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

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

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

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

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 and Galois theory

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

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 $X$, or just the geometric realization for short.

Note

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

Definition

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

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 on ∞-bundles, 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 mophism $\mathbf{Pi}(X) \to A$ may be thought of as assigning

• to each point of $X$ a fiber in $A$;

• to each path in $X$ an equivalence between these fibers;

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

• and so on.

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

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.

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.

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

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

Proposition

This construction 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

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

Observation

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 an 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 ∞-connetion 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 necessrily trivial, while in degree 1 we find that it reproduces closed 1-forms, not divided out by exact forms. This difference to ordinary de Rham cohomology in the lowest two degrees may be interpreted in terms of the obstruction-theoretic meaning of de Rham cohomology by which we essentially characterized it above: we have that the intrinsic $H_{dR}^n(X,K)$ 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.

Exponentiated $\infty$-Lie algebras

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 $Disc$ isfull 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 [[unit of an adjunction|counit]9 $\exp Lie \mathbf{B}G \to \mathbf{B}G$.

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 $X$

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

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 $\mathbf{B}^n A$.

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. \ref{TwistedCohomologyInOvertopos}, 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 a 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)

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

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 $\mathbf{H}_{diff}(X, \mathbf{B}^{n_i}A)$.

Higher holonomy and Chern-Simons functional

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

Definition

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

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

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 $\Sigma$.

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.

References

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

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

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

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

• Carlos Simpson, Constantin Teleman, deRham theorem for $\infty$-stacks (pdf)

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

• Carlos Simpson, The topological realization of a simplicial presheaf , arXiv:q-alg/9609004.

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

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

• Maxim Kontsevich, Alexander Rosenberg, Noncommutative spaces, preprint MPI-2004-35 (ps, dvi)

in the context of Q-categories .

The material presented here is also in section 2 of

• Urs Schreiber, differential cohomology in a cohesive topos

A commented list of further related references is at

• differential cohomology in a cohesive topos -- references