This is a sub-section of the entry cohesive (∞,1)-topos . See there for background and context
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.
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.
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
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.)
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
is an equivalence.
This follows by general facts discussed at reflective sub-(∞,1)-category.
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.
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.
For $X \in \mathbf{H}$ and $n \in \mathbb{N}$, the $(n+1)$-concretification of $X$ is the morphism
that is the left factor in the decomposition with respect to the n-connected/n-truncated factorization system of the $(\Gamma \dashv coDisc)$-unit
By that very n-connected/n-truncated factorization system we have that $conc_{n+1} X$ is an $(n+1)$-concrete object.
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.
For $X \in \mathbf{H}$ an object and $x : * \to X$ a point, the loop space object of $X$ is the (∞,1)-pullback $\Omega_x X := * \times_X *$:
This object $\Omega_x X$ is canonically equipped with the structure of an ∞-group obect.
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 $\leq 0$.
The operation of forming loop space objects in $\mathbf{H}$ establishes an equivalence of (∞,1)-categories
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.
By the discussion at delooping.
We write
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 $\mathbf{H}(\mathbf{B}G, \mathbf{B}H)$.
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.
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 $X := {*} \times_Z Y$
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
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.
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
which is fibrant in $[C^{op}, sSet]_{proj}$;
the corresponding delooping object is presented by the presheaf
which is given over each $U \in C$ by $\bar W (G(U))$ (see simplicial group for the notation).
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
In particular its unit is a weak equivalence
for every $Y \in sSet_0 \hookrightarrow sSet_{Quillen}$ and $\bar W \Omega Y$ is always a Kan complex. Therefore
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
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).
There is an intrinsic notion of cohomology and of principal ∞-bundles in any (∞,1)-topos $\mathbf{H}$.
For $X,A \in \mathbf{H}$ two objects, we say that
is the cohomology set of $X$ with coefficients in $A$. If $A = G$ is an ∞-group we write
for cohomology with coefficients in its delooping. Generally, if $K \in \mathbf{H}$ has a $p$-fold delooping, we write
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 2-morphism : $g \Rightarrow h$ is a coboundary between cocycles.
a morphism $c : A \to B$ represents the characteristic class
For every morphism $c : \mathbf{B}G \to \mathbf{B}H \in \mathbf{H}$ define the long fiber sequence to the left
to be the given by the consecutive pasting diagrams of (∞,1)-pullbacks
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
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
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
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.
(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 $\Delta^{op} \to \mathbf{H}$
in $\mathbf{H}$.
We say that the (∞,1)-colimit
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.
The $G$-principal action as defined above satisfies the principality condition in that we have an equivalence of groupoid objects
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.
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 $X$.
By the above we need to show that we have a morphism of simplicial diagrams
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
To compute this, we may attach the defining $(\infty,1)$-pullback square of $P$ to obtain the pasting diagram
and use the pasting law for pullbacks, to conclude that $P \times_X P$ is the pullback
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
This finally may be computed as the pasting of two pullbacks
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
We may again paste this to obtain
and use from the previous induction step that
to conclude the induction step with the same arguments as before.
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
Of special interest are principal $\infty$-bundles of the form $P \to \mathbf{B}G$:
We say a sequence of cohesive ∞-groups
exhibits $\hat G$ as an extension of $G$ by $A$ if the corresponding delooping sequence
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
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.
This follows from repeated application of the pasting law for (∞,1)-pullbacks: consider the following diagram in $\mathbf{H}$
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.
For the moment see the discussion at ∞-gerbe .
A slight variant of cohomology is often relevant: twisted cohomology.
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
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 .
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 $A$.
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
in ∞Grpd.
This is an application of the general pullback-formula for hom-spaces in an over-(∞,1)-category. See there for details.
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
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:
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 $\phi$.
Let $P_\phi := X \times_C B$ be (∞,1)-pullback of the characteristic class $\mathbf{c}$ along the twisting cocycle $\phi$
Then the $\phi$-twisted $A$-cohomology of $X$ is equivalently the space of sections $\Gamma_X(P_\phi)$ of $P_\phi$ over $X$:
where on the right we have the (∞,1)-pullback
Consider the pasting diagram
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. .
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
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.
With a characteristic class $[\mathbf{c}] \in H(B,C)$ with homotopy fiber $A$ understood, we write
for the union of all twisted cohomology cocycle $\infty$-groupoids.
We have that $\mathbf{H}_{tw}(X,A)$ is the (∞,1)-pullback
where the right vertical morphism in any section of the projection from $C$-cocycles to $C$-cohomology.
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
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.
The material to go here is at Schreiber, section 2.3.7.
(…)
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
This should yield an (∞,1)-category $\tilde \mathbf{H}$ with the same objects as $\mathbf{H}$ and hom-$\infty$-groupoids defined by
We have that
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.
