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

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

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

A cohesive $(\infty,1)$-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 $(\infty,1)$-topos.

Concrete objects

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

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

Proposition

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

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

Proof

The full and faithfulness of $Disc$ and $coDisc$ follows as in the discussion at ∞-connected (∞,1)-topos, Since $\Gamma$ is also a right adjoint it preserves in particular finite (∞,1)-limits, so that $(\Gamma \dashv \mathrm{coDisc})$ 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 $\mathbf{H}$ on those objects $X \in \mathbf{H}$ for which the unit

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

is an equivalence.

Proof

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

Definition

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

If a 0-truncated object $X$ is $0$-concrete, we call it just concrete.

See also concrete (∞,1)-sheaf.

Proposition

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

Definition

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

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

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

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

Remark

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

Cohesive $\infty$-Groups

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

Definition

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

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

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

Remark

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

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

Proposition

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

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

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

Proof

By the discussion at delooping.

We write

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

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

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

Observation

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

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

Definition

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

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

Observation

Suppose that also $Y$ is pointed and $f$ 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

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

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

Proposition

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

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

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

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

• the corresponding delooping object is presented by the presheaf

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

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

Proof

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

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

$$(\Omega \dashv \bar W) : sGrp \stackrel{\overset{\Omega}{\leftarrow}}{\underset{\bar W}{\to}} sSet_0 \,.$$

In particular its unit is a weak equivalence

$$\bar W \Omega Y \stackrel{\simeq}{\to} Y$$

for every $Y \in sSet_0 \hookrightarrow sSet_{Quillen}$ and $\bar W \Omega Y$ is always a Kan complex. Therefore

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

is an equivalent representative for $X$, 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

$$\array{ \Omega X &\to& W \Omega X \\ \downarrow && \downarrow \\ * &\to& \bar W \Omega X }$$

is

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

• a homotopy pullback of the point along itself (since $W G \to \bar W G$ 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 $\mathbf{H}$.

Definition

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

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

is the cohomology set of $X$ with coefficients in $A$. If $A = G$ is an ∞-group we write

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

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

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

In the context of cohomology on $X$ with coefficients in $A$ we we say that

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

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

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

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

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

Definition

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

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

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

$$\array{ \vdots && \vdots \\ \Omega H &\to& G &\to& * \\ \downarrow &&\downarrow && \downarrow \\ * &\to& H &\to& \mathbf{B}F &\to& * \\ &&\downarrow && \downarrow && \downarrow \\ && * &\to& \mathbf{B}G & \stackrel{c}{\to} & \mathbf{B}H } \,.$$

Proposition

• The long fiber sequence to the left of $c : \mathbf{B}G \to \mathbf{B}H$ becomes constant on the point after $n$ iterations if $H$ is $n$-truncated.

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

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

Proof

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

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

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

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 $g : X \to \mathbf{B}G$ is canonically associated its homotopy fiber $P \to X$, the (∞,1)-pullback

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

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

Definition

(principal $G$-action)

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

$$\array{ \vdots && \vdots \\ P \times G \times G &\stackrel{(p_2, p_3)}{\to}& G \times G \\ \downarrow\downarrow\downarrow && \downarrow\downarrow\downarrow \\ P \times G &\stackrel{p_2}{\to}& G \\ \downarrow\downarrow && \downarrow\downarrow \\ P &\stackrel{}{\to}& {*} }$$

in $\mathbf{H}$.

We say that the (∞,1)-colimit

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

is the base space defined by this action.

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

Proposition

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

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

Proof

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

Proposition

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

Proof

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

$$\array{ \vdots && \vdots && \vdots \\ P \times_X P \times_X P &\stackrel{\simeq}{\to}& P \times G \times G &\to& G \times G \\ \downarrow\downarrow\downarrow && \downarrow\downarrow\downarrow && \downarrow\downarrow\downarrow \\ P \times_X P &\stackrel{\simeq}{\to}& P \times G &\stackrel{p_2}{\to}& G \\ \downarrow\downarrow && \downarrow\downarrow && \downarrow\downarrow \\ P &\stackrel{=}{\to}& P &\stackrel{}{\to}& {*} \\ \downarrow && \downarrow && \downarrow \\ X &\stackrel{=}{\to}& X &\stackrel{g}{\to}& \mathbf{B}G } \,,$$

