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

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

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

A cohesive (,1)(\infty,1)-topos is a general context for higher geometry with good cohomology and homotopy properties. We list fundamental structures and constructions that exist in every cohesive (,1)(\infty,1)-topos.

Concrete objects

The cohesive structure on an object in a cohesive (,1)(\infty,1)-topos need not be supported by points. We discuss a general abstract characterization of objects that do have an interpretation as bare nn-groupoids equipped with cohesive structure.

Compare with the section Quasitoposes of concrete objects at cohesive topos.

Proposition

On a cohesive (,1)(\infty,1)-topos H\mathbf{H} both Disc\mathrm{Disc} and coDisc\mathrm{coDisc} are full and faithful (∞,1)-functors and coDisc\mathrm{coDisc} exhibits ∞Grpd as a sub-(,1)(\infty,1)-topos of H\mathbf{H} by an
(∞,1)-geometric embedding

GrpdcoDiscΓH. \infty Grpd \stackrel{\overset{\Gamma}{\leftarrow}}{\underset{coDisc}{\hookrightarrow}} \mathbf{H} \,.
Proof

The full and faithfulness of DiscDisc and coDisccoDisc follows as in the discussion at ∞-connected (∞,1)-topos, Since Γ\Gamma is also a right adjoint it preserves in particular finite (∞,1)-limits, so that (ΓcoDisc)(\Gamma \dashv \mathrm{coDisc}) is indeed an (∞,1)-geometric morphism. (See the general discussion at local (∞,1)-topos.)

Proposition

The (∞,1)-topos ∞Grpd is equivalent to the full sub-(∞,1)-category of H\mathbf{H} on those objects XHX \in \mathbf{H} for which the unit

XcoDiscΓX X \to \mathrm{coDisc}\Gamma X

is an equivalence.

Proof

This follows by general facts discussed at reflective sub-(∞,1)-category.

Definition

We say an object XX is nn-concrete if the canonical morphism XcoDiscΓXX \to coDisc \Gamma X is (n-1)-truncated.

If a 0-truncated object XX is 00-concrete, we call it just concrete.

See also concrete (∞,1)-sheaf.

Proposition

For CC an ∞-cohesive site, a 0-truncated object in the (∞,1)-topos over CC is concrete precisely if it is a concrete sheaf in the traditional sense.

Definition

For XHX \in \mathbf{H} and nn \in \mathbb{N}, the (n+1)(n+1)-concretification of XX is the morphism

Xconc n+1X X \to conc_{n+1} X

that is the left factor in the decomposition with respect to the n-connected/n-truncated factorization system of the (ΓcoDisc)(\Gamma \dashv coDisc)-unit

conc n+1X X coDiscΓX. \array{ && conc_{n+1} X \\ & \nearrow && \searrow \\ X &&\to&& coDisc \Gamma X } \,.
Remark

By that very n-connected/n-truncated factorization system we have that conc n+1Xconc_{n+1} X is an (n+1)(n+1)-concrete object.

Cohesive \infty-Groups

Every (∞,1)-topos H\mathbf{H} comes with a notion of ∞-group objects that generalizes the traditional notion of grouplike A A_\infty spaces in Top \simeq ∞Grpd. For more details on the following see also looping and delooping.

Definition

For XHX \in \mathbf{H} an object and x:*Xx : * \to X a point, the loop space object of XX is the (∞,1)-pullback Ω xX:=*× X*\Omega_x X := * \times_X *:

Ω xX * x * x X. \array{ \Omega_x X & \to & {*} \\ \downarrow && \downarrow^{\mathrlap{x}} \\ {*} & \stackrel{x}{\to} & X } \,.

This object Ω xX\Omega_x X is canonically equipped with the structure of an ∞-group obect.

Remark

Notice that every 0-connected object AA in the cohesive (,1)(\infty,1)-topos H\mathbf{H} does have a global point (then necessarily essentially unique) *A* \to A.

This follows from the above proposition which says that H\mathbf{H} necessarily has homotopy dimension 0\leq 0.

Proposition

The operation of forming loop space objects in H\mathbf{H} establishes an equivalence of (∞,1)-categories

Ω:PointedConnected(H)Grp(H) \Omega : PointedConnected(\mathbf{H}) \stackrel{\simeq}{\to} Grp(\mathbf{H})

between the (∞,1)-category of group objects in H\mathbf{H} and the full sub-(∞,1)-category of pointed objects */H*/\mathbf{H} on those that are 0-connected.

Proof

By the discussion at delooping.

We write

B:Grpd(H)PointedConnected(H) \mathbf{B} : Grpd(\mathbf{H}) \to PointedConnected(\mathbf{H})

for the inverse to Ω\Omega. For GGrp(H)G \in Grp(\mathbf{H}) we call BGPointedConnected(H)H\mathbf{B}G \in PointedConnected(\mathbf{H}) \hookrightarrow \mathbf{H} the delooping of GG.

Notice that since the cohesive (,1)(\infty,1)-topos H\mathbf{H} has homotopy dimension 00 by the above proposition every 0-connected object has an essentially unique point, but nevertheless the homotopy type of */H(BG,BH)*/\mathbf{H}(\mathbf{B}G, \mathbf{B}H) may differ from that of H(BG,BH)\mathbf{H}(\mathbf{B}G, \mathbf{B}H).

Observation

The delooping object BGH\mathbf{B}G \in \mathbf{H} is concrete precisely if GG is.

We may therefore unambiguously speak of concrete cohesive \infty-groups.

Definition

For f:YZf : Y \to Z any morhism in H\mathbf{H} and z:*Zz : * \to Z a point, the ∞-fiber or of ff over this point is the (∞,1)-pullback X:=*× ZY X := {*} \times_Z Y

X * z Y f Z. \array{ X & \to & {*} \\ \downarrow && \downarrow^{\mathrlap{z}} \\ Y & \stackrel{f}{\to} & Z } \,.
Observation

Suppose that also YY is pointed and ff is a morphism of pointed objects. Then the \infty-fiber of an \infty-fiber is the loop space object of the base.

This means that we have a diagram

Ω zZ X * * Y f Z, \array{ \Omega_z Z & \to& X & \to & {*} \\ \downarrow && \downarrow && \downarrow \\ {*} & \to & Y & \stackrel{f}{\to} & Z } \,,

where the outer rectangle is an (∞,1)-pullback if the left square is an (∞,1)-pullback. This follows from the pasting law for (,1)(\infty,1)-pullbacks in any (∞,1)-category.

Proposition

If the cohesive (,1)(\infty,1)-topos H\mathbf{H} has an ∞-cohesive site of definition CC, then

  • every ∞-group object has a presentation by a presheaf of simplicial groups

    G[C op,sGrp]U[C op,sSet] G \in [C^{op}, sGrp] \stackrel{U}{\to} [C^{op}, sSet]

    which is fibrant in [C op,sSet] proj[C^{op}, sSet]_{proj};

  • the corresponding delooping object is presented by the presheaf

    W¯G[C op,sSet 0][C op,sSet] \bar W G \in [C^{op}, sSet_0] \hookrightarrow [C^{op}, sSet]

    which is given over each UCU \in C by W¯(G(U))\bar W (G(U)) (see simplicial group for the notation).

