Higher geometric structures on manifolds]Higher Geometric Structures on Manifolds
and the Gauge Theory of Deligne Cohomology

S. Bunk]S. Bunk Mathematical Institute, University of Oxford, United Kingdom

C. S. Shahbazi]C. S. Shahbazi Departamento de Matemáticas, Universidad UNED - Madrid, Reino de España
Fakultät für Mathematik, Universität Hamburg, Bundesrepublik Deutschland.

2020 MSC. Primary: 53C08. Secondary: 18N50, 18N60, 53Z05, 58D27.

We study smooth higher symmetry groups and moduli $\infty$-stacks of generic higher geometric structures on manifolds.
Symmetries are automorphisms which cover non-trivial diffeomorphisms of the base manifold.
We construct the smooth higher symmetry group of any geometric structure on $M$ and show that this completely classifies—via a universal property—equivariant structures on the higher geometry.
We construct moduli stacks of higher geometric data as $\infty$-categorical quotients by the action of the higher symmetries, extract information about the homotopy types of these moduli $\infty$-stacks, and prove a helpful sufficient criterion for when two such higher moduli stacks are equivalent.

In the second part of the paper we study higher $\U(1)$-connections.
First, we observe that higher connections come organised into higher groupoids, which further carry affine actions by Baez-Crans-type higher vector spaces.
We compute a presentation of the higher gauge actions for $n$-gerbes with $k$-connection, comment on the relation to higher-form symmetries, and present a new String group model.
We construct smooth moduli $\infty$-stacks of higher Maxwell and Einstein-Maxwell solutions, correcting previous such considerations in the literature, and compute the homotopy groups of several moduli $\infty$-stacks of higher $\U(1)$-connections.
Finally, we show that a discrepancy between two approaches to the differential geometry of NSNS supergravity (via generalised and higher geometry, respectively) vanishes at the level of moduli $\infty$-stacks of NSNS supergravity solutions.

§ INTRODUCTION

In [35] Gajer presented simultaneously a geometric description and a categorification of the Deligne cohomology groups $\rmH^n(M; \scD(n))$ of any manifold $M$, for each $n \in \NN$.
The seminal insight was that these groups arise as isomorphism classes of $\rmB^{n-1} \U(1)$-bundles with $n$-connection <cit.>.
For $k \in \NN_0$, a $k$-connection on a $\rmB^{n-1} \U(1)$-bundle consists of a tower of local connection forms of degrees $1,2, \ldots, k$ on the base manifold
[Note that we have adopted a slightly different terminology from [35].].
It was soon realised that this structure can be resolved further:
there are Kan complexes (in fact symmetric monoidal $n$-groupoids) of $\rmB^{n-1} \U(1)$-bundles with $k$-connection on any manifold $M$.
They can be obtained via the Dold-Kan correspondence from Čech resolutions of the Deligne complex $\scD(n)$—see, for instance, [29]—or modelled in terms of differential function spectra [49] (which is more in line with the Cheeger-Simons, rather than Deligne, description of differential cohomology).
In the modern literature, a $\rmB^n \U(1)$-bundle usually goes by the term $n$-gerbe.
Shifting indices accordingly, we can describe $\rmH^{n+2}(M; \scD(n{+}2))$ as the connected components of a simplicial abelian group of $n$-gerbes with $(n{+}1)$-connection on the manifold $M$.

Connections on (higher) gerbes play important roles in string theory and the various supergravity theories arising as its low-energy limits [30, 24, 31, 72, 2, 68, 32].
Nevertheless, despite the success of connections on classical principal bundles in geometry, topology and physics, very little is currently known about the mathematical gauge theory of higher connections and, in particular, their moduli theory (for first steps, see, for instance, [58, 28, 29, 72]).
In part, this is due to the fact that treating moduli of higher geometric structures genuinely requires higher-categorical technology, whose development is only a recent—and still ongoing—achievement in mathematics.

