \to \mathbf{H}(X,A)

precisely if there is a lift \nabla in the diagram

A X g A. \array{ && \mathbf{\flat}A \\ & {}^{\nabla}\nearrow& \downarrow \\ X &\stackrel{g}{\to}& A } \,.

We call A\mathbf{\flat}A the coefficient object for flat AA-connections.

Proposition

For G:=DiscG 0HG := Disc G_0 \in \mathbf{H} a discrete ∞-group the canonical morphism H flat(X,BG)H(X,BG)\mathbf{H}_{flat}(X,\mathbf{B}G) \to \mathbf{H}(X,\mathbf{B}G) is an equivalence.

Proof

Since DiscDisc is a full and faithful (∞,1)-functor we have that the unit IdΓDiscId \to \Gamma Disc is a natural equivalence. It follows that on DiscG 0Disc G_0 also the counit DiscΓDiscG 0DiscG 0Disc \Gamma Disc G_0 \to Disc G_0 is a weak equivalence (since by the triangle identity we have that DiscG 0DiscΓDiscG 0DiscG 0Disc G_0 \stackrel{\simeq}{\to} Disc \Gamma Disc G_0 \to Disc G_0 is the identity).

Remark

This says that for discrete structure ∞-groups GG there is an essentially unique flat \infty-connection on any GG-principal ∞-bundle. Moreover, the further equivalence

H(Π(X),BG)H flat(X,BG)H(X,BG) \mathbf{H}(\mathbf{\Pi}(X), \mathbf{B}G) \simeq \mathbf{H}_{flat}(X, \mathbf{B}G) \simeq \mathbf{H}(X, \mathbf{B}G)

may be read as saying that the GG-principal \infty-bundle is entirely characterized by the flat higher parallel transport of this unique \infty-connection.

de Rham cohomology

In every locally ∞-connected (∞,1)-topos H\mathbf{H} there is an intrinsic notion of nonabelian de Rham cohomology.

Definition

For XHX \in \mathbf{H} an object, write Π dRX:=* XΠX\mathbf{\Pi}_{dR}X := * \coprod_X \mathbf{\Pi} X for the (∞,1)-pushout

X * Π(X) Π dRX. \array{ X &\to& * \\ \downarrow && \downarrow \\ \mathbf{\Pi}(X) &\to& \mathbf{\Pi}_{dR}X } \,.

For *A* \to A any pointed object in H\mathbf{H}, write dRA:* AA\mathbf{\flat}_{dR} A : * \prod_A \mathbf{\flat}A for the (∞,1)-pullback

dRA A * A. \array{ \mathbf{\flat}_{dR} A &\to& \mathbf{\flat} A \\ \downarrow && \downarrow \\ * &\to& A } \,.
Proposition

This construction yields a pair of adjoint (∞,1)-functors

(Π dR dR):*/H dRΠ dRH. (\mathbf{\Pi}_{dR} \dashv \mathbf{\flat}_{dR} ) : */\mathbf{H} \stackrel{ \overset{\mathbf{\Pi}_{dR}}{\leftarrow} }{ \underset{\mathbf{\flat}_{dR}}{\to} } \mathbf{H} \,.
Proof

We check the defining natural hom-equivalence

*/H(Π dRX,A)H(X, dRA). {*}/\mathbf{H}(\mathbf{\Pi}_{dR}X,A) \simeq \mathbf{H}(X, \mathbf{\flat}_{dR}A) \,.

The hom-space in the under-(∞,1)-category */H*/\mathbf{H} is (as discussed there), computed by the (∞,1)-pullback

*/H(Π dRX,A) H(Π dRX,A) * pt A H(*,A). \array{ */\mathbf{H}(\mathbf{\Pi}_{dR}X, A) &\to& \mathbf{H}(\mathbf{\Pi}_{dR}X, A) \\ \downarrow && \downarrow \\ * &\stackrel{pt_A}{\to}& \mathbf{H}(*,A) } \,.

By the fact that the hom-functor H(,):H op×HGrpd\mathbf{H}(-,-) : \mathbf{H}^{op} \times \mathbf{H} \to \infty Grpd preserves limits in both arguments we have a natural equivalence

H(Π dRX,A) :=H(* XΠ(X),A) H(*,A) H(X,A)H(Π(X),A). \begin{aligned} \mathbf{H}(\mathbf{\Pi}_{dR} X, A) & := \mathbf{H}( *\coprod_{X} \mathbf{\Pi}(X), A ) \\ & \simeq \mathbf{H}(*,A) \prod_{\mathbf{H}(X,A)} \mathbf{H}(\mathbf{\Pi}(X),A) \end{aligned} \,.

We paste this pullback to the above pullback diagram to obtain

*/H(Π dRX,A) H(Π dRX,A) H(Π(X),A) * pt A H(*,A) H(X,A). \array{ */\mathbf{H}(\mathbf{\Pi}_{dR}X, A) &\to& \mathbf{H}(\mathbf{\Pi}_{dR}X, A) &\to& \mathbf{H}(\mathbf{\Pi}(X),A) \\ \downarrow && \downarrow && \downarrow \\ * &\stackrel{pt_A}{\to}& \mathbf{H}(*,A) &\to& \mathbf{H}(X,A) } \,.

By the pasting law for (∞,1)-pullbacks the outer diagram is still a pullback. We may evidently rewrite the bottom composite as in

*/H(Π dRX,A) H(Π(X),A) * H(X,*) (pt A) * H(X,A). \array{ */\mathbf{H}(\mathbf{\Pi}_{dR}X, A) &\to& &\to& \mathbf{H}(\mathbf{\Pi}(X),A) \\ \downarrow && && \downarrow \\ * &\stackrel{\simeq}{\to}& \mathbf{H}(X,*) &\stackrel{(pt_A)_*}{\to}& \mathbf{H}(X,A) } \,.