Proof

Let *X[C op,sSet] proj,loc* \to X \in [C^{op}, sSet]_{proj,loc} be a locally fibrant representative of *BG* \to \mathbf{B}G. Since the terminal object ** is indeed presented by the presheaf constant on the point we have functorial choices of basepoints in all the X(U)X(U) for all UCU \in C and by assumption that XX is connected all the X(U)X(U) are connected. Hence without loss of generality we may assume that XX is presented by a presheaf of reduced simplicial sets X[C op,sSet 0]X[C op,sSet]X \in [C^{op}, sSet_0] \hookrightarrow X \in [C^{op}, sSet].

Then notice the Quillen equivalence between the model structure on reduced simplicial sets and the model structure on simplicial groups

(ΩW¯):sGrpW¯ΩsSet 0. (\Omega \dashv \bar W) : sGrp \stackrel{\overset{\Omega}{\leftarrow}}{\underset{\bar W}{\to}} sSet_0 \,.

In particular its unit is a weak equivalence

W¯ΩYY \bar W \Omega Y \stackrel{\simeq}{\to} Y

for every YsSet 0sSet QuillenY \in sSet_0 \hookrightarrow sSet_{Quillen} and W¯ΩY\bar W \Omega Y is always a Kan complex. Therefore

W¯ΩX[C op,sSet] proj \bar W \Omega X \in [C^{op}, sSet]_{proj}

is an equivalent representative for XX, fibrant at least in the global model structure. Since the finite (∞,1)-limit involved in forming loop space objects is equivalently computed in the global model structure, it is sufficient to observe that

ΩX WΩX * W¯ΩX \array{ \Omega X &\to& W \Omega X \\ \downarrow && \downarrow \\ * &\to& \bar W \Omega X }

is

  • a pullback diagram in [C op,sSet][C^{op}, sSet] (because it is so over each UCU \in C by the general discussion at simplicial group);

  • a homotopy pullback of the point along itself (since WGW¯GW G \to \bar W G is objectwise a fibration resolution of the point inclusion).

Cohomology and principal \infty-bundles

There is an intrinsic notion of cohomology and of principal ∞-bundles in any (∞,1)-topos H\mathbf{H}.

Definition

For X,AHX,A \in \mathbf{H} two objects, we say that

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

is the cohomology set of XX with coefficients in AA. If A=GA = G is an ∞-group we write

H 1(X,G):=π 0H(X,BG) H^1(X,G) := \pi_0 \mathbf{H}(X, \mathbf{B}G)

for cohomology with coefficients in its delooping. Generally, if KHK \in \mathbf{H} has a pp-fold delooping, we write

H p(X,K):=π 0H(X,B pK). H^p(X,K) := \pi_0 \mathbf{H}(X, \mathbf{B}^p K) \,.

In the context of cohomology on XX with coefficients in AA we we say that

  • the hom-space H(X,A)\mathbf{H}(X,A) is the cocycle \infty-groupoid;

  • a morphism g:XAg : X \to A is a cocycle;

  • a 2-morphism : ghg \Rightarrow h is a coboundary between cocycles.

  • a morphism c:ABc : A \to B represents the characteristic class

    [c]:H(,A)H(,B). [c] : H(-,A) \to H(-,B) \,.
Definition

For every morphism c:BGBHHc : \mathbf{B}G \to \mathbf{B}H \in \mathbf{H} define the long fiber sequence to the left

ΩGΩHFGHBFBGcBH \cdots \to \Omega G \to \Omega H \to F \to G \to H \to \mathbf{B} F \to \mathbf{B}G \stackrel{c}{\to} \mathbf{B}H

to be the given by the consecutive pasting diagrams of (∞,1)-pullbacks

ΩH G * * H BF * * BG c BH. \array{ \vdots && \vdots \\ \Omega H &\to& G &\to& * \\ \downarrow &&\downarrow && \downarrow \\ * &\to& H &\to& \mathbf{B}F &\to& * \\ &&\downarrow && \downarrow && \downarrow \\ && * &\to& \mathbf{B}G & \stackrel{c}{\to} & \mathbf{B}H } \,.
Proposition
  • The long fiber sequence to the left of c:BGBHc : \mathbf{B}G \to \mathbf{B}H becomes constant on the point after nn iterations if HH is nn-truncated.

  • For every object XHX \in \mathbf{H} we have a long exact sequence of pointed sets

    H 0(X,G)H 0(X,H)H 1(X,F)H 1(X,G)H 1(X,H). \cdots \to H^0(X,G) \to H^0(X,H) \to H^1(X,F) \to H^1(X,G) \to H^1(X,H) \,.
Proof

The first statement follows from the observation that a loop space object Ω xA\Omega_x A is a fiber of the free loop space object A\mathcal{L} A and that this may equivalently be computed by the (∞,1)-powering A S 1A^{S^1}, where S 1TopGrpdS^1 \in Top \simeq \infty Grpd is the circle. (See Hochschild cohomology for details.)

The second statement follows by observing that the \infty-hom-functor H(X,)\mathbf{H}(X,-) preserves all (∞,1)-limits, so that we have (∞,1)-pullbacks

H(X,F) * H(X,G) H(X,H) \array{ \mathbf{H}(X,F) &\to &* \\ \downarrow && \downarrow \\ \mathbf{H}(X,G) &\to& \mathbf{H}(X,H) }

etc. in ∞Grpd at each stage of the fiber sequence. The statement then follows with the familiar long exact sequence for homotopy groups in Top \simeq ∞Grpd.

To every cocycle g:XBGg : X \to \mathbf{B}G is canonically associated its homotopy fiber PXP \to X, the (∞,1)-pullback

P * X g BG.. \array{ P &\to& {*} \\ \downarrow && \downarrow \\ X &\stackrel{g}{\to}& \mathbf{B}G \,. } \,.

We discuss now that PP canonically has the structure of a GG-principal ∞-bundle and that BG\mathbf{B}G is the fine moduli space for GG-principal \infty-bundles.

Definition

(principal GG-action)

Let GG be a group object in the (∞,1)-topos H\mathbf{H}. A principal action of GG on an object PHP \in \mathbf{H} is a groupoid object in the (∞,1)-topos P//GP//G that sits over *//G*//G in that we have a morphism of simplicial diagrams Δ opH\Delta^{op} \to \mathbf{H}

P×G×G (p 2,p 3) G×G P×G p 2 G P * \array{ \vdots && \vdots \\ P \times G \times G &\stackrel{(p_2, p_3)}{\to}& G \times G \\ \downarrow\downarrow\downarrow && \downarrow\downarrow\downarrow \\ P \times G &\stackrel{p_2}{\to}& G \\ \downarrow\downarrow && \downarrow\downarrow \\ P &\stackrel{}{\to}& {*} }

in H\mathbf{H}.

We say that the (∞,1)-colimit

X:lim (P//G:Δ opH) X : \lim_\to (P//G : \Delta^{op} \to \mathbf{H})

is the base space defined by this action.

We may think of P//GP//G as the action groupoid of the GG-action on PP. For us it defines this GG-action.