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 $P \times_X P$ is

$$\array{ P \times_X P &\to& P \\ \downarrow && \downarrow \\ P &\to& X }$$

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

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

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

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

By defnition of $P$ as the homotopy fiber of $X \to \mathbf{B}G$, the lower horizontal morphism is equivalent to $P \to {*} \to \mathbf{B}G$, so that $P \times_X P$ is equivalent to the pullback

$$\array{ P \times_X P &\to& &\to& {*} \\ \downarrow && && \downarrow \\ P &\to& {*} &\to& \mathbf{B}G \,. }$$

This finally may be computed as the pasting of two pullbacks

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

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

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

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

We may again paste this to obtain

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

and use from the previous induction step that

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

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

Definition

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

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

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

Definition

We say a sequence of cohesive ∞-groups

$$A \to \hat G \to G$$

exhibits $\hat G$ as an extension of $G$ by $A$ if the corresponding delooping sequence

$$\mathbf{B}A \to \mathbf{B}\hat G \to \mathbf{B}G$$

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

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

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

Proposition

Let $A \to \hat G \to G$ be an extension of $\infty$-groups, def. in $\mathbf{H}$ and let $P \to X$ be a $G$-principal ∞-bundle.

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

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 $\mathbf{H}$

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

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

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

Now choose any point $x : {*} \to X$ 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 $G \simeq \Omega \mathbf{B}G \simeq * \prod_{\mathbf{B}G} *$ the diagram completes by $(\infty,1)$-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 $\mathbf{H}$ an (∞,1)-topos let $\mathbf{c} : B \to C$ a morphism representing a characteristic class $[\mathbf{c}] \in H(B,C)$. Let $C$ be pointed and write $A \to B$ for its homotopy fiber.

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

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

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

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

For short we often say twist for twisting cocycle .

Proposition

We have the following immediate properties of twisted cohomology:

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

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

Proposition

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

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

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, $\phi = 0$ (meaning that it factors as $\phi : X \to * \to C$ through the point of the pointed object $C$), the corresponding twisted $A$-cohomology is equivalent to ordinary $A$-cohomology

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

Proof

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

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

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 $\phi : X \to C$ as the cocycle for an $\Omega C$-principal ∞-bundle over $X$, being the (∞,1)-pullback of the point inclusion $* \to C$ along $\phi$, where the point is the homotopy-incarnation of the universal $\Omega C$-principal $\infty$-bundle. The characteristic class $B \to C$ in turn we may think of as an $\Omega A$-bundle associated to this universal bundle. Accordingly the pullback of $P_\phi := X \times_C B$ is the associated $\Omega A$-bundle over $X$ classified by $\phi$.

Proposition

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

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

Then the $\phi$-twisted $A$-cohomology of $X$ is equivalently the space of sections $\Gamma_X(P_\phi)$ of $P_\phi$ over $X$:

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

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

$$\array{ \Gamma_X(P_\phi) &\to& * \\ \downarrow && \downarrow^{\mathrm{id}} \\ \mathbf{H}(X,P_\phi) &\stackrel{p_*}{\to}& \mathbf{H}(X,X) } \,.$$

Proof

Consider the pasting diagram

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

By the fact that the hom-functor $\mathbf{H}(X,-)$ 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 $* \stackrel{\mathbf{\phi}}{\to} \mathbf{H}(X,C)$ the claim follows with prop. .

Note

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

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

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

Definition

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

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

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

Observation

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

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

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

Note

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

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

where on the left we have the simplicially constant object on the vertices of $\mathbf{H}(X,C)_{simp}$. This morphism, in turn, presents a morphism in $\infty Grpd$ that in general contains a multitude of copies of the components of any $H(X,C) \to \mathbf{H}(X,C)$: 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 $(\infty,1)$-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 $\mathbf{H}_{tw}(X,A)$ as defined above.