This exhibits the hom-space as the pullback

*/H(Π dR(X),A)H(X,*) H(X,A)H(X,A), \begin{aligned} */\mathbf{H}(\mathbf{\Pi}_{dR}(X),A) \simeq \mathbf{H}(X,*) \prod_{\mathbf{H}(X,A)} \mathbf{H}(X,\mathbf{\flat} A) \end{aligned} \,,

where we used the (Π)(\mathbf{\Pi} \dashv \mathbf{\flat})-adjunction. Now using again that H(X,)\mathbf{H}(X,-) preserves pullbacks, this is

H(X,* AA)H(X, dRA). \cdots \simeq \mathbf{H}(X, * \prod_A \mathbf{\flat}A ) \simeq \mathbf{H}(X , \mathbf{\flat}_{dR}A) \,.
Observation

If H\mathbf{H} is also local, then there is a further right adjoint Γ dR\mathbf{\Gamma}_{dR}

(Π dR dRΓ dR):HΓ dR dRΠ dR*/H (\mathbf{\Pi}_{dR} \dashv \mathbf{\flat}_{dR} \dashv \mathbf{\Gamma}_{dR}) : \mathbf{H} \stackrel{\overset{\mathbf{\Pi}_{dR}}{\to}}{\stackrel{\stackrel{\mathbf{\flat}_{dR}}{\leftarrow}}{\underset{\mathbf{\Gamma}_{dR}}{\to}}} */\mathbf{H}

given by

Γ dRX:=* XΓ(X), \mathbf{\Gamma}_{dR} X {:=} * \coprod_{X} \mathbf{\Gamma}(X) \,,

where (ΠΓ):HH(\mathbf{\Pi} \dashv \mathbf{\flat} \dashv \mathbf{\Gamma}) : \mathbf{H} \to \mathbf{H} is the triple of adjunctions discussed at Paths.

Proof

This follows by the same kind of argument as above.

Definition

For X,AHX, A \in \mathbf{H} we write

H dR(X,A):=H(Π dRX,A)H(X, dRA). \mathbf{H}_{dR}(X,A) := \mathbf{H}(\mathbf{\Pi}_{dR}X, A) \simeq \mathbf{H}(X, \mathbf{\flat}_{dR} A) \,.

A cocycle ω:X dRA\omega : X \to \mathbf{\flat}_{dR}A we call an flat AA-valued differential form on XX.

We say that H dR(X,A):=π 0H dR(X,A)H_{dR}(X,A) {:=} \pi_0 \mathbf{H}_{dR}(X,A) is the de Rham cohomology of XX with coefficients in AA.

Observation

A cocycle in de Rham cohomology

ω:Π dRXA \omega : \mathbf{\Pi}_{dR}X \to A

is precisely a flat ∞-connetion on a trivializable AA-principal \infty-bundle. More precisely, H dR(X,A)\mathbf{H}_{dR}(X,A) is the homotopy fiber of the forgetful functor from \infty-bundles with flat \infty-connection to \infty-bundles: we have an (∞,1)-pullback

H dR(X,A) * H flat(X,A) H(X,A). \array{ \mathbf{H}_{dR}(X,A) &\to& * \\ \downarrow && \downarrow \\ \mathbf{H}_{flat}(X,A) &\to& \mathbf{H}(X,A) } \,.
Proof

This follows by the fact that the hom-functor H(X,)\mathbf{H}(X,-) preserves the defining (∞,1)-pullback for dRA\mathbf{\flat}_{dR} A.

Just for emphasis, notice the dual description of this situation: by the universal property of the (∞,1)-colimit that defines Π dRX\mathbf{\Pi}_{dR} X we have that ω\omega corresponds to a diagram

X * Π(X) ω A. \array{ X &\to& * \\ \downarrow &\swArrow_{\simeq}& \downarrow \\ \mathbf{\Pi}(X) &\stackrel{\omega}{\to}& A } \,.

The bottom horizontal morphism is a flat connection on the \infty-bundle given by the cocycle XΠ(X)ωAX \to \mathbf{\Pi}(X) \stackrel{\omega}{\to} A. The diagram says that this is equivalent to the trivial bundle given by the trivial cocycle X*AX \to * \to A.

Proposition

The de Rham cohomology with coefficients in discrete objects is trivial: for all SGrpdS \in \infty Grpd we have

dRDiscS*. \mathbf{\flat}_{dR} Disc S \simeq * \,.
Proof

Using that in a ∞-connected (∞,1)-topos the functor DiscDisc is a full and faithful (∞,1)-functor so that the unit IdΓDiscId \to \Gamma Disc is an equivalence and using that by the zig-zag identity we have then that the counit component DiscS:=DiscΓDiscSDiscS\mathbf{\flat} Disc S := Disc \Gamma Disc S \to Disc S is also an equivalence, we have

dRDiscS :=* DiscSDiscS * DiscSDiscS *, \begin{aligned} \mathbf{\flat}_{dR} Disc S & {:=} * \prod_{Disc S} \mathbf{\flat} Disc S \\ & \simeq * \prod_{Disc S} Disc S \\ & \simeq * \end{aligned} \,,

since the pullback of an equivalence is an equivalence.

Proposition

In a cohesive H\mathbf{H} pieces have points precisely if for all XHX \in \mathbf{H}, the de Rham coefficient object Π dRX\mathbf{\Pi}_{dR} X is globally connected in that π 0H(*,Π dRX)=*\pi_0 \mathbf{H}(*, \mathbf{\Pi}_{dR}X) = *.

If XX has at least one point (π 0(ΓX)\pi_0(\Gamma X) \neq \emptyset ) and is geometrically connected (π 0(ΠX)=*\pi_0 (\Pi X) = {*}) then Π dR(X)\mathbf{\Pi}_{\mathrm{dR}}(X) is also locally connected: τ 0Π dRX*H\tau_0 \mathbf{\Pi}_{\mathrm{dR}}X \simeq {*} \in \mathbf{H}.

Proof

Since Γ\Gamma preserves (∞,1)-colimits in a cohesive (,1)(\infty,1)-topos we have

H(*,Π dRX) ΓΠ dRX * ΓXΓΠX * ΓXΠX, \begin{aligned} \mathbf{H}(*, \mathbf{\Pi}_{dR}X) & \simeq \Gamma \mathbf{\Pi}_{dR} X \\ & \simeq * \coprod_{\Gamma X} \Gamma \mathbf{\Pi}X \\ & \simeq * \coprod_{\Gamma X} \Pi X \end{aligned} \,,

where in the last step we used that DiscDisc is a full and faithful, so that there is an equivalence ΓΠX:=ΓDiscΠXΠX\Gamma \mathbf{\Pi}X := \Gamma Disc \Pi X \simeq \Pi X.

To analyse this (∞,1)pushout we present it by a homotopy pushout in the standard model structure on simplicial sets sSet Quillen\mathrm{sSet}_{\mathrm{Quillen}}. Denoting by ΓX\Gamma X and ΠX\Pi X any representatives in sSet Quillen\mathrm{sSet}_{\mathrm{Quillen}} of the objects of the same name in Grpd\infty \mathrm{Grpd}, this may be computed by the ordinary pushout in sSet

ΓX (ΓX)×Δ[1] ΓX* ΠX Q, \array{ \Gamma X &\to& (\Gamma X) \times \Delta[1] \coprod_{\Gamma X} {*} \\ \downarrow && \downarrow \\ \Pi X &\to & Q } \,,

where on the right we have inserted the cone on ΓX\Gamma X in order to turn the top morphism into a cofibration. From this ordinary pushout it is clear that the connected components of QQ are obtained from those of ΠX\Pi X by identifying all those in the image of a connected component of ΓX\Gamma X. So if the left morphism is surjective on π 0\pi_0 then π 0(Q)=*\pi_0(Q) = *. This is precisely the condition that pieces have points in H\mathbf{H}.