In the present paper we move beyond the pure structural theory of higher bundles with connections (such as how they organise into higher groupoids, their local-to-global properties, classification, parallel transport, etc.) and initiate the study of their actual gauge theory.
This requires us, in the first place, to develop a general formalism for constructing smooth higher symmetry groups and moduli $\infty$-(pre)stacks of generic higher geometric structures on manifolds
[We emphasise that the focus of the present paper lies on finite, or global, aspects of higher gauge theoretic moduli stacks, rather than infinitesimal moduli in the sense of deformation theory; the latter will be explored in later works.].
Importantly, the action of symmetries includes the action of diffeomorphisms on the higher geometric data.
We show that these smooth higher symmetry groups completely classify equivariant structures of higher geometric structures via a universal property, extract information about the homotopy types of the spaces underlying our higher moduli stacks, and prove a helpful sufficient criterion for when two moduli stacks are equivalent.

This theory is applicable, for instance, to higher bundles with connection in full generality, but in the second part of this paper, we focus on the geometry of higher $\U(1)$-connections, and thus Deligne differential cocycles.
This produces numerous insights into the gauge theory of higher connections, including that higher connections come organised into higher groupoids, rather than sets, and carry an affine action by higher Baez-Crans vector spaces.
We compute an explicit presentation of the higher gauge actions for $n$-gerbes with $k$-connection arising from the formalism in Part <ref>, comment on the relation of these smooth higher gauge groups to higher-form symmetries, and show that a particular higher symmetry group of a $1$-gerbe with $1$-connection produces a new String group model (in the sense of [53, 70, 13]).
The proof of the latter uses our theorem on equivalences of moduli stacks together with further results on the topology of moduli stacks.
We present and analyse smooth higher stacks of higher $\U(1)$-connections, including moduli stacks of higher Maxwell and Einstein-Maxwell theory, on fixed background data in a fully homotopical manner, correcting previous such considerations in the literature.
Finally, we address two approaches to the geometry of NSNS supergravity, originating in higher and generalised geometry, respectively.
We show that a discrepancy between two approaches to the B-field in the differential geometry of NSNS supergravity vanishes at the level of higher moduli stacks.
Resolving this discrepancy was one of the main original motivations for this paper.

§ OVERVIEW OF MAIN RESULTS

§.§ Moduli $\infty$-(pre-)stacks of higher geometric structures on manifolds

This paper is split into two main parts, plus a third part of appendices.
In Part <ref> we treat higher geometric structures on manifolds, as well as their smooth higher symmetries and moduli at a general level.
In Part <ref> we analyse in detail various geometric and gauge theoretic problems which feature higher $\U(1)$-connections.

§.§.§ Higher geometric data: smooth families and diffeomorphism actions

We begin by developing techniques which produce global higher geometric structures on manifolds from local data and incorporate both smooth families of geometric structures and smooth diffeomorphism actions.
We briefly sketch the key constructions here, before describing our main results in the following sections.
At the most general level, higher geometric structures are modelled as morphisms of $\infty$-presheaves or $\infty$-sheaves [57, 68].
In our case, these are functors out of nerves of 1-categories, so that we can describe them in terms of model categories of simplicial presheaves.
Having this explicit description available is beneficial for computational purposes.

Concretely, let $\Cart$ denote the category of cartesian spaces
[A cartesian space is a manifold $c$ which is diffeomorphic to $\RR^n$ for some $n \in \NN_0$.] and smooth maps; all smooth families of geometric data in this paper are parameterised over objects of $\Cart$.
A local model for smooth families of a higher geometric structure is a simplicial homotopy sheaf on a certain category $\Cartfam$ (Definition <ref>), whose objects are smooth families of cartesian spaces
[This approach enhances the formalism from, for instance, [68, 29] to describe smooth families of higher geometric data on a given manifold.].
In order to also treat the smooth action of the diffeomorphism group $\Diff(M)$ on smooth families of geometric data, we introduce the following category $\rmB\scD$:
its objects are cartesian spaces, and its morphisms are commutative squares of smooth maps

\begin{equation}
\label{eq:intro:BD square}
\begin{tikzcd}
c_0 \times M \ar[r, "\hat{\varphi}"] \ar[d, "\pr"']
& c_1 \times M \ar[d, "\pr"]
\\
c_0 \ar[r, "f"']
& c_1
\end{tikzcd}
\end{equation}

where $\hat{\varphi}$ restricts to a diffeomorphism on all fibres and satisfies that its fibrewise inverse is also smooth.
We understand functors $\rmB\scD^\opp \to \sSet$ as describing smooth families of higher geometric data on $M$, subject to the smooth action of $\Diff(M)$ via pullback.
We present a functor