$\infty$-Group representations and associated $\infty$-bundles

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

(…)

Concordance

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

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

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

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

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

We have that

$$\tilde \mathbf{H}(X,\mathbf{B}G) = \Pi(\mathbf{B}G^X)$$

is the $\infty$-groupoid whose objects are $G$-principal ∞-bundles on $X$ and whose morphisms have the interpretaton of $G$-principal bundles on the cylinder $X \times I$. 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 $\mathbf{H}$ a locally ∞-connected (∞,1)-topos and $X \in \mathbf{H}$ an object, we call $\Pi X \in$ ∞Grpd the fundamental ∞-groupoid in a locally ∞-connected (∞,1)-topos of $X$.

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

$$\pi_\bullet^{geom}(X) := \pi_\bullet(\Pi (X)) \,.$$

Definition

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

$$\vert X \vert := \vert \Pi X \vert \in Top$$

and call this the topological geometric realization of cohesive ∞-groupoids of $X$, or just the geometric realization for short.

Note

In presentations of $\mathbf{H}$ 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 $f,g : X \to Y$ in $\mathbf{H}$ is a diagram

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

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

Proposition

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

Proof

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

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

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

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

in ∞Grpd.

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

Proposition

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

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

$$\Pi_X(X \in \mathbf{H}/X) \simeq \Pi(X \in \mathbf{H}) \,.$$

Proof

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

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

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

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

Definition

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

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

• A1-homotopy theory

• continuum

Galois theory

We discuss a canonical internal notion of Galois theory in $\mathbf{H}$.

Definition

For $\kappa$ a regular cardinal write

$$Core \infty Grpd_\kappa \in \infty Grpd$$

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

Remark

We have

$$Core \infty Grpd_\kappa \simeq \coprod_i \mathbf{B} Aut(F_i) \,,$$

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

Definition

For $X \in \mathbf{H}$ write

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

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

Observation

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

$$Disc Core \infty Grpd_\kappa \simeq \coprod_i \mathbf{B} Disc Aut(F_i) \,.$$

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

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

• a natural equivalence

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

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

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

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

Proof

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

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

Using this and that $\Pi$ preserves the terminal object, so that the adjunct of $(* \to X \to Disc Core \infty Grpd_\kappa)$ is $(* \to \Pi(X) \to \infty Grpd_\kappa)$

the second statement follows with an iterated application of the (∞,1)-Yoneda lemma (this is pure Tannaka duality as discussed there):

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

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

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

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

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

$$\begin{aligned} End x^* & \simeq End_{PSh(PSh(\Pi(X)^{op})^{op})} (\tilde j(j (x))) \\ & \simeq End(PSh(\Pi(X))^{op}) (j(x)) \\ & \simeq End_{\Pi(X)^{op}}(x,x) \\ & \simeq Aut_x \Pi(X) \\ & =: \Omega_x \Pi(X) \end{aligned} \,.$$

van Kampen theorem

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

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

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

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

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

Observation

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

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

$$\Pi(X) \simeq {\lim_\to} \Pi(U_\bullet) \,.$$

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

Proposition

Let $X$ be a paracompact or locally contractible topological spaces and $U_1 \hookrightarrow X$, $U_2 \hookrightarrow X$ a covering by two open subsets.

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

$$\array{ Sing(U_1) \cap Sing(U_2) &\to& Sing(U_2) \\ \downarrow && \downarrow \\ Sing(U_1) &\to& Sing(X) } \,.$$

Proof

We inject the topological space via the external Yoneda embedding

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

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

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

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

$$\array{ U_1 \cap U_2 &\to& U_2 \\ \downarrow && \downarrow \\ U_1 &\to& X }$$

in $Sh(TopBalls, sSet)_{inj,loc}$ 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 $(\infty,1)$-pushouts, so that indeed $X$ is the $(\infty,1)$-pushout

$$X \simeq U_1 \coprod_{U_1 \cap U_2} U_2 \in \mathbf{H} \,.$$

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

Paths and geometric Postnikov towers

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

Definition

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

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

We say

• $\mathbf{\Pi}(X)$ is the path $\infty$-groupoid of $X$ – the reflection of the fundamental ∞-groupoid in a locally ∞-connected (∞,1)-topos back into the cohesive context of $\mathbf{H}$;

• $\mathbf{\flat} A$ (“flat $A$”) is the coefficient object for flat differential A-cohomology or for $A$-local systems

Write

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

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

$$\mathbf{\tau}_n : \mathbf{H} \stackrel{\tau_n}{\to} \mathbf{H}_{\leq n} \hookrightarrow \mathbf{H}$$

for the truncation-localization funtor.