For the local analysis we consider the same setup objectwise in the injective model structure on simplicial presheaves [C op,sSet] inj,loc[C^{\mathrm{op}}, \mathrm{sSet}]_{\mathrm{inj},\mathrm{loc}}. For any UCU \in C we then have the pushout Q UQ_U in

X(U) (X(U))×Δ[1] X(U)* sSet(Γ(U),ΠX) Q U, \array{ X(U) &\to & (X(U)) \times \Delta[1] \coprod_{X(U)} {*} \\ \downarrow && \downarrow \\ \mathrm{sSet}(\Gamma(U), \Pi X) & \to & Q_U } \,,

as a model for the value of the simplicial presheaf presenting Π dR(X)\mathbf{\Pi}_{\mathrm{dR}}(X). If XX is geometrically connected then π 0sSet(Γ(U),Π(X))=*\pi_0 \mathrm{sSet}(\Gamma(U), \Pi(X)) = * and hence for the left morphism to be surjective on π 0\pi_0 it suffices that the top left object is not empty. Since the simplicial set X(U)X(U) contains at least the vertices U*XU \to * \to X of which there is by assumption at least one, this is the case.

Remark

In summary this means that in a cohesive (,1)(\infty,1)-topos the objects Π dRX\mathbf{\Pi}_{dR} X have the abstract properties of pointed geometric de Rham homotopy types.

In the Examples we will see that, indeed, the intrinsic de Rham cohomology H dR(X,A):=π 0H(Π dRX,A)H_{dR}(X, A) {:=} \pi_0 \mathbf{H}(\mathbf{\Pi}_{dR} X, A) reproduces ordinary de Rham cohomology in degree d>1d\gt 1.

In degree 0 the intrinsic de Rham cohomology is necessrily trivial, while in degree 1 we find that it reproduces closed 1-forms, not divided out by exact forms. This difference to ordinary de Rham cohomology in the lowest two degrees may be interpreted in terms of the obstruction-theoretic meaning of de Rham cohomology by which we essentially characterized it above: we have that the intrinsic H dR n(X,K)H_{dR}^n(X,K) is the home for the obstructions to flatness of B n2K\mathbf{B}^{n-2}K-principal ∞-bundles. For n=1n = 1 this are groupoid-principal bundles over the groupoid with KK as its space of objects. But the 1-form curvatures of groupoid bundles are not to be regarded modulo exact forms. More details on this are at circle n-bundle with connection.

Exponentiated \infty-Lie algebras

Definition

For a connected object Bexp(𝔤)\mathbf{B}\exp(\mathfrak{g}) in H\mathbf{H} that is geometrically contractible

Π(Bexp(𝔤))* \Pi (\mathbf{B}\exp(\mathfrak{g})) \simeq *

we call its loop space object exp(𝔤):=Ω *Bexp(𝔤)\exp(\mathfrak{g}) := \Omega_* \mathbf{B}\exp(\mathfrak{g}) the Lie integration of an ∞-Lie algebra in H\mathbf{H}.

Definition

Set

expLie:=Π dR dR:*/H*/H. \exp Lie := \mathbf{\Pi}_{dR} \circ \mathbf{\flat}_{dR} : */\mathbf{H} \to */\mathbf{H} \,.
Observation

If H\mathbf{H} is cohesive, then expLie\exp Lie is a left adjoint.

Proof

When H\mathbf{H} is cohesive we have the de Rham triple of adjunction (Π dR dRΓ dR)(\mathbf{\Pi}_{dR} \dashv \mathbf{\flat}_{dR} \dashv \mathbf{\Gamma}_{dR}). Accordingly then LieLie is part of an adjunction

(expLieΓ dR dR). (\exp Lie \dashv \mathbf{\Gamma}_{dR}\mathbf{\flat}_{dR}) \,.
Proposition/Example

For all XX the object Π dR(X)\mathbf{\Pi}_{dR}(X) is geometrically contractible.

Proof

Since on the locally ∞-connected (∞,1)-topos and ∞-connected H\mathbf{H} the functor Π\Pi preserves (∞,1)-colimits and the terminal object, we have

ΠΠ dRX :=Π(*) ΠXΠΠX * ΠXΠDiscΠX * ΠXΠX *, \begin{aligned} \Pi \mathbf{\Pi}_{dR} X & {:=} \Pi (*) \coprod_{\Pi X} \Pi \mathbf{\Pi} X \\ & \simeq * \coprod_{\Pi X} \Pi Disc \Pi X \\ & \simeq * \coprod_{\Pi X} \Pi X & \simeq * \end{aligned} \,,

where we used that in the ∞-connected H\mathbf{H} the functor DiscDisc isfull and faithful.

Corollary

We have for every BG\mathbf{B}G that expLieBG\exp Lie \mathbf{B}G is geometrically contractible.

We shall write Bexp(𝔤)\mathbf{B}\exp(\mathfrak{g}) for expLieBG\exp Lie \mathbf{B}G, when the context is clear.

Proposition

Every de Rham cocycle ω:Π dRXBG\omega : \mathbf{\Pi}_{dR} X \to \mathbf{B}G factors through the ∞-Lie algebra of GG

Bexp(𝔤) Π dRX ω BG. \array{ && \mathbf{B}\exp(\mathfrak{g}) \\ & \nearrow & \downarrow \\ \mathbf{\Pi}_{dR}X &\stackrel{\omega}{\to}& \mathbf{B}G } \,.
Proof

By the universality of the counit of (Π dR dR)(\mathbf{\Pi}_{dR} \dashv \mathbf{\flat}_{dR}) we have that ω\omega factors through the [[unit of an adjunction|counit]9 expLieBGBG\exp Lie \mathbf{B}G \to \mathbf{B}G.

Therefore instead of speaking of a GG-valued de Rham cocycle, it is less redundant to speak of an exp(𝔤)\exp(\mathfrak{g})-valued de Rham cocycle. In particular we have the following.