\begin{equation}
\label{eq:intro: (-)^scD}
\Fun(\Cartfam^\opp, \sSet) \longrightarrow \Fun(\rmB\scD^\opp, \sSet)\,,
\end{equation}

which takes a local model for higher geometric structures and produces simplicial presheaves on $\rmB\scD$ in a particularly well-behaved way (Theorem <ref>).
As in the non-family case, the idea is to resolve $M$ by a good open covering by cartesian spaces and then evaluate a simplicial presheaf on the Čech nerve of this covering (compare, for instance, [29]).
The difficulty in the enhanced formalism here lies in combining this idea with smooth parameterisations of coverings and geometric data, as well as the action of diffeomorphisms of $M$:
we resolve this by passing through an intermediate category of families of coverings of $M$, which includes families of diffeomorphisms in its morphisms, and then average over all good open coverings by means of a left Kan extension (Proposition <ref>).
Our construction can also be used to produce explicit concretifications in the sense of [68, 6, 28] (see, in particular, Section <ref>).

We devote a section to showing that this averaging construction has a deeper homotopical meaning:
on the left fibrations associated to our simplicial presheaves, the above left Kan extension amounts precisely to an $\infty$-categorical localisation at all refinements of open coverings (Theorem <ref>).
We are thus able to produce, in a computable fashion, smooth families of geometric structures on $M$ from purely local data, and these families carry a natural smooth action of the diffeomorphism group of $M$.

§.§.§ Smooth higher groups of automorphisms and symmetries

With this construction set up we move on to our main goal of investigating the smooth higher symmetry groups and moduli stacks of higher geometric data on $M$.
First, we study their symmetry groups:
let $G^\scD \colon \rmB\scD^\opp \to \sSet$ be a simplicial presheaf.
We fix a vertex $\cG \in G^\scD(\RR^0)$; for instance, if $G$ classifies higher bundles on cartesian spaces, then $\cG$ is the datum of a higher bundle on $M$.
We let $\Diff_{[\cG]}(M) \subset \Diff(M)$ denote the subgroup of all those diffeomorphisms whose action on $M$ admits a lift to $\cG$; that is, it consists of those diffeomorphisms $\varphi$ for which there is an equivalence $\cG \simeq \varphi^* \cG$ between $\cG$ and the pullback of $\cG$ along the action of $\varphi$.
Let $\rmB\scD[\cG] \subset \rmB\scD$ denote the category objects are cartesian spaces and whose morphisms are commutative squares (<ref>) where $\hat{\varphi}$ restricts to an element of $\Diff_{[\cG]}(M)$ on each fibre.

A smooth $\infty$-group is a group object $\bbGamma$ in the $\infty$-category $\scP(N\Cart)$ of presheaves of spaces
[In this introduction we will freely pass between the $\infty$-category $\scP(N\scC)$ of presheaves of spaces on the nerve $N\scC$ of a 1-category $\scC$ and its presentations by the projective model category of simplicial presheaves on $\scC$ and the covariant model category $\sSet_{/N\scC^\opp}$.
In the main text, we employ the technology in Appendix <ref> to transition between these models.]
on $\Cart$; smooth $\infty$-groups form an $\infty$-category $\SmGrp$.
Any smooth group action $\Phi \colon \bbGamma \to \Diff(M)$ on $M$ for which there exists an equivariant structure on $\cG$ must factor through $\Diff_{[\cG]}(M)$.
For any such action, we construct a smooth $\infty$-group $\scSym_\Phi(\cG)$, consisting of all possible ways to lift the action of smooth families in $\bbGamma$ from $M$ to $\cG$.
In particular, if $\bbGamma = *$ is the trivial group, this produces the smooth $\infty$-group $\scAut(\cG)$ of automorphisms of $\cG$.
For instance, if $\cG$ is a higher principal bundle on $M$, then $\scAut(\cG)$ is its smooth higher gauge group.
We show (Corollaries <ref> and <ref>):