We discuss canonical internal realizations of the notions of étale homotopy, geometric homotopy groups in an (infinity,1)-topos and local systems .
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 $X$
For $\vert - \vert :$ ∞Grpd $\stackrel{\simeq}{\to}$ Top the homotopy hypothesis-equivalence we write
and call this the topological geometric realization of cohesive ∞-groupoids of $X$, or just the geometric realization for short.
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).
We say a geometric homotopy between two morphism $f,g : X \to Y$ in $\mathbf{H}$ is a diagram
such that $I$ is geometrically connected, $\pi_0^{geom}(I) = *$.
If $f,g : X\to Y$ are geometrically homotopic in $\mathbf{H}$, then their images $\Pi(f), \Pi(g)$ are equivalent in $\infty Grpd$.
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
Now since $\Pi(I)$ is connected by assumption, there is a diagram
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 $\infty Grpd$.
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
By the general facts recalled at étale geometric morphism we have a composite essential geometric morphism
and $X_!$ is given by sending $(Y \to X) \in \mathbf{H}/X$ to $Y \in \mathbf{H}$.
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 $\{ X \times \mathbb{A}^1 \to X \}_{X \in \mathbf{H}}$.
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).
The analogous notion in infinitesimal cohesion is discussed in infinitesimal cohesion – infinitesimal A1-homotopy-topos+–+infinitesimal+cohesion#InfinitesimalA1Homotopy).
See also at
We discuss a canonical internal notion of Galois theory in $\mathbf{H}$.
For $\kappa$ a regular cardinal write
for the ∞-groupoid of $\kappa$-small ∞-groupoids: the core of the full sub-(∞,1)-category of ∞Grpd on the $\kappa$-small ones.
We have
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 $[F_i]$.
For $X \in \mathbf{H}$ write
We call this the $\infty$-groupoid of locally constant ∞-stacks on $X$.
Since $Disc$ is left adjoint and right adjoint it commutes with coproducts and with delooping and therefore
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.
For $\mathbf{H}$ locally ∞-connected and ∞-connected, we have
a natural equivalence
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 $x$
The first statement is just the adjunction $(\Pi \dashv Disc)$.
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:
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
With this, we compute the endomorphisms of $x^*$ by applying the (∞,1)-Yoneda lemma two more times:
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
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
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
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:
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
We inject the topological space via the external Yoneda embedding
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
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
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 $\Pi X \simeq Sing X$.
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 $\mathbf{H}$.
For $\mathbf{H}$ a locally ∞-connected (∞,1)-topos define the composite adjoint (∞,1)-functors
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
for the reflective sub-(∞,1)-category of n-truncated objects and
for the truncation-localization funtor.
We say
is the homotopy path n-groupoid functor.
We say that the (truncated) components of the $(\Pi \dashv Disc)$-unit
are the constant path inclusions. Dually we have canonical morphism
If $\mathbf{H}$ is cohesive, then $\mathbf{\flat}$ has a right adjoint $\mathbf{\Gamma}$
and this makes $\mathbf{H}$ be $\infty$-connected and locally $\infty$-connected over itself.
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
The first statement is proven at locally ∞-connected (∞,1)-topos, the second at local (∞,1)-topos.
In a cohesive $(\infty,1)$-topos $\mathbf{H}$, if $X$ is small-projective then so is its path ∞-groupoid $\mathbf{\Pi}(X)$.
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
where in the last step we used that $X$ is small-projective by assumption.
For $X \in \mathbf{H}$ we say that the geometric Postnikov tower of $X$ is the Postnikov tower in an (∞,1)-category of $\mathbf{\Pi}(X)$:
We discuss an intrinsic notion of Whitehead towers in a locally ∞-connected ∞-connected (∞,1)-topos $\mathbf{H}$.
For $X \in \mathbf{H}$ a pointed object, the geometric Whitehead tower of $X$ is the sequence of objects
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.
Here the morphisms $X^{\mathbf{(n+1)}} \to X^{\mathbf{n}}$ are those induced from this pasting diagram of (∞,1)-pullbacks
where the object $\mathbf{B}^n \mathbf{\pi}_n(X)$ is defined as the homotopy fiber of the bottom right morphism.
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.
Consider the diagram
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.
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.
We apply $\Pi(-)$ to the above diagram over a single vertex $s$ and attach the $(\Pi \dashv Disc)$-counit to get
Then the bottom morphism is an equivalence by the $(\Pi \dashv Disc)$-zig-zag-identity.
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
and the $(Disc \dashv \Gamma)$-counit with components
For $X, A \in \mathbf{H}$ we write
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
we say is the forgetful functor that forgets flat connections.
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.
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.