Corollary

Every morphism expLieBHBG\exp Lie \mathbf{B}H \to \mathbf{B}G from an exponentiated \infty-Lie algebra to an \infty-group factors through the exponentiated \infty-Lie algebra of that \infty-group

Bexp(𝔥) Bexp(𝔤) BG. \array{ \mathbf{B}\exp(\mathfrak{h}) &\to& \mathbf{B}\exp(\mathfrak{g}) \\ & \searrow& \downarrow \\ && \mathbf{B}G } \,.
Proposition

If H\mathbf{H} is cohesive then we have

expLieexpLieexpLieΣΩ. \exp Lie \circ \exp Lie \simeq \exp Lie \circ \Sigma \circ \Omega \,.
Proof

First observe that for all A*/HA \in */\mathbf{H} we have

dRA* \mathbf{\flat} \mathbf{\flat}_{dR} A \simeq *

This follows using

by computing

dRA *× AA *× AA *, \begin{aligned} \mathbf{\flat} \mathbf{\flat}_{dR} A & * \times_{\mathbf{\flat}A} \mathbf{\flat}\mathbf{\flat}A \\ & \simeq * \times_{\mathbf{\flat}A} \mathbf{\flat}A \\ & \simeq * \end{aligned} \,,

using that the (∞,1)-pullback of an equivalence is an equivalence.

From this we deduce that

dR dR dRΩ. \mathbf{\flat}_{dR} \circ \mathbf{\flat}_{dR} \simeq \mathbf{\flat}_{dR} \circ \Omega \,.

by computing for all AHA \in \mathbf{H}

dR dRA *× dRA dRA *× dRA* dR(*× A*) dRΩA. \begin{aligned} \mathbf{\flat}_{dR} \circ \mathbf{\flat}_{dR} A & \simeq * \times_{\mathbf{\flat}_{dR} A} \mathbf{\flat}\mathbf{\flat}_{dR} A \\ & \simeq * \times_{\mathbf{\flat}_{dR} A} * \\ & \simeq \mathbf{\flat}_{dR}( * \times_A * ) \\ & \simeq \mathbf{\flat}_{dR} \Omega A \end{aligned} \,.

Also observe that by a proposition above we have

dRΠX* \mathbf{\flat}_{dR} \mathbf{\Pi} X \simeq *

for all XHX \in \mathbf{H}.

Finally to obtain expLieexpLie\exp Lie \circ \exp Lie we do one more computation of this sort, using that

We compute:

expLieexpLieA expLieΠ dR dRA * expLie dRAexpLieΠ dRA * expLie dRAΠ dR dRΠ dRA * expLie dRA* * Π dR dR dRA* * Π dR dRΩA* * expLieΩA* expLie(* ΩA*) expLieΣΩA. \begin{aligned} \exp Lie \exp Lie A & \simeq \exp Lie \mathbf{\Pi}_{dR} \mathbf{\flat}_{dR} A \\ & \simeq * \coprod_{\exp Lie \mathbf{\flat}_{dR} A} \exp Lie \mathbf{\Pi} \mathbf{\flat}_{dR} A \\ & \simeq * \coprod_{\exp Lie \mathbf{\flat}_{dR} A} \mathbf{\Pi}_{dR} \mathbf{\flat}_{dR} \mathbf{\Pi} \mathbf{\flat}_{dR} A \\ & \simeq * \coprod_{\exp Lie \mathbf{\flat}_{dR} A} * \\ & \simeq * \coprod_{\mathbf{\Pi}_{dR} \mathbf{\flat}_{dR} \mathbf{\flat}_{dR} A} * \\ & \simeq * \coprod_{\mathbf{\Pi}_{dR} \mathbf{\flat}_{dR} \Omega A} * \\ & \simeq * \coprod_{\exp Lie \Omega A} * \\ & \simeq \exp Lie ( * \coprod_{\Omega A} * ) \\ & \simeq \exp Lie \Sigma \Omega A \end{aligned} \,.

Maurer-Cartan forms and curvature characteristic forms

In the intrinsic de Rham cohomology of a locally ∞-connected ∞-connected there exist canonical cocycles that we may identify with Maurer-Cartan forms and with universal curvature characteristic forms.

Definition

For GHG \in \mathbf{H} an ∞-group, write

θ:G dRBG \theta : G \to \mathbf{\flat}_{dR} \mathbf{B}G

for the 𝔤\mathfrak{g}-valued de Rham cocycle on GG which is induced by the (∞,1)-pullback pasting

G * θ dRBG BG * BG \array{ G &\to& * \\ {}^{\mathllap{\theta}}\downarrow && \downarrow \\ \mathbf{\flat}_{dR} \mathbf{B}G &\to& \mathbf{\flat}\mathbf{B}G \\ \downarrow && \downarrow \\ * &\to& \mathbf{B}G }

and the above proposition.

We call θ\theta the Maurer-Cartan form on GG.

Remark

By postcomposition the Maurer-Cartan form sends GG-valued functions on XX to 𝔤\mathfrak{g}-valued forms on XX

θ *:H(X,G)H dR 1(X,G). \theta_* : \mathbf{H}(X,G) \to \mathbf{H}^1_{dR}(X,G) \,.
Definition

For G=B nAG = \mathbf{B}^n A an Eilenberg-MacLane object, we also write

curv:B nA dRB n+1A curv : \mathbf{B}^n A \to \mathbf{\flat}_{dR} \mathbf{B}^{n+1} A

for the intrinsic Maurer-Cartan form and call this the intrinsic universal curvature characteristic form on B nA\mathbf{B}^n A.

Differential cohomology

In every locally ∞-connected ∞-connected (∞,1)-topos there is an intrinsic notion of ordinary differential cohomology.

Fix a 0-truncated abelian group object Aτ 0HHA \in \tau_{\leq 0} \mathbf{H} \hookrightarrow \mathbf{H}. For all nNn \in \mathbf{N} we have then the Eilenberg-MacLane object B nA\mathbf{B}^n A.

Definition

For XHX \in \mathbf{H} any object and n1n \geq 1 write

H diff(X,B nA):=H(X,B nA) H dR(X,B nA)H dR n+1(X,A) \mathbf{H}_{diff}(X,\mathbf{B}^n A) := \mathbf{H}(X,\mathbf{B}^n A) \prod_{\mathbf{H}_{dR}(X,\mathbf{B}^n A)} H_{dR}^{n+1}(X,A)

for the cocycle \infty-groupoid of twisted cohomology, def. \ref{TwistedCohomologyInOvertopos}, of XX with coefficients in AA and with twist given by the canonical curvature characteristic morphism curv:B nA dRB n+1Acurv : \mathbf{B}^n A \to \mathbf{\flat}_{dR}\mathbf{B}^{n+1} A. This is the (∞,1)-pullback

H diff(X,B nA) [F] H dR n+1(X,A) η H(X,B nA) curv * H dR(X,B n+1A), \array{ \mathbf{H}_{diff}(X,\mathbf{B}^n A) &\stackrel{[F]}{\to}& H_{dR}^{n+1}(X,A) \\ {}^{\mathllap{\eta}}\downarrow && \downarrow \\ \mathbf{H}(X,\mathbf{B}^n A) &\stackrel{curv_*}{\to}& \mathbf{H}_{dR}(X,\mathbf{B}^{n+1} A) } \,,

where the right vertical morphism H dR n+1(X)=π 0H dR(X,B n+1A)H dR(X,B n+1A)H^{n+1}_{dR}(X) = \pi_0 \mathbf{H}_{dR}(X,\mathbf{B}^{n+1}A) \to \mathbf{H}_{dR}(X,\mathbf{B}^{n+1}A) is any choice of cocycle representative for each cohomology class: a choice of point in every connected component.