Proposition

The GG-principal action as defined above satisfies the principality condition in that we have an equivalence of groupoid objects

P× XP× XP P×G×G P× XP P×G P P. \array{ \vdots && \vdots \\ P \times_X P \times_X P &\stackrel{\simeq}{\to}& P \times G \times G \\ \downarrow\downarrow\downarrow && \downarrow\downarrow\downarrow \\ P \times_X P &\stackrel{\simeq}{\to}& P \times G \\ \downarrow\downarrow && \downarrow\downarrow \\ P &\stackrel{\simeq}{\to}& P } \,.
Proof

This principality condition asserts that the groupoid object P//GP//G is effective. By Giraud's axioms characterizing (∞,1)-toposes, every groupoid object in H\mathbf{H} is effective.

Proposition

For XBGX \to \mathbf{B}G any morphism, its homotopy fiber PXP \to X is canonically equipped with a principal GG-action with base space XX.

Proof

By the above we need to show that we have a morphism of simplicial diagrams

P× XP× XP P×G×G G×G P× XP P×G p 2 G P = P * X = X g BG, \array{ \vdots && \vdots && \vdots \\ P \times_X P \times_X P &\stackrel{\simeq}{\to}& P \times G \times G &\to& G \times G \\ \downarrow\downarrow\downarrow && \downarrow\downarrow\downarrow && \downarrow\downarrow\downarrow \\ P \times_X P &\stackrel{\simeq}{\to}& P \times G &\stackrel{p_2}{\to}& G \\ \downarrow\downarrow && \downarrow\downarrow && \downarrow\downarrow \\ P &\stackrel{=}{\to}& P &\stackrel{}{\to}& {*} \\ \downarrow && \downarrow && \downarrow \\ X &\stackrel{=}{\to}& X &\stackrel{g}{\to}& \mathbf{B}G } \,,

where the left horizontal morphisms are equivalences, as indicated. We proceed by induction through on the height of this diagram.

The defining (∞,1)-pullback square for P× XPP \times_X P is

P× XP P P X \array{ P \times_X P &\to& P \\ \downarrow && \downarrow \\ P &\to& X }

To compute this, we may attach the defining (,1)(\infty,1)-pullback square of PP to obtain the pasting diagram

P× XP P * P X BG. \array{ P \times_X P &\to& P &\to& {*} \\ \downarrow && \downarrow && \downarrow \\ P &\to& X &\to& \mathbf{B}G \,. }

and use the pasting law for pullbacks, to conclude that P× XPP \times_X P is the pullback

P× XP * P X BG. \array{ P \times_X P &\to& &\to& {*} \\ \downarrow && && \downarrow \\ P &\to& X &\to& \mathbf{B}G \,. }

By defnition of PP as the homotopy fiber of XBGX \to \mathbf{B}G, the lower horizontal morphism is equivalent to P*BGP \to {*} \to \mathbf{B}G, so that P× XPP \times_X P is equivalent to the pullback

P× XP * P * BG. \array{ P \times_X P &\to& &\to& {*} \\ \downarrow && && \downarrow \\ P &\to& {*} &\to& \mathbf{B}G \,. }

This finally may be computed as the pasting of two pullbacks

P× XP P×G G * P * BG. \array{ P \times_X P &\simeq& P \times G &\to& G &\to& {*} \\ &&\downarrow && \downarrow && \downarrow \\ &&P &\to& {*} &\to& \mathbf{B}G \,. }

of which the one on the right is the defining one for GG and the remaining one on the left is just an (∞,1)-product.

Proceeding by induction from this case we find analogousy that P × X n+1P×G × nP^{\times_X^{n+1}} \simeq P \times G^{\times_n}: suppose this has been shown for (n1)(n-1), then the defining pullback square for P × X n+1P^{\times_X^{n+1}} is

P× XP × X n P P × X n X. \array{ P \times_X P^{\times_X^n} &\to& P \\ \downarrow && \downarrow \\ P^{\times_X^n}&\to& X } \,.

We may again paste this to obtain

P× XP × X n P * P × X n X BG \array{ P \times_X P^{\times_X^n} &\to& P &\to& * \\ \downarrow && \downarrow && \downarrow \\ P^{\times_X^n}&\to& X &\to& \mathbf{B}G }

and use from the previous induction step that

(P × X nXBG)(P × X n*BG) (P^{\times_X^n} \to X \to \mathbf{B}G) \simeq (P^{\times_X^n} \to * \to \mathbf{B}G)

to conclude the induction step with the same arguments as before.

Definition

We say a GG-principal action of GG on PP over XX is a GG-principal ∞-bundle if the colimit over P//G*//GP//G \to *//G produces a pullback square: the bottom square in

P×G×G G×G P×G p 2 G P * X=lim (P×G ) g BG=lim (G ). \array{ \vdots && \vdots \\ P \times G \times G &\to& G \times G \\ \downarrow\downarrow\downarrow && \downarrow\downarrow\downarrow \\ P \times G &\stackrel{p_2}{\to}& G \\ \downarrow\downarrow && \downarrow\downarrow \\ P &\stackrel{}{\to}& {*} \\ \downarrow && \downarrow \\ X = \lim_\to (P \times G^\bullet) &\stackrel{g}{\to}& \mathbf{B}G = \lim_\to( G^\bullet) } \,.

Of special interest are principal \infty-bundles of the form PBGP \to \mathbf{B}G:

Definition

We say a sequence of cohesive ∞-groups

AG^G A \to \hat G \to G

exhibits G^\hat G as an extension of GG by AA if the corresponding delooping sequence

BABG^BG \mathbf{B}A \to \mathbf{B}\hat G \to \mathbf{B}G

if a fiber sequence. If this fiber sequence extends one step further to the right to a morphism ϕ:BGB 2A\phi : \mathbf{B}G \to \mathbf{B}^2 A, we have by def. 8 that BG^BG\mathbf{B}\hat G \to \mathbf{B}G is the BA\mathbf{B}A-principal ∞-bundle classified by the cocycle ϕ\phi; and BABG^\mathbf{B}A \to \mathbf{B}\hat G is its fiber over the unique point of BG\mathbf{B}G.

Given an extension and a a GG-principal ∞-bundle PXP \to X in H\mathbf{H} we say a lift P^\hat P of PP to a G^\hat G-principal \infty-bundle is a factorization of its classifying cocycle g:XBGg : X \to \mathbf{B}G through the extension

BG^ g^ X g BG. \array{ && \mathbf{B}\hat G \\ & {}^{\mathllap{\hat g}}\nearrow & \downarrow \\ X &\stackrel{g}{\to}& \mathbf{B}G } \,.
Proposition

Let AG^GA \to \hat G \to G be an extension of \infty-groups, def. 9 in H\mathbf{H} and let PXP \to X be a GG-principal ∞-bundle.

Then a G^\hat G-extension P^X\hat P \to X of PP is in particular also an AA-principal \infty-bundle P^P\hat P \to P over PP with the property that its restriction to any fiber of PP is equivalent to G^G\hat G \to G.

We may summarize this as saying:

An extension of \infty-bundles is an \infty-bundle of extensions.