By the $(\mathbf{\Pi} \dashv \mathbf{\flat})$-adjunction we have a natural equivalence
A cocycle $g : X \to A$ for a principal ∞-bundle on $X$ is in the image of
precisely if there is a lift $\nabla$ in the diagram
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.
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.
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).
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
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.
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 $\mathbf{\flat}A$:
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
$\mathbf{B}G$ is the moduli ∞-stack of $G$-principal ∞-bundles;
$\mathbf{\flat} \mathbf{B}G$ is the moduli $\infty$-stack of $G$-principal $\infty$-bundles equipped with a flat $\infty$-connection.
Therefore
is the ∞-groupoid of $G$-principal $\infty$-bundles over the point (the terminal object in $\mathbf{H}$). Similarly
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, $\Pi(*) \simeq *$.
In every locally ∞-connected (∞,1)-topos $\mathbf{H}$ there is an intrinsic notion of nonabelian de Rham cohomology.
For $X \in \mathbf{H}$ an object, write $\mathbf{\Pi}_{dR}X := * \coprod_X \mathbf{\Pi} X$ for the (∞,1)-pushout
For $* \to A$ any pointed object in $\mathbf{H}$, write $\mathbf{\flat}_{dR} A := * \prod_A \mathbf{\flat}A$ for the (∞,1)-pullback
We also say $\flat_{dR}$ is the dR-flat modality and $\Pi_{dR}$ is the dR-shape modality.
The construction in def. yields a pair of adjoint (∞,1)-functors
We check the defining natural hom-equivalence
The hom-space in the under-(∞,1)-category $*/\mathbf{H}$ is (as discussed there), computed by the (∞,1)-pullback
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
We paste this pullback to the above pullback diagram to obtain
By the pasting law for (∞,1)-pullbacks the outer diagram is still a pullback. We may evidently rewrite the bottom composite as in
This exhibits the hom-space as the pullback
where we used the $(\mathbf{\Pi} \dashv \mathbf{\flat})$-adjunction. Now using again that $\mathbf{H}(X,-)$ preserves pullbacks, this is
If $\mathbf{H}$ is also local, then there is a further right adjoint $\mathbf{\Gamma}_{dR}$
given by
where $(\mathbf{\Pi} \dashv \mathbf{\flat} \dashv \mathbf{\Gamma}) : \mathbf{H} \to \mathbf{H}$ is the triple of adjunctions discussed at Paths.
This follows by the same kind of argument as above.
For $X, A \in \mathbf{H}$ we write
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 $A$.
A cocycle in de Rham cohomology
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
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
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 $X \to * \to A$.
The de Rham cohomology with coefficients in discrete objects is trivial: for all $S \in \infty Grpd$ we have
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
since the pullback of an equivalence is an equivalence.
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: $\tau_0 \mathbf{\Pi}_{\mathrm{dR}}X \simeq {*} \in \mathbf{H}$.
Since $\Gamma$ preserves (∞,1)-colimits in a cohesive $(\infty,1)$-topos we have
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
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
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.
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.
See at integration of differential forms – In cohesive homotopy-type theory
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
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.
For a connected object $\mathbf{B}\exp(\mathfrak{g})$ in $\mathbf{H}$ that is geometrically contractible
we call its loop space object $\exp(\mathfrak{g}) := \Omega_* \mathbf{B}\exp(\mathfrak{g})$ the Lie integration of an ∞-Lie algebra in $\mathbf{H}$.
Set
If $\mathbf{H}$ is cohesive, then $\exp Lie$ is a left adjoint.
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
For all $X$ the object $\mathbf{\Pi}_{dR}(X)$ is geometrically contractible.
Since on the locally ∞-connected (∞,1)-topos and ∞-connected $\mathbf{H}$ the functor $\Pi$ preserves (∞,1)-colimits and the terminal object, we have
where we used that in the ∞-connected $\mathbf{H}$ the functor $Disc$ is full and faithful.
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.
Every de Rham cocycle $\omega : \mathbf{\Pi}_{dR} X \to \mathbf{B}G$ factors through the ∞-Lie algebra of $G$
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.
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
If $\mathbf{H}$ is cohesive then we have
First observe that for all $A \in */\mathbf{H}$ we have
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
using that the (∞,1)-pullback of an equivalence is an equivalence.
From this we deduce that
by computing for all $A \in \mathbf{H}$
Also observe that by a proposition above we have
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:
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.
For $G \in \mathbf{H}$ an ∞-group, write
for the $\mathfrak{g}$-valued de Rham cocycle on $G$ which is induced by the (∞,1)-pullback pasting
and the above proposition.
We call $\theta$ the Maurer-Cartan form on $G$.