We call

H diff n(X,A):=π 0H diff(X,B nA) H_{diff}^n(X,A) {:=} \pi_0 \mathbf{H}_{diff}(X, \mathbf{B}^{n} A)

the degree-nn differential cohomology of XX with coefficient in AA.

For H diff(X,B nA)\nabla \in \mathbf{H}_{diff}(X,\mathbf{B}^n A) a cocycle, we call

  • [η()]H n(X,A)[\eta(\nabla)] \in H^n(X,A) the class of the underlying B n1A\mathbf{B}^{n-1} A-principal ∞-bundle;

  • F()H dR n+1(X,A)F(\nabla) \in H_{dR}^{n+1}(X,A) the curvature class of cc.

We also say \nabla is an \infty-connection on η()\eta(\nabla) (see below).

Observation

The differential cohomology H diff n(X,A)H_{diff}^n(X,A) does not depend on the choice of morphism H dR n+1(X,A)H dR(X,B n+1A)H_{dR}^{n+1}(X,A) \to \mathbf{H}_{dR}(X, \mathbf{B}^{n+1}A) (as long as it is an isomorphism on π 0\pi_0, as required). In fact, for different choices the corresponding cocycle ∞-groupoids H diff(X,B nA)\mathbf{H}_{diff}(X,\mathbf{B}^n A) are equivalent.

Proof

The set

H dR n+1(X,A)= H dR n+1(X,A)* H_{dR}^{n+1}(X,A) = \coprod_{H_{dR}^{n+1}(X,A)} {*}

is, as a 0-truncated ∞-groupoid, an (∞,1)-coproduct of the terminal object in ∞Grpd. By universal colimits in this (∞,1)-topos we have that (∞,1)-colimits are preserved by (∞,1)-pullbacks, so that H diff(X,B nA)\mathbf{H}_{diff}(X, \mathbf{B}^n A) is the coproduct

H diff(X,B nA) H dR n+1(X,A)(H(X,B nA) H dR(X,B n+1A)*) \mathbf{H}_{diff}(X,\mathbf{B}^n A) \simeq \coprod_{H_{dR}^{n+1}(X,A)} \left( \mathbf{H}(X,\mathbf{B}^n A) \prod_{\mathbf{H}_{dR}(X,\mathbf{B}^{n+1}A)} {*} \right)

of the homotopy fibers of curv *curv_* over each of the chosen points *H dR(X,B n+1A)* \to \mathbf{H}_{dR}(X,\mathbf{B}^{n+1}A). These homotopy fibers only depend, up to equivalence, on the connected component over which they are taken.

Proposition

When restricted to vanishing curvature, differential cohomology coincides with flat differential cohomology:

H diff n(X,A)| [F]=0H flat(X,B nA). H_{diff}^n (X,A)|_{[F] = 0} \simeq H_{flat}(X,\mathbf{B}^n A) \,.

Moreover this is true at the level of cocycle ∞-groupoids

(H diff(X,B nA) H dR n+1(X,A){[F]=0})H flat(X,B nA). \left( \mathbf{H}_{diff}(X, \mathbf{B}^n A) \prod_{H_{dR}^{n+1}(X,A)} \{[F] = 0\} \right) \simeq \mathbf{H}_{flat}(X,\mathbf{B}^n A) \,.
Proof

By the pasting law for (∞,1)-pullbacks the claim is equivalently that we have a an (,1)(\infty,1)-pullback diagram

H flat(X,B nA) * {[F]=0} H diff(X,B nA) [F] H dR n+1(X,A) η H(X,B nA) curv * H dR(X,B n+1A). \array{ \mathbf{H}_{flat}(X, \mathbf{B}^n A) &\to& {*} \\ \downarrow && \downarrow^{\mathrlap{\{[F] = 0\}}} \\ \mathbf{H}_{diff}(X,\mathbf{B}^n A) &\stackrel{[F]}{\to}& H_{dR}^{n+1}(X,A) \\ {}^{\mathllap{\eta}}\downarrow && \downarrow \\ \mathbf{H}(X,\mathbf{B}^n A) &\stackrel{curv_*}{\to}& \mathbf{H}_{dR}(X,\mathbf{B}^{n+1} A) } \,.

By definition of flat cohomology and of intrinsic de Rham cohomology in H\mathbf{H}, the outer rectangle is

H(X,B nA) * H(X,B nA) curv * H(X, dRB n+1A). \array{ \mathbf{H}(X,\mathbf{\flat}\mathbf{B}^n A) &\to& {*} \\ \downarrow && \downarrow \\ \mathbf{H}(X, \mathbf{B}^n A) &\stackrel{curv_*}{\to}& \mathbf{H}(X, \mathbf{\flat}_{dR}\mathbf{B}^{n+1} A) } \,.

Since the hom-functor H(X,)\mathbf{H}(X,-) preserves (∞,1)-limits this is a pullback if

B nA * B nA curv dRB n+1A \array{ \mathbf{\flat} \mathbf{B}^n A &\to& * \\ \downarrow && \downarrow \\ \mathbf{B}^n A &\stackrel{curv}{\to}& \mathbf{\flat}_{dR} \mathbf{B}^{n+1} A }

is. Indeed, this is one step in the fiber sequence

B nAB nAcurv dRB n+1AB n+1AB n+1A \cdots \to \mathbf{\flat} \mathbf{B}^n A \to \mathbf{B}^n A \stackrel{curv}{\to} \mathbf{\flat}_{dR} \mathbf{B}^{n+1}A \to \mathbf{\flat} \mathbf{B}^{n+1} A \to \mathbf{B}^{n+1} A

that defines curvcurv (using that \mathbf{\flat} preserves limits and hence looping and delooping)

Proposition

The differential cohomology group H diff n(X,A)H_{diff}^n(X,A) fits into a short exact sequence of abelian groups

0H dR n(X,A)/H n1(X,A)H diff n(X,A)H n(X,A)0. 0 \to H_{dR}^n(X,A)/H^{n-1}(X,A) \to H_{diff}^n(X,A) \to H^n(X,A) \to 0 \,.
Proof

This is a general statement about the definition of twisted cohomology. We claim that for all n1n \geq 1 we have a fiber sequence

H(X,B n1A)H dR(X,B nA)H diff(X,B nA)H(X,B nA) \mathbf{H}(X, \mathbf{B}^{n-1}A) \to \mathbf{H}_{dR}(X, \mathbf{B}^n A) \to \mathbf{H}_{diff}(X, \mathbf{B}^n A) \to \mathbf{H}(X, \mathbf{B}^n A)

in ∞Grpd. This implies the short exact sequence using that by construction the last morphism is surjective on connected components (because in the defining (,1)(\infty,1)-pullback for H diff\mathbf{H}_{diff} the right vertical morphism is by assumption surjective on connected components).