Proof

This follows from repeated application of the pasting law for (∞,1)-pullbacks: consider the following diagram in H\mathbf{H}

G^ P^ * G P q BA * * x X g^ BG^ BG. \array{ \hat G &\to& \hat P &\to& * \\ \downarrow && \downarrow && \downarrow \\ G &\to& P &\stackrel{q}{\to}& \mathbf{B}A &\to& {*} \\ \downarrow && \downarrow && \downarrow && \downarrow \\ {*} &\stackrel{x}{\to}& X &\stackrel{\hat g}{\to}& \mathbf{B}\hat G &\stackrel{}{\to}& \mathbf{B}G } \,.

The bottom composite g:XBGg : X \to \mathbf{B}G is a cocycle for the given GG-principal \infty-bundle PXP \to X and it factors through g^:XBG^\hat g : X \to \mathbf{B}\hat G by assumption of the existence of the extension P^P\hat P \to P.

Since also the bottom right square is an \infty-pullback by the given \infty-group extension, the pasting law asserts that the square over g^\hat g is also a pullback, and then that so is the square over qq. This exhibits P^\hat P as an AA-principal \infty-bundle over PP.

Now choose any point x:*Xx : {*} \to X of the base space as on the left of the diagram. Pulling this back upwards through the diagram and using the pasting law and the definition of loop space objects GΩBG* BG*G \simeq \Omega \mathbf{B}G \simeq * \prod_{\mathbf{B}G} * the diagram completes by (,1)(\infty,1)-pullback squares on the left as indicated, which proves the claim.

\infty-Gerbes

For the moment see the discussion at ∞-gerbe .

Twisted cohomology and section

A slight variant of cohomology is often relevant: twisted cohomology.

Definition

For H\mathbf{H} an (∞,1)-topos let c:BC\mathbf{c} : B \to C a morphism representing a characteristic class [c]H(B,C)[\mathbf{c}] \in H(B,C). Let CC be pointed and write ABA \to B for its homotopy fiber.

We say that the twisted cohomology with coefficients in AA relative to c\mathbf{c} is the intrinsic cohomology of the over-(∞,1)-topos H/C\mathbf{H}/C with coefficients in ff.

If c\mathbf{c} is understood and ϕ:XB\phi : X \to B is any morphism, we write

H ϕ(X,A):=H/C(ϕ,c) \mathbf{H}_{\phi}(X, A) := \mathbf{H}/C(\phi, \mathbf{c})

and speak of the cocycle ∞-groupoid of twisted cohomology on XX with coefficients in AA and twisting cocycle ϕ\phi relative to [c][\mathbf{c}] .

For short we often say twist for twisting cocycle .

Proposition

We have the following immediate properties of twisted cohomology:

  • The ϕ\phi-twisted cohomology relative to c\mathbf{c} depends, up to equivalence, only on the characteristic class [c]H(B,C)[\mathbf{c}] \in H(B,C) represented by c\mathbf{c} and also only on the equivalence class [ϕ]H(X,C)[\phi] \in H(X,C) of the twist.

  • If the characteristic class is terminal, c:B*\mathbf{c} : B \to * we have ABA \simeq B and the corresponding twisted cohomology is ordinary cohomology with coefficients in AA.

Proposition

Let the characteristic class c:BC\mathbf{c} : B \to C and a twist ϕ:XC\phi : X \to C be given. Then the cocycle \infty-groupoid of twisted AA-cohomology on XX is given by the (∞,1)-pullback

H ϕ(X,A) * ϕ H(X,B) c * H(X,C) \array{ \mathbf{H}_{\phi}(X,A) &\to& {*} \\ \downarrow && \downarrow^{\mathrlap{\phi}} \\ \mathbf{H}(X,B) &\stackrel{\mathbf{c}_*}{\to}& \mathbf{H}(X,C) }

in ∞Grpd.

Proof

This is an application of the general pullback-formula for hom-spaces in an over-(∞,1)-category. See there for details.

Proposition

If the twist is trivial, ϕ=0\phi = 0 (meaning that it factors as ϕ:X*C\phi : X \to * \to C through the point of the pointed object CC), the corresponding twisted AA-cohomology is equivalent to ordinary AA-cohomology

H ϕ=0(X,A)H(X,A). \mathbf{H}_{\phi = 0}(X,A) \simeq \mathbf{H}(X,A) \,.
Proof

In this case we have that the characterizing (,1)(\infty,1)-pullback diagram from prop. 12 is the image under the hom-functor H(X,):HGrpd\mathbf{H}(X,-) : \mathbf{H} \to \infty Grpd of the pullback diagram BcC*B \stackrel{\mathbf{c}}{\to} C \leftarrow *. By definition of AA as the homotopy fiber of c\mathbf{c}, its pullback is AA. Since the hom-functor H(X,)\mathbf{H}(X,-) preserves (∞,1)-pullbacks the claim follows:

H ϕ=0(X,A) H(X,B) H(X,C)H(X,*) H(X,B C*) H(X,A). \begin{aligned} \mathbf{H}_{\phi = 0 }(X,A) & \simeq \mathbf{H}(X,B) \prod_{\mathbf{H}(X,C)} \mathbf{H}(X,*) \\ & \simeq \mathbf{H}(X, B \prod_C *) \\ & \simeq \mathbf{H}(X,A) \end{aligned} \,.

Often twisted cohomology is formulated in terms of homotopy classes of sections of a bundle. The following asserts that this is equivalent to the above definition.

By the discussion at Cohomology and principal ∞-bundles we may understand the twist ϕ:XC\phi : X \to C as the cocycle for an ΩC\Omega C-principal ∞-bundle over XX, being the (∞,1)-pullback of the point inclusion *C* \to C along ϕ\phi, where the point is the homotopy-incarnation of the universal ΩC\Omega C-principal \infty-bundle. The characteristic class BCB \to C in turn we may think of as an ΩA\Omega A-bundle associated to this universal bundle. Accordingly the pullback of P ϕ:=X× CBP_\phi := X \times_C B is the associated ΩA\Omega A-bundle over XX classified by ϕ\phi.

Proposition

Let P ϕ:=X× CBP_\phi := X \times_C B be (∞,1)-pullback of the characteristic class c\mathbf{c} along the twisting cocycle ϕ\phi

P ϕ B p c X ϕ C. \array{ P_\phi &\to& B \\ {}^{\mathllap{p}}\downarrow && \downarrow^{\mathrlap{\mathbf{c}}} \\ X &\stackrel{\phi}{\to}& C } \,.