By postcomposition the Maurer-Cartan form sends $G$-valued functions on $X$ to $\mathfrak{g}$-valued forms on $X$
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
and
respectively, and they identify the homotopy quotients of the action as
and
respectively.
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.
By the discussion at ∞-action the equivariant structure in question is a morphism of the form
such that it induces $\theta \colon G \to \flat_{dR}\mathbf{B}G$ on homotopy fibers.
By remark the above diagram is equivalently
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
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.
For $G = \mathbf{B}^n A$ an Eilenberg-MacLane object, we also write
for the intrinsic Maurer-Cartan form and call this the intrinsic universal curvature characteristic form on $\mathbf{B}^n A$.
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
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
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
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
on the $\infty$-group $G$, discussed above.
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 $\mathbf{B}^n A$.
For $X \in \mathbf{H}$ any object and $n \geq 1$ write
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
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
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).
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.
The set
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
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.
When restricted to vanishing curvature, differential cohomology coincides with flat differential cohomology:
Moreover this is true at the level of cocycle ∞-groupoids
By the pasting law for (∞,1)-pullbacks the claim is equivalently that we have an $(\infty,1)$-pullback diagram
By definition of flat cohomology and of intrinsic de Rham cohomology in $\mathbf{H}$, the outer rectangle is
Since the hom-functor $\mathbf{H}(X,-)$ preserves (∞,1)-limits this is a pullback if
is. Indeed, this is one step in the fiber sequence
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.
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
Apply the long exact sequence of homotopy groups to the fiber sequence
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
The differential cohomology group $H_{diff}^n(X,A)$ fits into a short exact sequence of abelian groups
This is a general statement about the definition of twisted cohomology. We claim that for all $n \geq 1$ we have a fiber sequence
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
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
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)
and
since both $\mathbf{H}(X,-)$ as well as $\mathbf{\flat}_{dR}$ preserve (∞,1)-limits and hence formation of loop space objects, the claim follows.
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:
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 $H^{n+1}_{sing}({|\Pi X|},\mathbb{Z})$.
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
denote a choice of effective epimorphism out of a 0-truncated object which we suggestively denote by $\Omega^{n+1}_{cl}(-,A)$.
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
is itself an effective epimorphism (of ∞-groupoid)s.
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
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.
Write $\mathbf{B}^n A_{conn}$ for the $\infty$-pullback
We say that this is the differential coefficient object of $\mathbf{B}^n A$.
For every dR-projective $X \in \mathbf{H}$ there is a canonical monomorphism
Consider the diagram
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.
For cohesive stable homotopy types the above discussion may be refined and stream-lined considerably. For more on this see at differential cohomology diagram.
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
for all $n \geq 1$.
For $G$ an ∞-group and
a representative of a characteristic class $[\mathbf{c}] \in H^n(\mathbf{B}G, A)$ we say that the composite
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
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.
For $X \in \mathbf{H}$ any object, define the ∞-groupoid $\mathbf{H}_{conn}(X,\mathbf{B}G)$ as the (∞,1)-pullback
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
is the (full/refined) ∞-Chern-Weil homomorphism induced by the characteristic class $[\mathbf{c}]$.
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.
This is due to the existence of the pasting composite
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
(…)
The notion of intrinsic ∞-connections in a cohesive $(\infty,1)$-topos induces a notion of higher holonomy
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
Let $dim(\Sigma)$ be the maximum $n$ for which this is true.
If $\Sigma \in \mathbf{H}$ has cohomological dimension $\leq n$ then its intrinsic de Rham cohomology vanishes in degree $k \gt n$
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
Let now again $A$ be fixed as above.
Let $\Sigma \in \mathbf{H}$, $n \in \mathbf{N}$ with $dim \Sigma \leq n$.
We say that the composite
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
(using the above proposition) for the morphism on $(\infty,1)$-pullbacks induced by the morphism of diagrams
The holonomy of $\nabla$ over $\sigma$ is the flat holonomy of $\phi^* \nabla$
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
$[-,-]$ 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
In this case we say that for
a differential characteristic map, that the composite
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.
(…)
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.
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
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…)
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.
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
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.
(shape modality $\dashv$ flat modality $\dashv$ sharp modality)
$(\esh \dashv \flat \dashv \sharp )$
dR-shape modality$\dashv$ dR-flat modality
$\esh_{dR} \dashv \flat_{dR}$
(reduction modality $\dashv$ infinitesimal shape modality $\dashv$ infinitesimal flat modality)
$(\Re \dashv \Im \dashv \&)$
fermionic modality$\dashv$ bosonic modality $\dashv$ rheonomy modality
$(\rightrightarrows \dashv \rightsquigarrow \dashv Rh)$
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
For formalizations of some structures in cohesive $(\infty,1)$-toposes in terms of homotopy type theory see cohesive homotopy type theory.