To see that we do have the fiber sequence as claimed consider the pasting composite of (∞,1)-pullbacks

H dR(X,B n1A) H diff(X,B nA) H dR(X,B n+1A) * H(X,B nA) curv H dR(X,B n+1A). \array{ \mathbf{H}_{dR}(X,\mathbf{B}^{n-1} A) &\to& \mathbf{H}_{diff}(X,\mathbf{B}^n A) &\to& H_{dR}(X, \mathbf{B}^{n+1} A) \\ \downarrow && \downarrow && \downarrow \\ {*} &\to& \mathbf{H}(X, \mathbf{B}^n A) &\stackrel{curv}{\to}& \mathbf{H}_{dR}(X, \mathbf{B}^{n+1} A) } \,.

The square on the right is a pullback by the above definition. Since also the square on the left is assumed to be an (,1)(\infty,1)-pullback it follows by the pasting law for (∞,1)-pullbacks that the top left object is the (,1)(\infty,1)-pullback of the total rectangle diagram. That total diagram is

ΩH(X, dRB n+1A) H(X, dRB n+1A) * H(X, dRB n+1A), \array{ \Omega \mathbf{H}(X, \mathbf{\flat}_{dR} \mathbf{B}^{n+1}A) &\to& H(X,\mathbf{\flat}_{dR} \mathbf{B}^{n+1} A) \\ \downarrow && \downarrow \\ * &\to& \mathbf{H}(X,\mathbf{\flat}_{dR} \mathbf{B}^{n+1} A) } \,,

because, as before, this (,1)(\infty,1)-pullback is the coproduct of the homotopy fibers, and they are empty over the connected components not in the image of the bottom morphism and are the loop space object over the single connected component that is in the image.

Finally using that (as discussed at cohomology and at fiber sequence)

ΩH(X, dRB n+1A)H(X,Ω dRB n+1A) \Omega \mathbf{H}(X,\mathbf{\flat}_{dR} \mathbf{B}^{n+1}A) \simeq \mathbf{H}(X,\Omega \mathbf{\flat}_{dR} \mathbf{B}^{n+1}A)

and

Ω dRB n+1A dRΩB n+1A \Omega \mathbf{\flat}_{dR} \mathbf{B}^{n+1}A \simeq \mathbf{\flat}_{dR} \Omega \mathbf{B}^{n+1}A

since both H(X,)\mathbf{H}(X,-) as well as dR\mathbf{\flat}_{dR} preserve (∞,1)-limits and hence formation of loop space objects, the claim follows.

Remark

This is essentially the short exact sequence whose form is familiar from the traditional definition of ordinary differential cohomology only up to the following slight nuances in notation:

  1. The cohomology groups of the short exact sequence above denote the groups obtained in the given (∞,1)-topos H\mathbf{H}, not in Top. Notably for H=\mathbf{H} = ∞LieGrpd, A=U(1)=/A = U(1) =\mathbb{R}/\mathbb{Z} the circle group and |X|Top|X| \in Top the geometric realization of a paracompact manifold XX, we have that H n(X,/)H^n(X,\mathbb{R}/\mathbb{Z}) above is H sing n+1(|ΠX|,)H^{n+1}_{sing}({|\Pi X|},\mathbb{Z}).

  2. The fact that on the left of the short exact sequence for differential cohomology we have the de Rham cohomology set H dR n(X,A)H_{dR}^n(X,A) instead of something like the set of all flat forms as familiar from ordinary differential cohomology is because the latter has no intrinsic meaning but depends on a choice of model. After fixing a specific presentation of H\mathbf{H} by a model category CC we can consider instead of H dR n+1(X,A)H dR(X,B n+1A)H_{dR}^{n+1}(X,A) \to \mathbf{H}_{dR}(X, \mathbf{B}^{n+1}A) the inclusion of the set of objects Ω cl n+1(X,A):=Hom C(X,B n+1A) 0Hom C(X,B n+1A)\Omega_{cl}^{n+1}(X,A) {:=} \mathbb{R}Hom_C(X, \mathbf{B}^{n+1}A )_0 \hookrightarrow \mathbb{R}Hom_C(X, \mathbf{B}^{n+1}A ). However, by the above observation this only adds multiple copies of the homotopy types of the connected components of H diff(X,B nA)\mathbf{H}_{diff}(X, \mathbf{B}^n A).

Chern-Weil homomorphism and \infty-connections

Induced by the intrinsic differential cohomology in any ∞-connected and locally ∞-connected (∞,1)-topos is an intrinsic notion of Chern-Weil homomorphism.

Let AA be the chosen abelian ∞-group as above. Recall the universal curvature characteristic class

curv:B nA dRB n+1A curv : \mathbf{B}^n A \to \mathbf{\flat}_{dR}\mathbf{B}^{n+1}A

for all n1n \geq 1.

Definition

For GG an ∞-group and

c:BGB nA \mathbf{c} : \mathbf{B}G \to \mathbf{B}^n A

a representative of a characteristic class [c]H n(BG,A)[\mathbf{c}] \in H^n(\mathbf{B}G, A) we say that the composite

c dR:BGcB nAcurv dRB n+1A \mathbf{c}_{dR} : \mathbf{B}G \stackrel{\mathbf{c}}{\to} \mathbf{B}^n A \stackrel{curv}{\to} \mathbf{\flat}_{dR} \mathbf{B}^{n+1} A

represents the corresponding differential characteristic class or curvature characteristic class [c dR]H dR n+1(BG,A)[\mathbf{c}_{dR}] \in H_{dR}^{n+1}(\mathbf{B}G, A).

The induced map on cohomology

(c dR) *:H 1(,G)H dR n+1(,A) (\mathbf{c}_{dR})_* : H^1(-,G) \to H^{n+1}_{dR}(-,A)

we call the (unrefined) ∞-Chern-Weil homomorphism induced by c\mathbf{c}.

The following construction universally lifts the \infty-Chern-Weil homomorphism from taking values in intrinsic de Rham cohomology to values in intrinsic differential cohomology.