Then the ϕ\phi-twisted AA-cohomology of XX is equivalently the space of sections Γ X(P ϕ)\Gamma_X(P_\phi) of P ϕP_\phi over XX:

H tw,ϕ(X,A)Γ X(P ϕ), \mathbf{H}_{tw,\phi}(X,A) \simeq \Gamma_X(P_\phi) \,,

where on the right we have the (∞,1)-pullback

Γ X(P ϕ) * id H(X,P ϕ) p * H(X,X). \array{ \Gamma_X(P_\phi) &\to& * \\ \downarrow && \downarrow^{\mathrm{id}} \\ \mathbf{H}(X,P_\phi) &\stackrel{p_*}{\to}& \mathbf{H}(X,X) } \,.
Proof

Consider the pasting diagram

H ϕ(X,A) Γ ϕ(X) * id H(X,P ϕ) p * H(X,X) ϕ * H(X,B) c * H(X,C). \array{ \mathbf{H}_{\phi}(X,A) \simeq & \Gamma_\phi(X) &\to& {*} \\ & \downarrow && \downarrow^{\mathrlap{id}} \\ & \mathbf{H}(X,P_{\phi}) &\stackrel{p_*}{\to}& \mathbf{H}(X,X) \\ & \downarrow && \downarrow^{\mathrlap{\phi}_*} \\ & \mathbf{H}(X,B) &\stackrel{\mathbf{c}_*}{\to}& \mathbf{H}(X,C) } \,.

By the fact that the hom-functor H(X,)\mathbf{H}(X,-) preserves (∞,1)-limits the bottom square is an (∞,1)-pullback. By the pasting law for (∞,1)-pullbacks so is then the total outer diagram. Noticing that the right vertical composite is *ϕH(X,C)* \stackrel{\mathbf{\phi}}{\to} \mathbf{H}(X,C) the claim follows with prop. 12.

Note

In applications one is typically interested in situations where the characteristic class [c][\mathbf{c}] and the domain XX is fixed and the twist ϕ\phi varies. Since by prop. 11 only the equivalence class [ϕ]H(X,C)[\phi] \in H(X,C) matters, it is sufficient to pick one representative ϕ\phi in each equivalence class. Such as choice is equivalently a choice of section

H(X,C):=π 0H(X,C)H(X,C) H(X,C) := \pi_0 \mathbf{H}(X,C) \to \mathbf{H}(X,C)

of the 0-truncation projection H(X,C)H(X,C)\mathbf{H}(X,C) \to H(X,C) from the cocycle \infty-groupoid to the set of cohomology classes. This justifies the following terminology.

Definition

With a characteristic class [c]H(B,C)[\mathbf{c}] \in H(B,C) with homotopy fiber AA understood, we write

H tw(X,A):= [ϕ]H(X,C)H tw,ϕ(X,A) \mathbf{H}_{tw}(X,A) := \coprod_{[\phi] \in H(X,C)} \mathbf{H}_{tw, \phi}(X,A)

for the union of all twisted cohomology cocycle \infty-groupoids.

Observation

We have that H tw(X,A)\mathbf{H}_{tw}(X,A) is the (∞,1)-pullback

H tw(X,A) tw H(X,C) H(X,B) c * H(X,C), \array{ \mathbf{H}_{tw}(X,A) &\stackrel{tw}{\to}& H(X,C) \\ \downarrow && \downarrow \\ \mathbf{H}(X,B) &\stackrel{\mathbf{c}_*}{\to}& \mathbf{H}(X,C) } \,,

where the right vertical morphism in any section of the projection from CC-cocycles to CC-cohomology.

Note

When the (∞,1)-topos H\mathbf{H} is presented by a model structure on simplicial presheaves and model for XX and CC is chosen, then the cocycle ∞-groupoid H(X,C)\mathbf{H}(X,C) is presented by an explicit simplicial presheaf H(X,C) simpsSet\mathbf{H}(X,C)_{simp} \in sSet. Once these choices are made, there is therefore the inclusion of simplicial presheaves

const(H(X,C) simp) 0H(X,C) simp, const (\mathbf{H}(X,C)_{simp})_0 \to \mathbf{H}(X,C)_{simp} \,,

where on the left we have the simplicially constant object on the vertices of H(X,C) simp\mathbf{H}(X,C)_{simp}. This morphism, in turn, presents a morphism in Grpd\infty Grpd that in general contains a multitude of copies of the components of any H(X,C)H(X,C)H(X,C) \to \mathbf{H}(X,C): a multitude of representatives of twists for each cohomology class of twists. Since by the above the twisted cohomology does not depend, up to equivalence, on the choice of representative, the coresponding (,1)(\infty,1)-pullback yields in general a larger coproduct of \infty-groupoids as the corresponding twisted cohomology. This however just contains copies of the homotopy types already present in H tw(X,A)\mathbf{H}_{tw}(X,A) as defined above.

\infty-Group representations and associated \infty-bundles

The material to go here is at Schreiber, section 2.3.7.

(…)

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 / étale homotopy

We discuss canonical internal realizations of the notions of étale homotopy, geometric homotopy groups in an (infinity,1)-topos and local systems .

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 cohesive ∞-groupoids of XX, or just the geometric realization for short.

Note

In presentations of H\mathbf{H} by a model structure on simplicial presheaves – as discussed at ∞-cohesive site – 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}.

Cohesive 𝔸 1\mathbb{A}^1-homotopy / The Continuum

Definition

An object 𝔸 1H\mathbb{A}^1 \in \mathbf{H} is called a line object exhibiting the cohesion of H\mathbf{H} if the shape modality ʃʃ (hence the reflector Π:HGrpd\Pi : \mathbf{H} \to \infty Grpd) exhibits the localization of an (∞,1)-category of H\mathbf{H} at the class of morphisms {X×𝔸 1X} XH\{ X \times \mathbb{A}^1 \to X \}_{X \in \mathbf{H}}.

Example

The cohesion of Smooth∞Grpd is exhibited (in the sense defined here) by the real line (the standard continuum) under the canonical embedding \mathbb{R} \in SmoothMfd \hookrightarrow Smooth∞Grpd.

This is (dcct, 3.9.1).

Remark

The analogous notion in infinitesimal cohesion is discussed in infinitesimal cohesion – infinitesimal A1-homotopy-topos+–+infinitesimal+cohesion#InfinitesimalA1Homotopy).

See also at

Galois theory

We discuss a canonical internal notion of Galois theory in H\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, 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 xXx \in X a fiber P xP_x in AA;

  • to each path γ:x 1x 2\gamma : x_1 \to x_2 in XX an equivalence (γ):P x 1P x 2\nabla(\gamma) : P_{x_1} \to P_{x_2} between these fibers (the parallel transport along γ\gamma);

  • to each disk Σ\Sigma in XX a 2-equivalalence (Σ)\nabla(\Sigma) between these equivaleces associated to its boundary (the higher parallel transport)

  • and so on.

P x 2 A (γ 1) (Σ) (γ 2) P x 1 (γ 3) P x 3 x 2 Π(X) γ 1 Σ γ 2 x 1 γ 3 x 3 \array{ && && P_{x_2} \\ A && & {}^{\mathllap{\nabla(\gamma_1)}}\nearrow & \Downarrow^{\nabla(\Sigma)} & \searrow^{\mathrlap{\nabla(\gamma_2)}} \\ \uparrow^{\mathrlap{\nabla}} && P_{x_1} &&\underset{\nabla(\gamma_3)}{\to}&& P_{x_3} \\ && && x_2 \\ \mathbf{\Pi}(X) && & {}^{\mathllap{\gamma_1}}\nearrow & \Downarrow^{\Sigma} & \searrow^{\mathrlap{\gamma_2}} \\ && x_1 &&\underset{\gamma_3}{\to}&& x_3 }

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.