We say

$$\mathbf{\Pi}_n : \mathbf{H} \stackrel{\mathbf{\Pi}_n}{\to} \mathbf{H} \stackrel{\mathbf{\tau}_n}{\to} \mathbf{H}$$

is the homotopy path n-groupoid functor.

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

$$X \to \mathbf{\Pi}(X)$$

are the constant path inclusions. Dually we have canonical morphism

$$\mathbf{\flat}A \to A \,.$$

Observation

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

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

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

Proposition

Let $\mathbf{H}$ be a locally ∞-connected (∞,1)-topos. If $X \in \mathbf{H}$ is small-projective then the over-(∞,1)-topos $\mathbf{H}/X$ 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 $(\infty,1)$-topos $\mathbf{H}$, if $X$ is small-projective then so is its path ∞-groupoid $\mathbf{\Pi}(X)$.

Proof

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

$$\begin{aligned} \mathbf{H}(\mathbf{\Pi}(X), {\lim_\to}_i A_i) & \simeq \mathbf{H}(X, \mathbf{\flat}{\lim_\to}_i A_i) \\ & \simeq \mathbf{H}(X, {\lim_\to}_i \mathbf{\flat} A_i) \\ & \simeq {\lim_\to}_i \mathbf{H}(X, \mathbf{\flat} A_i) \end{aligned} \,,$$

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

Definition

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

$$\mathbf{\Pi}(X) \to \cdots \to \mathbf{\Pi}_2(X) \to \mathbf{\Pi}_1(X) \to \mathbf{\Pi}_0(X) \,.$$

Universal coverings and geometric Whitehead towers

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

Definition

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

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

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

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

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

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

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

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

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

Proposition

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

Proof

Consider the diagram

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

The bottom morphism is the constant path inclusion, the $(\Pi \dashv Disc)$-unit. The right morphism is the equivalence in an (∞,1)-category that is the image under $Disc$ of the decomposition ${\lim_\to}_S * \stackrel{\simeq}{\to} S$ 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 $i$ of this equivalence, hence itself an equivalence. By universal colimits in the (∞,1)-topos $\mathbf{H}$ the top left object is the (∞,1)-colimit over the single homotopy fibers $i^* *_s$ of the form $X^{\mathbf{(\infty)}}$ as indicated.

Proposition

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

Proof

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

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

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

Flat $\infty$-connections and local systems

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

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

$$X \to \mathbf{\Pi}(X)$$

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

$$\mathbf{\flat} A \to A \,.$$

Definition

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

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

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

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

The induced morphism

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

we say is the forgetful functor that forgets flat connections.

Remark

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

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

• to each point $x \in X$ a fiber $P_x$ in $A$;

• to each path $\gamma : x_1 \to x_2$ in $X$ an equivalence $\nabla(\gamma) : P_{x_1} \to P_{x_2}$ between these fibers (the parallel transport along $\gamma$);

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

• and so on.

$$\array{ && && P_{x_2} \\ A && & {}^{\mathllap{\nabla(\gamma_1)}}\nearrow & \Downarrow^{\nabla(\Sigma)} & \searrow^{\mathrlap{\nabla(\gamma_2)}} \\ \uparrow^{\mathrlap{\nabla}} && P_{x_1} &&\underset{\nabla(\gamma_3)}{\to}&& P_{x_3} \\ && && x_2 \\ \mathbf{\Pi}(X) && & {}^{\mathllap{\gamma_1}}\nearrow & \Downarrow^{\Sigma} & \searrow^{\mathrlap{\gamma_2}} \\ && x_1 &&\underset{\gamma_3}{\to}&& x_3 }$$