Definition

For XHX \in \mathbf{H} any object, define the ∞-groupoid H conn(X,BG)\mathbf{H}_{conn}(X,\mathbf{B}G) as the (∞,1)-pullback

H conn(X,BG) (c^ i) i [c i]H n i(BG,A);i1H diff(X,B n iA) η H(X,BG) (c i) i [c i]H n i(BG,A);i1H(X,B n iA). \array{ \mathbf{H}_{conn}(X, \mathbf{B}G) &\stackrel{(\hat \mathbf{c}_i)_i}{\to}& \prod_{[\mathbf{c}_i] \in H^{n_i}(\mathbf{B}G,A); i \geq 1} \mathbf{H}_{diff}(X,\mathbf{B}^{n_i} A) \\ {}^{\mathllap{\eta}}\downarrow && \downarrow \\ \mathbf{H}(X, \mathbf{B}G) &\stackrel{( \mathbf{c}_i)_i}{\to}& \prod_{[\mathbf{c}_i] \in H^{n_i}(\mathbf{B}G,A); i \geq 1} \mathbf{H}(X,\mathbf{B}^{n_i} A) } \,.

We say

  • a cocycle in H conn(X,BG)\nabla \in \mathbf{H}_{conn}(X, \mathbf{B}G) is an ∞-connection

  • on the principal ∞-bundle η()\eta(\nabla);

  • a morphism in H conn(X,BG)\mathbf{H}_{conn}(X, \mathbf{B}G) is a gauge transformation of connections;

  • for each [c]nH n(BG,A)[\mathbf{c}] \n H^n(\mathbf{B}G, A) the morphism

    [c^]:H conn(X,BG)H diff n(X,A) [\hat \mathbf{c}] : H_{conn}(X,\mathbf{B}G) \to H_{diff}^n(X, A)

    is the (full/refined) ∞-Chern-Weil homomorphism induced by the characteristic class [c][\mathbf{c}].

Observation

Under the curvature projection [F]:H diff n(X,A)H dR n+1(X,A)[F] : H_{diff}^n (X,A) \to H_{dR}^{n+1}(X,A) the refined Chern-Weil homomorphism for c\mathbf{c} projects to the unrefined Chern-Weil homomorphism.

Proof

This is due to the existence of the pasting composite

H conn(X,BG) (c^ i) i [c i]H n i(BG,A);i1H diff(X,B n iA) [F] [c i]H n i(BG,A);i1H dR n i+1(X,A) η H(X,BG) (c i) i [c i]H n i(BG,A);i1H(X,B n iA) curv * [c i]H n i(BG,A);i1H dR(X,B n i+1,A) \array{ \mathbf{H}_{conn}(X, \mathbf{B}G) &\stackrel{(\hat \mathbf{c}_i)_i}{\to}& \prod_{[\mathbf{c}_i] \in H^{n_i}(\mathbf{B}G,A); i \geq 1} \mathbf{H}_{diff}(X,\mathbf{B}^{n_i} A) &\stackrel{[F]}{\to}& \prod_{[\mathbf{c}_i] \in H^{n_i}(\mathbf{B}G,A); i \geq 1} H_{dR}^{n_i+1}(X,A) \\ {}^{\mathllap{\eta}}\downarrow && \downarrow && \downarrow \\ \mathbf{H}(X, \mathbf{B}G) &\stackrel{(\mathbf{c}_i)_i}{\to}& \prod_{[\mathbf{c}_i] \in H^{n_i}(\mathbf{B}G,A); i \geq 1} \mathbf{H}(X,\mathbf{B}^{n_i} A) &\stackrel{curv_*}{\to}& \prod_{[\mathbf{c}_i] \in H^{n_i}(\mathbf{B}G,A); i \geq 1} \mathbf{H}_{dR}(X, \mathbf{B}^{n_i+1},A) }

of the defining (,1)(\infty,1)-pullback for H conn(X,BG)\mathbf{H}_{conn}(X,\mathbf{B}G) with the products of the defining (,1)(\infty,1)-pullbacks for the H diff(X,B n iA)\mathbf{H}_{diff}(X, \mathbf{B}^{n_i}A).

Higher holonomy and Chern-Simons functional

The notion of intrinsic ∞-connections in a cohesive (,1)(\infty,1)-topos induces a notion of higher holonomy and Chern-Simons functionals.

Definition

We say an object ΣH\Sigma \in \mathbf{H} has cohomological dimension n\leq n \in \mathbb{N} if for all nn-connected coefficient objects and (n++1)(n++1)-truncated objects B n+1A\mathbf{B}^{n+1}A the corresponding cohomology on Σ\Sigma is trivial

H(Σ,B n+1A)*. H(\Sigma, \mathbf{B}^{n+1}A ) \simeq * \,.

Let dim(Σ)dim(\Sigma) be the maximum nn for which this is true.

Observation

If Σ\Sigma has cohomological dimension n\leq n then its intrinsic de Rham cohomology vanishes in degree k>nk \gt n

H dR k>n(Σ,A)*. H_{dR}^{k \gt n}(\Sigma, A) \simeq * \,.
Proof

Since \mathbf{\flat} is a right adjoint it preserves delooping and hence B kAB kA\mathbf{\flat} \mathbf{B}^k A \simeq \mathbf{B}^k \mathbf{\flat}A. It follows that

H dR k(Σ,A) :=π 0H(Σ, dRB kA) π 0H(Σ,* B kAB kA) π 0(H(Σ,*) H(Σ,B kA)H(Σ,B kA)) π 0(*). \begin{aligned} H_{dR}^{k}(\Sigma,A) & := \pi_0 \mathbf{H}(\Sigma, \mathbf{\flat}_{dR} \mathbf{B}^k A) \\ & \simeq \pi_0 \mathbf{H}(\Sigma, * \prod_{\mathbf{B}^k A} \mathbf{B}^k \mathbf{\flat}A) \\ & \simeq \pi_0 \left( \mathbf{H}(\Sigma,*) \prod_{\mathbf{H}(\Sigma, \mathbf{B}^k A)} \mathbf{H}(\Sigma, \mathbf{B}^k \mathbf{\flat}A) \right) \\ & \simeq \pi_0 (*) \end{aligned} \,.

Let now again AA be fixed as above.

Definition

Let ΣH\Sigma \in \mathbf{H}, nNn \in \mathbf{N} with dimΣndim \Sigma \leq n.