For a non-flat \infty-connection the parallel transport (γ 3 1γ 2γ 1)\nabla(\gamma_3^{-1}\circ \gamma_2\circ \gamma_1) around a contractible loop as above need not be equivalent to the identity. We will obtain a formal notion of non-flat parallel transport below.

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.

The following lists some basic properties of objects of the form A\mathbf{\flat}A and their interpretation in terms of flat \infty-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.

Remark

Since (DiscΓ)(Disc \dashv \Gamma) is a coreflection, we have that for any cohesive \infty-groupoid AA the underlying discrete ∞-groupoid ΓA\Gamma A coincides with the underlying \infty-groupoid ΓA\Gamma \mathbf{\flat}A of A\mathbf{\flat}A:

ΓAΓA. \Gamma \mathbf{\flat} A \stackrel{\simeq}{\to} \Gamma A \,.

To interpret this it is useful to think of AA as a moduli stack for principal \infty-bundles. This is most familiar in the case that AA is connected, in which case by the above we write it A=BGA = \mathbf{B}G for some cohesive ∞-group GG.

In terms of this we may say that

  1. BG\mathbf{B}G is the moduli ∞-stack of GG-principal ∞-bundles;

  2. BG\mathbf{\flat} \mathbf{B}G is the moduli \infty-stack of GG-principal \infty-bundles equipped with a flat \infty-connection.

Therefore

ΓBGH(*,BG) \Gamma \mathbf{B}G \simeq \mathbf{H}(*, \mathbf{B}G)

is the ∞-groupoid of GG-principal \infty-bundles over the point (the terminal object in H\mathbf{H}). Similarly

ΓBGH(*,BG) \Gamma \mathbf{\flat}\mathbf{B}G \simeq \mathbf{H}(*, \mathbf{\flat}\mathbf{B}G)

is the \infty-groupoid of flat GG-principal \infty-bundles over the point.

So the equivalence ΓBGΓBG\Gamma \mathbf{\flat}\mathbf{B}G \simeq \Gamma \mathbf{B}G says that over the point every GG-principal \infty-bundle carries an essentially unique flat \infty-connection. This is certainly what one expects, and certainly the case for ordinary connections on ordinary principal bundles.

Notice here that the axioms of cohesion imply in particular that the terminal object *H* \in \mathbf{H} really behaves like a geometric point: it has underlying it a single point, Γ**\Gamma * \simeq *, and its geometric homotopy type is that of the point, Π(*)*\Pi(*) \simeq *.

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 } \,.

We also say dR\flat_{dR} is the dR-flat modality and Π dR\Pi_{dR} is the dR-shape modality.

Proposition

The construction in def. 24 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) \,.
Proposition

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.

Integration of differential forms and Stokes lemma

See at integration of differential forms – In cohosive homotopy-type theory

Exponentiated \infty-Lie algebras

We now use the intrinsic non-abelian de Rham cohomology in the cohesive (,1)(\infty,1)-topos H\mathbf{H} discussed above to see that there is also an intrinsic notion of exponentiated higher Lie algebra objects in H\mathbf{H}. (The fact that for H=\mathbf{H} = Smooth∞Grpd these abstractly defined objects are indeed presented by L-∞ algebras is disucssed at smooth ∞-groupoid -- structures.)

The idea is that for GGrp(H)G \in Grp(\mathbf{H}) an ∞-group, a GG-valued differential form on some XHX \in \mathbf{H}, which by the above is given by a morphism

A:Π dR(X)BG A : \mathbf{\Pi}_{dR}(X) \to \mathbf{B}G

maps “infitesimal paths” to elements of GG, and hence only hits “infinitesimal elements” in GG. Therefore the object that such forms universally factor through we write Bexp(𝔤)\mathbf{B} \exp(\mathfrak{g}) and think of as the formal Lie integration of the \infty-Lie algebra of GG.

The reader should note here that all this is formulated without an explicit (“synthetic”) notion of infinitesimals. Instead, it is infinitesimal in the same sense that Π dR(X)\mathbf{\Pi}_{dR}(X) is the schematic de Rham homotopy type of XX, as discussed above. But if we add a bit more structure to the cohesive (,1)(\infty,1)-topos H\mathbf{H}, then these infinitesimals can be realized also synthetically. That extra structure is that of infinitesimal cohesion. See there for more details.

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) \,.
Remark

For GG an ∞-group, there are canonical GG-∞-actions on GG and on dRBG\flat_{dR} \mathbf{B}G. By the discussion at ∞-action these are exhibited by the defining homotopy fiber sequences

G * BG \array{ G &\longrightarrow& \ast \\ && \downarrow \\ && \mathbf{B}G }

and

dRBG BG BG, \array{ \flat_{dR}\mathbf{B}G &\longrightarrow& \flat \mathbf{B}G \\ && \downarrow \\ && \mathbf{B}G } \,,

respectively, and they identify the homotopy quotients of the action as

*G/G \ast \simeq G/G

and

BG( dRBG)/G, \flat \mathbf{B}G \simeq (\flat_{dR}\mathbf{B}G)/G \,,

respectively.

Proposition

For GG an ∞-group, then the Maurer-Cartan form θ G:G dRBG\theta_G \colon G \to \flat_{dR}\mathbf{BG} of def. 27 naturally carries equivariance structure with respect to the GG-∞-actions of remark 12, hence the structure of a homomorphism/intertwiner of these ∞-actions.

Proof

By the discussion at ∞-action the equivariant structure in question is a morphism of the form

G/G θ/G ( dRBG)/G BG \array{ G/G &&\stackrel{\theta/G}{\longrightarrow}&& (\flat_{dR}\mathbf{B}G)/G \\ & \searrow && \swarrow \\ && \mathbf{B}G }

such that it induces θ:G dRBG\theta \colon G \to \flat_{dR}\mathbf{B}G on homotopy fibers.

By remark 12 the above diagram is equivalently

* θ/G BG BG. \array{ \ast &&\stackrel{\theta/G}{\longrightarrow}&& \flat\mathbf{B}G \\ & \searrow && \swarrow \\ && \mathbf{B}G } \,.

There is an essentially unique horizontal morphism θ/G\theta/G making this commute (up to homotopy). To see that this does induce the Maurer-Cartan form θ\theta on homotopy fibers, notice that the morphism on homotopy fibers is the universal one from the total homotopy pullback diagram to the bottom homotopy pullback diagram labeled θ\theta in

G * θ θ/G dRBG BG * BG \array{ G && \longrightarrow && \ast \\ & \searrow^{\mathrlap{\theta}} && && \searrow^{\mathrlap{\theta/G}} \\ \downarrow && \flat_{dR}\mathbf{B}G && \longrightarrow && \flat \mathbf{B}G \\ & \swarrow && && \swarrow \\ \ast && \longrightarrow && \mathbf{B}G }

The pasting law implies that also the top rectangle here, is a homotopy pullback, hence this identifies θ\theta in this diagram indeed as the MC form.

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.