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

For a non-flat $\infty$-connection the parallel transport $\nabla(\gamma_3^{-1}\circ \gamma_2\circ \gamma_1)$ 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 $(\mathbf{\Pi} \dashv \mathbf{\flat})$-adjunction we have a natural equivalence

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

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

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

precisely if there is a lift $\nabla$ in the diagram

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

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

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

Proposition

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

Proof

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

Remark

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

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

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

Remark

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

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

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

In terms of this we may say that

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

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

Therefore

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

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

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

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

So the equivalence $\Gamma \mathbf{\flat}\mathbf{B}G \simeq \Gamma \mathbf{B}G$ says that over the point every $G$-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 $* \in \mathbf{H}$ really behaves like a geometric point: it has underlying it a single point, $\Gamma * \simeq *$, and its geometric homotopy type is that of the point, $\Pi(*) \simeq *$.

de Rham cohomology

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

Definition

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

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

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

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

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

Proposition

The construction in def. yields a pair of adjoint (∞,1)-functors

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

Proof

We check the defining natural hom-equivalence

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

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

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

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

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

We paste this pullback to the above pullback diagram to obtain

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

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

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

This exhibits the hom-space as the pullback

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

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

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

Proposition

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

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

given by

$$\mathbf{\Gamma}_{dR} X {:=} * \coprod_{X} \mathbf{\Gamma}(X) \,,$$

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

Proof

This follows by the same kind of argument as above.

Definition

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

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

A cocycle $\omega : X \to \mathbf{\flat}_{dR}A$ we call a flat $A$-valued differential form on $X$.

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

Observation

A cocycle in de Rham cohomology

$$\omega : \mathbf{\Pi}_{dR}X \to A$$

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

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

Proof

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

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

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

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

Proposition

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

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

Proof

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

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

since the pullback of an equivalence is an equivalence.

Proposition

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

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

Proof

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

$$\begin{aligned} \mathbf{H}(*, \mathbf{\Pi}_{dR}X) & \simeq \Gamma \mathbf{\Pi}_{dR} X \\ & \simeq * \coprod_{\Gamma X} \Gamma \mathbf{\Pi}X \\ & \simeq * \coprod_{\Gamma X} \Pi X \end{aligned} \,,$$

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

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

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

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

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

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

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

Remark

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

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

In degree 0 the intrinsic de Rham cohomology is necessarily 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 $H_{dR}^n(X,K)$ is the home for the obstructions to flatness of $\mathbf{B}^{n-2}K$-principal ∞-bundles. For $n = 1$ this are groupoid-principal bundles over the groupoid with $K$ 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 cohesive homotopy-type theory

Exponentiated $\infty$-Lie algebras

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

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

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

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

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 $\mathbf{\Pi}_{dR}(X)$ is the schematic de Rham homotopy type of $X$, as discussed above. But if we add a bit more structure to the cohesive $(\infty,1)$-topos $\mathbf{H}$, 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 $\mathbf{B}\exp(\mathfrak{g})$ in $\mathbf{H}$ that is geometrically contractible

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

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

Definition

Set

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

Observation

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

Proof

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

$$(\exp Lie \dashv \mathbf{\Gamma}_{dR}\mathbf{\flat}_{dR}) \,.$$

Proposition/Example

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

Proof

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

$$\begin{aligned} \Pi \mathbf{\Pi}_{dR} X & {:=} \Pi (*) \coprod_{\Pi X} \Pi \mathbf{\Pi} X \\ & \simeq * \coprod_{\Pi X} \Pi Disc \Pi X \\ & \simeq * \coprod_{\Pi X} \Pi X & \simeq * \end{aligned} \,,$$

where we used that in the ∞-connected $\mathbf{H}$ the functor $Disc$ is full and faithful.

Corollary

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

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

Proposition

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

$$\array{ && \mathbf{B}\exp(\mathfrak{g}) \\ & \nearrow & \downarrow \\ \mathbf{\Pi}_{dR}X &\stackrel{\omega}{\to}& \mathbf{B}G } \,.$$

Proof

By the universality of the counit of $(\mathbf{\Pi}_{dR} \dashv \mathbf{\flat}_{dR})$ we have that $\omega$ factors through the counit $\exp Lie \mathbf{B}G \to \mathbf{B}G$.