We say that the composite

Σ:H flat(Σ,B nA)Gprd(Π(Σ),Π(B nA))τ ndim(Σ)τ ndim(Σ)Gprd(Π(Σ),Π(B nA)) \int_\Sigma : \mathbf{H}_{flat}(\Sigma, \mathbf{B}^n A) \stackrel{\simeq}{\to} \infty Gprd(\Pi(\Sigma), \Pi(\mathbf{B}^n A)) \stackrel{\tau_{\leq n-dim(\Sigma)}}{\to} \tau_{n-dim(\Sigma)} \infty Gprd(\Pi(\Sigma), \Pi(\mathbf{B}^n A))

of the adjunction equivalence followed by truncation is the flat holonomy operation on flat \infty-connections.

More generally, let

  • H diff(X,B nA)\nabla \in \mathbf{H}_{diff}(X, \mathbf{B}^n A) be a differential coycle on some XHX \in \mathbf{H}

  • ϕ:ΣX\phi : \Sigma \to X a morphism.

Write

ϕ *:H diff(X,B n+1A)H diff(Σ,B nA)H flat(Σ,B nA) \phi^* : \mathbf{H}_{diff}(X, \mathbf{B}^{n+1} A) \to \mathbf{H}_{diff}(\Sigma, \mathbf{B}^n A) \simeq \mathbf{H}_{flat}(\Sigma, \mathbf{B}^n A)

(using the above proposition) for the morphism on (,1)(\infty,1)-pullbacks induced by the morphism of diagrams

H(X,B nA) H dR(X,B n+1A) H dR n+1(X,A) ϕ * ϕ * H(Σ,B nA) H dR(X,B n+1A) * \array{ \mathbf{H}(X, \mathbf{B}^n A) &\to& \mathbf{H}_{dR}(X, \mathbf{B}^{n+1} A) &\leftarrow& H_{dR}^{n+1}(X, A) \\ \downarrow^{\mathrlap{\phi^*}} && \downarrow^{\mathrlap{\phi^*}} && \downarrow \\ \mathbf{H}(\Sigma, \mathbf{B}^n A) &\to& \mathbf{H}_{dR}(X, \mathbf{B}^{n+1} A) &\leftarrow& * }

The holonomomy of \nabla over σ\sigma is the flat holonomy of ϕ *\phi^* \nabla

ϕ:= Σϕ *. \int_\phi \nabla := \int_{\Sigma} \phi^* \nabla \,.
Definition

Let ΣH\Sigma \in \mathbf{H} be of cohomological dimension dimΣ=ndim\Sigma = n \in \mathbb{N} and let c:XB nA\mathbf{c} : X \to \mathbf{B}^n A a representative of a characteristic class [c]H n(X,A)[\mathbf{c}] \in H^n(X, A) for some object XX. We say that the composite

exp(S c()):H(Σ,X)c^H diff(Σ,B nA)H flat(Σ,B nA) Στ 0Grpd(Π(Σ),ΠB nA) \exp(S_{\mathbf{c}}(-)) : \mathbf{H}(\Sigma, X) \stackrel{\hat \mathbf{c}}{\to} \mathbf{H}_{diff}(\Sigma, \mathbf{B}^n A) \stackrel{\simeq}{\to} \mathbf{H}_{flat}(\Sigma, \mathbf{B}^n A) \stackrel{\int_\Sigma}{\to} \tau_{\leq 0} \infty Grpd(\Pi(\Sigma), \Pi \mathbf{B}^n A)

where c^\hat \mathbf{c} denotes the refined Chern-Weil homomorphism induced by c\mathbf{c}, is the extended Chern-Simons functional induced by c\mathbf{c} on Σ\Sigma.

Remark

In the language of sigma-model quantum field theory the ingredients of this definition have the following interpretation

  • Σ\Sigma is the worldvolume of a fundamental (dimΣ1)(dim\Sigma-1)-brane ;

  • XX is the target space;

  • c^\hat \mathbf{c} is the background gauge field on XX;

  • H conn(Σ,X)\mathbf{H}_{conn}(\Sigma,X) is the space of worldvolume field configurations ϕ:ΣX\phi : \Sigma \to X or trajectories of the brane in XX;

  • exp(S c(ϕ))= Σϕ *c^\exp(S_{\mathbf{c}}(\phi)) = \int_\Sigma \phi^* \hat \mathbf{c} is the value of the action functional on the field configuration ϕ\phi.

In suitable situations this construction refines to an internal construction.

Assume that H\mathbf{H} has a canonical line object 𝔸 1\mathbb{A}^1 and a natural numbers object \mathbb{Z}. Then the action functional exp(iS())\exp(i S(-)) may lift to the internal hom with respect to the canonical cartesian closed monoidal structure on any (∞,1)-topos to a morphism of the form

exp(iS c()):[Σ,BG conn]B ndimΣ𝔸 1/. \exp(i S_{\mathbf{c}}(-)) : [\Sigma,\mathbf{B}G_{conn}] \to \mathbf{B}^{n-dim \Sigma}\mathbb{A}^1/\mathbb{Z} \,.

We call [Σ,BG conn][\Sigma, \mathbf{B}G_{conn}] the configuration space of the ∞-Chern-Simons theory defined by c\mathbf{c} and exp(iS c())\exp(i S_\mathbf{c}(-)) the action functional in codimension (ndimΣ)(n-dim\Sigma) defined on it.

See ∞-Chern-Simons theory for more discussion.

References

The category-theoretic definition of cohesive topos was proposed by Bill Lawvere. See the references at cohesive topos.

The observation that the further left adjoint Π\Pi in a locally ∞-connected (∞,1)-topos defines an intrinsic notion of paths and geometric homotopy groups in an (∞,1)-topos was suggested by Richard Williamson.

The observation that the further right adjoint coDisccoDisc in a local (∞,1)-topos serves to characterize concrete (∞,1)-sheaves was amplified by David Carchedi.

Several aspects of the discussion here are, more or less explicitly, in

For instance something similar to the notion of ∞-connected site and the fundamental ∞-groupoid in a locally ∞-connected (∞,1)-topos is the content of section 2.16. The infinitesimal path ∞-groupoid adjunction (RedΠ inf inf)(\mathbf{Red} \dashv \mathbf{\Pi}_{inf} \dashv \mathbf{\flat}_{inf}) is essentially discussed in section 3. The notion of geometric realization, 2, is touched on around remark 2.22, referring to

But, more or less explicitly, the presentation of geometric realization of simplicial presheaves is much older, going back to Artin-Mazur. See geometric homotopy groups in an (∞,1)-topos for a detailed commented list of literature.

A characterization of infinitesimal extensions and formal smoothness by adjoint functors (discussed at infinitesimal cohesion) is considered in

in the context of Q-categories .

The material presented here is also in section 2 of

A commented list of further related references is at

Created on April 17, 2011 10:12:37 by Urs Schreiber (89.204.153.112)