Flat Ehresmann connections

We discuss now a general abstract notion of flat Ehresmann connections in a cohesive (,1)(\infty,1)-topos H\mathbf{H}.

Let GGrp(H)G \in Grp(\mathbf{H}) be an ∞-group. For g:XBGg : X \to \mathbf{B}G a cocycle that modulates a GG-principal ∞-bundle PXP \to X, we saw above that lifts

BG X g BG \array{ && \flat \mathbf{B}G \\ & {}^{\mathllap{\nabla}}\nearrow & \downarrow \\ X &\stackrel{g}{\to}& \mathbf{B}G }

modulate flat \infty-connections \nabla in PXP \to X.

We can think of :XBG\nabla : X \to \flat \mathbf{B}G as the cocycle datum for the connection on base space, in generalization of the discussion at connection on a bundle. On the other hand, there is the classical notion of an Ehresmann connection, which instead encodes such connection data in terms of differential form data on the total space PP.

We may now observe that such differential form data on PP is identified with the twisted ∞-bundle induced by the lift, with respect to the local coefficient ∞-bundle given by the fiber sequence

dRBG BG BG \array{ \flat_{dR} \mathbf{B}G &\to& \flat \mathbf{B}G \\ && \downarrow \\ && \mathbf{B}G }

that defines the de Rham coefficient object, discussed above.

Notice also that the dRBG\flat_{dR}\mathbf{B}G-twisted cohomology defined by this local coefficient bundle says that: flat \infty-connections are locally flat Lie(G)Lie(G)-valued forms that are globally twisted by by a GG-principal \infty-bundle.

By the general discussion at twisted ∞-bundle we find that the flat connection \nabla induces on PP the structure

G P A dRBG * x X BG BG \array{ G &\to& P &\stackrel{A}{\to}& \flat_{dR} \mathbf{B}G \\ \downarrow && \downarrow && \downarrow \\ * &\stackrel{x}{\to}& X &\stackrel{\nabla}{\to}& \flat \mathbf{B}G &\stackrel{}{\to}& \mathbf{B}G }

consisting of

  • a (flat) Lie(G)Lie(G)-valued form datum A:P dRBGA : P \to \flat_{dR}\mathbf{B}G on the total space PP

  • such that this intertwines the GG-actions on PP and on dRBG\flat_{dR}\mathbf{B}G.

In the model H\mathbf{H} = Smooth∞Grpd one finds that the last condition reduces indeed to that of an Ehresmann connection for AA on PP (this is discussed here). One of the two Ehresmann conditions is manifest already abstractly: for every point x:*Xx : * \to X of base space, the restriction of AA to the fiber of PP over XX is the Maurer-Cartan form

θ:GPA dRBG \theta : G \to P \stackrel{A}{\to} \flat_{dR} \mathbf{B}G

on the \infty-group GG, discussed above.

Differential cohomology

Ordinary 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. 10, 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 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).

The following establishes the characteristic short exact sequences that characterizes intrinsic differential cohomology as an extension of curvature forms by flat \infty-bundles and of bare \infty-bundles by connection forms.

Proposition

Let imFH dR n+1(X,A)im F \subset H_{dR}^{n+1}(X, A) be the image of the curvatures. Then the differential cohomology group H diff n(X,A)H_{diff}^n(X,A) fits into a short exact sequence

0H flat n(X,A)H diff n(X,A)imF0 0 \to H^n_{flat}(X, A) \to H^n_{diff}(X,A) \to im F \to 0
Proof

Apply the long exact sequence of homotopy groups to the fiber sequence

H flat(X,B nA)H diff(X,B nA)[F]H dR n+1(X,A) \mathbf{H}_{flat}(X, \mathbf{B}^n A) \to \mathbf{H}_{diff}(X, \mathbf{B}^n A) \stackrel{[F]}{\to} H_{dR}^{n+1}(X,A)

of prop. 32 and use that H dR n+1(X,A)H_{dR}^{n+1}(X,A) is, as a set, a homotopy 0-type to get the short exact sequence

π 1(H dR(X,A)) π 0(H flat(X,B nA)) π 0(H diff(X,B nA)) [F] π 0(H dR n+1(X,A)) = = = 0 H flat n(X,A) H diff n(X,A) im[F]. \array{ \pi_1(H_{dR}(X,A)) &\to& \pi_0(\mathbf{H}_{flat}(X, \mathbf{B}^n A)) &\to& \pi_0(\mathbf{H}_{diff}(X, \mathbf{B}^n A)) &\stackrel{[F]}{\to}& \pi_0(H_{dR}^{n+1}(X,A)) \\ = && = && = && \downarrow \\ 0 &\to& H_{flat}^n(X, A) &\to& H_{diff}^n(X,A) &\to& im [F] } \,.
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).

For a detailed discussion of the relation to ordinary differential cohomology see at smooth ∞-groupoid the section Abstract properties of differential cohomology.

In view of the second of these points one can make a choice of cover in order to present the twisting cocycles functorially. To that end, let

Ω cl n+1(,A) dRB n+1A \Omega^{n+1}_{cl}(-,A) \to \mathbf{\flat}_{dR} \mathbf{B}^{n+1}A

denote a choice of effective epimorphism out of a 0-truncated object which we suggestively denote by Ω cl n+1(,A)\Omega^{n+1}_{cl}(-,A).

Definition

With a choice Ω cl n+1(,A) dRB n+1A\Omega^{n+1}_{cl}(-,A) \to \mathbf{\flat}_{dR} \mathbf{B}^{n+1}A fixed, we say an object XHX \in \mathbf{H} is dR-projective if the induced morphism

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

is itself an effective epimorphism (of ∞-groupoid)s.

Remark

A morphism of \infty-groupoids is an effective epimorphism precisely if it is surjective on π 0\pi_0 (see here). Since Ω cl n+1(,A)\Omega^{n+1}_{cl}(-,A) is assumed to be 0-truncated, also

Ω cl n+1(X,A):=H(X,Ω cl n+1(,A)) \Omega^{n+1}_{cl}(X,A) := \mathbf{H}(X, \Omega^{n+1}_{cl}(-,A))

is 0-truncated. Hence XX is dR-projective precisely if the set Ω cl n+1(X,A)\Omega^{n+1}_{cl}(X,A) contains representatives of all intrinsic de Rham cohomology classes of XX.

In terms of hypercohomology this may be thought of as saying that XX is dR-projective if every de Rham hypercohomology class on XX has a representative by a globally defined differential form. In models of cohesion we typically have that manifolds are dR-projective, but nontrivial orbifolds are not.

Definition

Write B nA conn\mathbf{B}^n A_{conn} for the \infty-pullback

B nA conn Ω cl n+1(,A) B nA curv dRB n+1A. \array{ \mathbf{B}^n A_{conn} & \stackrel{}{\to} & \Omega^{n+1}_{cl}(-,A) \\ \downarrow && \downarrow \\ \mathbf{B}^n A & \stackrel{curv}{\to} & \mathbf{\flat}_{dR}\mathbf{B}^{n+1}A } \,.

We say that this is the differential coefficient object of B nA\mathbf{B}^n A.