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

Corollary

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

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

Proposition

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

$$\exp Lie \circ \exp Lie \simeq \exp Lie \circ \Sigma \circ \Omega \,.$$

Proof

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

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

This follows using

• $\mathbf{\flat}$ is a right adjoint and hence preserves (∞,1)-pullbacks;

• $\mathbf{\flat} \mathbf{\flat} := Disc \Gamma Disc \Gamma \simeq Disc \Gamma =: \mathbf{\flat}$ by the fact that $Disc$ is a full and faithful (∞,1)-functor;

• the counit $\mathbf{\flat} \mathbf{\flat} A \to \mathbf{\flat} A$ is equivalent to the identity, by the zig-zag-identity of the adjunction and using that equivalences satisfy 2-out-of-3.

by computing

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

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

From this we deduce that

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

by computing for all $A \in \mathbf{H}$

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

Also observe that by a proposition above we have

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

for all $X \in \mathbf{H}$.

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

• $\exp Lie := \mathbf{\Pi}_{dR} \mathbf{\flat}_{dR}$ preserves the terminal object (since $\mathbf{H}$ is locally ∞-connected and ∞-connected)

• and that it is a left adjoint by the above, since $\mathbf{H}$ is assumed to be cohesive.

We compute:

$$\begin{aligned} \exp Lie \exp Lie A & \simeq \exp Lie \mathbf{\Pi}_{dR} \mathbf{\flat}_{dR} A \\ & \simeq * \coprod_{\exp Lie \mathbf{\flat}_{dR} A} \exp Lie \mathbf{\Pi} \mathbf{\flat}_{dR} A \\ & \simeq * \coprod_{\exp Lie \mathbf{\flat}_{dR} A} \mathbf{\Pi}_{dR} \mathbf{\flat}_{dR} \mathbf{\Pi} \mathbf{\flat}_{dR} A \\ & \simeq * \coprod_{\exp Lie \mathbf{\flat}_{dR} A} * \\ & \simeq * \coprod_{\mathbf{\Pi}_{dR} \mathbf{\flat}_{dR} \mathbf{\flat}_{dR} A} * \\ & \simeq * \coprod_{\mathbf{\Pi}_{dR} \mathbf{\flat}_{dR} \Omega A} * \\ & \simeq * \coprod_{\exp Lie \Omega A} * \\ & \simeq \exp Lie ( * \coprod_{\Omega A} * ) \\ & \simeq \exp Lie \Sigma \Omega A \end{aligned} \,.$$

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 $G \in \mathbf{H}$ an ∞-group, write

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

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

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

and the above proposition.

We call $\theta$ the Maurer-Cartan form on $G$.

Remark

By postcomposition the Maurer-Cartan form sends $G$-valued functions on $X$ to $\mathfrak{g}$-valued forms on $X$

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

Remark

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

$$\array{ G &\longrightarrow& \ast \\ && \downarrow \\ && \mathbf{B}G }$$

and

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

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

$$\ast \simeq G/G$$

and

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

respectively.

Proposition

For $G$ an ∞-group, then the Maurer-Cartan form $\theta_G \colon G \to \flat_{dR}\mathbf{BG}$ of def. naturally carries equivariance structure with respect to the $G$-∞-actions of remark , 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

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

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

By remark the above diagram is equivalently

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

There is an essentially unique horizontal morphism $\theta/G$ making this commute (up to homotopy). To see that this does induce the Maurer-Cartan form $\theta$ 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 $\theta$ in

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

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

Definition

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

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

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

Flat Ehresmann connections

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

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

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

modulate flat $\infty$-connections $\nabla$ in $P \to X$.

We can think of $\nabla : X \to \flat \mathbf{B}G$ 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 $P$.

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

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

that defines the de Rham coefficient object, discussed above.

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

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

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

consisting of

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

• such that this intertwines the $G$-actions on $P$ and on $\flat_{dR}\mathbf{B}G$.