Remark

For every dR-projective XHX \in \mathbf{H} there is a canonical monomorphism

H diff(X,B nA)H(X,B nA conn), \mathbf{H}_{diff}(X,\mathbf{B}^n A) \to \mathbf{H}(X, \mathbf{B}^n A_{conn}) \,,
Proof

Consider the diagram

H(X,B nA conn) Ω cl n+1(,A) H diff(X,B nA) H dR n+1(X,A) H(X,B nA) H(X, dRB n+1A). \array{ \mathbf{H}(X, \mathbf{B}^n A_{\mathrm{conn}}) & \stackrel{}{\to} & \Omega^{n+1}_{cl}(-,A) \\ \downarrow && \downarrow \\ \mathbf{H}_{diff}(X,\mathbf{B}^n A) & \stackrel{}{\to} & H_{dR}^{n+1}(X,A) \\ \downarrow && \downarrow \\ \mathbf{H}(X, \mathbf{B}^n A) & \stackrel{}{\to} & \mathbf{H}(X, \mathbf{\flat}_{dR}\mathbf{B}^{n+1}A) } \,.

The bottom square is an ∞-pullback? by definition. A morphism as in the top right exists by assumption that XX is dR-prohective. Let also the top square be an \infty-pullback. Then by the pasting law so is the total rectangle, which identifies the top left object as indicated, since H(X,)\mathbf{H}(X,-) preserves \infty-pullbacks.

Since the top right morphism is in injection of sets, it is a monomorphism of \infty-groupoids. These are stable under \infty-pullback, which proves the claim.

Generalized differential cohomology

For cohesive stable homotopy types the above discussion may be refined and stream-lined considerably. For more on this see at differential cohomology diagram.

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).

As before for abelian coefficients, we introduce differential coefficient objects BG conn\mathbf{B}G_{conn} that represent these differential cohomology classes over dR-projective objects

BG c B n+1A BG conn c^ B n+1A conn BG c B n+1A. \array{ \mathbf{\flat}\mathbf{B}G & \stackrel{\mathbf{\flat}\mathbf{c}}{\to} & \mathbf{\flat}\mathbf{B}^{n+1} A \\ \downarrow && \downarrow \\ \mathbf{B}G_{conn} & \stackrel{\hat {\mathbf{c}}}{\to} & \mathbf{B}^{n+1}A_{conn} \\ \downarrow && \downarrow \\ \mathbf{B}G &\stackrel{\mathbf{c}}{\to}& \mathbf{B}^{n+1}A } \,.

(…)

Higher holonomy

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

Definition

We say an object ΣH\Sigma \in \mathbf{H} has cohomological dimension n\leq n \in \mathbb{N} if for all nn-connected 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 ΣH\Sigma \in \mathbf{H} 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 holonomy of \nabla over σ\sigma is the flat holonomy of ϕ *\phi^* \nabla

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

Transgression in differential cohomology

We discuss an intrinsic notion of transgression/fiber integration in ordinary differential cohomology internal to any cohesive (,1)(\infty,1)-topos. This generalizes the notion of higher holonomy discussed above.

Fix AA an abelian group object as above and B nA conn\mathbf{B}^n A_{conn} a corresponding differential coefficient object. Then for ΣH\Sigma \in \mathbf{H} of cohomological dimension knk \leq n consider the map

[Σ,B nA conn]conk kτ nkconk nkτ nk[Σ,B nA conn]. [\Sigma, \mathbf{B}^n A_{conn}] \stackrel{conk_k \circ \tau_{n-k}}{\to} conk_{n-k} \tau_{n-k} [\Sigma, \mathbf{B}^n A_{conn}] \,.

In typical models we have an equivalence

conk kτ nk[Σ,B nA conn]B n1A conn. conk_k \tau_{n-k} [\Sigma, \mathbf{B}^n A_{conn}] \simeq \mathbf{B}^{n-1} A_{conn} \,.

In this case we say that for

c^:BG connB nA conn \hat \mathbf{c} : \mathbf{B}G_{conn} \to \mathbf{B}^n A_{conn}

a differential characteristic map, that the composite

exp(i Σ()):[Σ,BG conn][Σ,B nA conn]B nkA conn \exp(i \int_{\Sigma}(-)) : [\Sigma, \mathbf{B} G_{conn}] \to [\Sigma, \mathbf{B}^n A_{conn}] \to \mathbf{B}^{n-k}A_{conn}

is the transgression of c^\hat \mathbf{c} to the mapping space [Σ,BG conn][\Sigma, \mathbf{B} G_{conn}].

For k=nk = n the reproduces, on the underlying \infty-groupoids, the higher holonomy discussed above.

(…)

Chern-Simons functional

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

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.

The cohesive refinement of this (…more discussion required…)

Σ:[Σ,X][Σ,B nA] diffconc ndimΣ[Σ,B nA] diffconc ndimΣ[Σ,B nA]τ ndimΣconc ndimΣ[Σ,B nA], \int_\Sigma : [\Sigma, X] \stackrel{}{\to} [\Sigma, \mathbf{B}^n A]_{diff} \stackrel{}{\to} conc_{n-dim \Sigma} [\Sigma, \mathbf{B}^n A]_{diff} \stackrel{\simeq}{\to} conc_{n-dim \Sigma} [\Sigma, \mathbf{\flat}\mathbf{B}^n A] \stackrel{}{\to} \tau_{n - \dim \Sigma} conc_{n-dim \Sigma} [\Sigma, \mathbf{\flat}\mathbf{B}^n A] \,,

where

  • [,][-,-] denotes the cartesian internal hom;

  • [Σ,B nA] diffconc ndimΣ[Σ,B nA] diff[\Sigma, \mathbf{B}^n A]_{diff} \stackrel{}{\to} conc_{n-dim \Sigma} [\Sigma, \mathbf{B}^n A]_{diff} is the concretification projection in degree ndimΣn - dim \Sigma

  • conc ndimΣ[Σ,B nA]τ ndimΣconc ndimΣ[Σ,B nA]conc_{n-dim \Sigma} [\Sigma, \mathbf{\flat}\mathbf{B}^n A] \stackrel{}{\to} \tau_{n - \dim \Sigma} conc_{n-dim \Sigma} [\Sigma, \mathbf{\flat} \mathbf{B}^n A] is the truncation projection in the same degree

we call the smooth extended Chern-Simons functional.

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.

cohesion

tangent cohesion

differential cohesion

id id ʃ inf inf ʃ * \array{ id & \dashv & id \\ \vee && \vee \\ \Re &\dashv& ʃ_{inf} &\dashv& \flat_{inf} \\ && \vee && \vee \\ && ʃ &\dashv& \flat &\dashv& \sharp \\ && && \vee && \vee \\ && && \emptyset &\dashv& \ast }

References

General

For general references on cohesive (∞,1)-toposes see there.

The above list of structures in any cohesive (,1)(\infty,1)-topos is the topic of section 2.3 of

Formulation in homotopy type theory

For formalizations of some structures in cohesive (,1)(\infty,1)-toposes in terms of homotopy type theory see cohesive homotopy type theory.

Revised on November 25, 2014 13:30:37 by Urs Schreiber (89.204.138.216)