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

$$\theta : G \to P \stackrel{A}{\to} \flat_{dR} \mathbf{B}G$$

on the $\infty$-group $G$, 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 $A \in \tau_{\leq 0} \mathbf{H} \hookrightarrow \mathbf{H}$. For all $n \in \mathbf{N}$ we have then the Eilenberg-MacLane object $\mathbf{B}^n A$.

Definition

For $X \in \mathbf{H}$ any object and $n \geq 1$ write

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

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

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

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

We call

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

the degree-$n$ differential cohomology of $X$ with coefficient in $A$.

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

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

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

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

Observation

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

Proof

The set

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

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 $\mathbf{H}_{diff}(X, \mathbf{B}^n A)$ is the coproduct

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

of the homotopy fibers of $curv_*$ over each of the chosen points $* \to \mathbf{H}_{dR}(X,\mathbf{B}^{n+1}A)$. 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:

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

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

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

Proof

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

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

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

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

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

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

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

$$\cdots \to \mathbf{\flat} \mathbf{B}^n A \to \mathbf{B}^n A \stackrel{curv}{\to} \mathbf{\flat}_{dR} \mathbf{B}^{n+1}A \to \mathbf{\flat} \mathbf{B}^{n+1} A \to \mathbf{B}^{n+1} A$$

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

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

Proof

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

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

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

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

Proposition

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

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

Proof

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

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

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

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

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

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

because, as before, this $(\infty,1)$-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)

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

and

$$\Omega \mathbf{\flat}_{dR} \mathbf{B}^{n+1}A \simeq \mathbf{\flat}_{dR} \Omega \mathbf{B}^{n+1}A$$

since both $\mathbf{H}(X,-)$ as well as $\mathbf{\flat}_{dR}$ 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 $\mathbf{H}$, not in Top. Notably for $\mathbf{H} =$ ?LieGrpd?, $A = U(1) =\mathbb{R}/\mathbb{Z}$ the circle group and $|X| \in Top$ the geometric realization of a paracompact manifold $X$, we have that $H^n(X,\mathbb{R}/\mathbb{Z})$ above is $H^{n+1}_{sing}({|\Pi X|},\mathbb{Z})$.

2. The fact that on the left of the short exact sequence for differential cohomology we have the de Rham cohomology set $H_{dR}^n(X,A)$ 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 $\mathbf{H}$ by a model category $C$ we can consider instead of $H_{dR}^{n+1}(X,A) \to \mathbf{H}_{dR}(X, \mathbf{B}^{n+1}A)$ the inclusion of the set of objects $\Omega_{cl}^{n+1}(X,A) {:=} \mathbb{R}Hom_C(X, \mathbf{B}^{n+1}A )_0 \hookrightarrow \mathbb{R}Hom_C(X, \mathbf{B}^{n+1}A )$. However, by the above observation this only adds multiple copies of the homotopy types of the connected components of $\mathbf{H}_{diff}(X, \mathbf{B}^n A)$.

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

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

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

Definition

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

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

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

Remark

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

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

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

In terms of hypercohomology this may be thought of as saying that $X$ is dR-projective if every de Rham hypercohomology class on $X$ 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 $\mathbf{B}^n A_{conn}$ for the $\infty$-pullback

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

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

Remark

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

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

Proof

Consider the diagram

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

The bottom square is an ∞-pullback? by definition. A morphism as in the top right exists by assumption that $X$ 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 $\mathbf{H}(X,-)$ 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 $A$ be the chosen abelian ∞-group as above. Recall the universal curvature characteristic class

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

for all $n \geq 1$.

Definition

For $G$ an ∞-group and

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

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

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

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

The induced map on cohomology

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

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

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

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

We say

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

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

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

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

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

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

Observation

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

Proof

This is due to the existence of the pasting composite

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

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

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

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

(…)

Higher holonomy

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

Definition

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

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

Let $dim(\Sigma)$ be the maximum $n$ for which this is true.

Observation

If $\Sigma \in \mathbf{H}$ has cohomological dimension $\leq n$ then its intrinsic de Rham cohomology vanishes in degree $k \gt n$

$$H_{dR}^{k \gt n}(\Sigma, A) \simeq * \,.$$

Proof

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

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

Let now again $A$ be fixed as above.

Definition

Let $\Sigma \in \mathbf{H}$, $n \in \mathbf{N}$ with $dim \Sigma \leq n$.

We say that the composite

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

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

More generally, let

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

• $\phi : \Sigma \to X$ a morphism.

Write

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

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

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

The holonomy of $\nabla$ over $\sigma$ is the flat holonomy of $\phi^* \nabla$

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

Transgression in differential cohomology

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

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

$$[\Sigma, \mathbf{B}^n A_{conn}] \stackrel{conk_k \circ \tau_{n-k}}{\to} conk_{n-k} \tau_{n-k} [\Sigma, \mathbf{B}^n A_{conn}] \,.$$

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

• $\tau_{n-k}$ denotes truncation in degree $n-k$;

• $conk_{n-k}$ denotes concretification in degree $(n-k)$.

In typical models we have an equivalence

$$conk_k \tau_{n-k} [\Sigma, \mathbf{B}^n A_{conn}] \simeq \mathbf{B}^{n-1} A_{conn} \,.$$

In this case we say that for

$$\hat \mathbf{c} : \mathbf{B}G_{conn} \to \mathbf{B}^n A_{conn}$$

a differential characteristic map, that the composite

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

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

For $k = n$ 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 $(\infty,1)$-topos induces an intrinsic notion of and higher Chern-Simons functionals.

Definition

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

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

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

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

$$\int_\Sigma : [\Sigma, X] \stackrel{}{\to} [\Sigma, \mathbf{B}^n A]_{diff} \stackrel{}{\to} conc_{n-dim \Sigma} [\Sigma, \mathbf{B}^n A]_{diff} \stackrel{\simeq}{\to} conc_{n-dim \Sigma} [\Sigma, \mathbf{\flat}\mathbf{B}^n A] \stackrel{}{\to} \tau_{n - \dim \Sigma} conc_{n-dim \Sigma} [\Sigma, \mathbf{\flat}\mathbf{B}^n A] \,,$$

where

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

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

• $conc_{n-dim \Sigma} [\Sigma, \mathbf{\flat}\mathbf{B}^n A] \stackrel{}{\to} \tau_{n - \dim \Sigma} conc_{n-dim \Sigma} [\Sigma, \mathbf{\flat} \mathbf{B}^n A]$ 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

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

• $X$ is the target space;

• $\hat \mathbf{c}$ is the background gauge field on $X$;

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

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

In suitable situations this construction refines to an internal construction.

Assume that $\mathbf{H}$ has a canonical line object $\mathbb{A}^1$ and a natural numbers object $\mathbb{Z}$. Then the action functional $\exp(i S(-))$ 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

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

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

See ∞-Chern-Simons theory for more discussion.

Related concepts

cohesion

• (shape modality $\dashv$ flat modality $\dashv$ sharp modality)

  $(\esh \dashv \flat \dashv \sharp )$

  • discrete object, codiscrete object, concrete object

  • points-to-pieces transform

  • structures in cohesion

• dR-shape modality$\dashv$ dR-flat modality

  $\esh_{dR} \dashv \flat_{dR}$

infinitesimal cohesion

• classical modality

tangent cohesion

• differential cohomology diagram

differential cohesion

• (reduction modality $\dashv$ infinitesimal shape modality $\dashv$ infinitesimal flat modality)

  $(\Re \dashv \Im \dashv \&)$

  • reduced object, coreduced object, formally smooth object

  • formally étale map

  • structures in differential cohesion

graded differential cohesion

• fermionic modality$\dashv$ bosonic modality $\dashv$ rheonomy modality

  $(\rightrightarrows \dashv \rightsquigarrow \dashv Rh)$

singular cohesion

$$\array{ && id &\dashv& id \\ && \vee && \vee \\ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \\ && \bot && \bot \\ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& \mathrm{R}\!\!\mathrm{h} & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& \esh &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast }$$

• concretification

References

General

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

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

• Urs Schreiber, differential cohomology in a cohesive topos

Formulation in homotopy type theory

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