Given any section $\cG \in G^\scD(\RR^0)$ and any smooth $\infty$-group action $\Phi \colon \bbGamma \to \Diff_{[\cG]}(M)$, there is an extension of group objects in $\scP(N\Cart)$

\begin{equation}
\label{eq:intro:Aut-Sym SES}
\begin{tikzcd}
\scAut(\cG) \ar[r]
& \scSym_\Phi(\cG) \ar[r]
& \bbGamma\,.
\end{tikzcd}
\end{equation}

We derive this fact using that smooth $\infty$-groups can be presented as left fibrations whose fibres are reduced simplicial sets [60, 20].
The significance of the smooth $\infty$-group $\scSym_\Phi(\cG)$ lies in the fact that it completely controls equivariant structures on the geometric object $\cG$.
To see this, we define the following functor

\begin{equation}
\Equivar(\cG) \colon (\SmGrp_{/\Diff(M)} )^\opp \longrightarrow \scS\,.
\end{equation}

Its domain is the $\infty$-category of smooth $\infty$-group actions on $M$, i.e. of morphisms of smooth $\infty$-groups $\Phi \colon \bbGamma \to \Diff(M)$.
Its target is the $\infty$-category of spaces.
The value of $\Equivar(\cG)$ on a pair $(\bbGamma, \Phi)$ is the space of equivariant structures on $\cG$ for the action $(\bbGamma, \Phi)$ on $M$.
We prove a significant generalisation of the main results in [10] (Theorem <ref> and Corollary <ref>):

The functor $\Equivar(\cG)$ is representable.
It is represented by the morphism

\begin{equation}
\scSym_\iota(\cG) \longrightarrow \Diff_{[\cG]}(M)
\end{equation}

from (<ref>), where $\bbGamma = \Diff_{[\cG]}(M)$ and $\iota \colon \Diff_{[\cG]}(M) \hookrightarrow \Diff(M)$ is the canonical inclusion.
Moreover, the are natural equivalences between the following spaces:

* the space $\Equivar_\Phi(\cG)$ of $(\bbGamma, \Phi)$-equivariant structures on $\cG$,

* the space of lifts of $\Phi \colon \bbGamma \to \Diff(M)$ through the morphism $\scSym_\iota(\cG) \longrightarrow \Diff_{[\cG]}(M)$,

* the space of splittings of the smooth $\infty$-group extension $\scAut(\cG) \longrightarrow \scSym_\Phi(\cG) \longrightarrow \bbGamma$, i.e. of sections of the morphism $\scSym_\Phi(\cG) \longrightarrow \bbGamma$ in $\SmGrp$.

§.§.§ Moduli $\infty$-(pre)stacks of higher geometric structures

The smooth higher automorphism and symmetry groups of higher geometric structures are key ingredients in the study of moduli:
they encode the redundancies we wish to divide out in our systems.
We put geometric moduli problems on the following general footing, which is motivated by physical field theories.
This formalism lies at the heart of the present paper.
Again, to maintain maximum availability of computational tools, we begin our presentation using simplicial presheaves, and only then pass to left fibrations and $\infty$-categorical language in order to formulate and prove our main results.

Consider a pair of morphisms of simplicial presheaves on $\rmB\scD$,

\begin{equation}
\label{eq:intro:Sol --> Conf --> Fix}
\begin{tikzcd}
\Sol^\scD \ar[r, hookrightarrow]
& \Conf^\scD \ar[r]
& \Fix^\scD\,.
\end{tikzcd}
\end{equation}

We think of $\Conf^\scD$ as describing all possible configurations of the higher geometric data we are interested in.
For example, this could consist of smooth families of higher bundles with connection on $M$, such as the higher $\U(1)$-bundles with connection that underlie the geometry of Deligne cohomology.
The morphism $\Sol^\scD \hookrightarrow \Conf^\scD$ is an inclusion of a full simplicial subpresheaf, which we think of as describing smooth families of only those configurations whose moduli we wish to study.
For instance, these could be families of solutions to a set of field equations (see Section <ref> for more on this perspective), such as higher bundles with connections on $M$ solving higher Yang-Mills equations.
Finally, the morphism $\Conf^\scD \to \Fix^\scD$ is an objectwise Kan fibration; we think of this as forgetting part of the geometric data, leaving only the part which we wish to keep fixed in the moduli problem.
An important example is where $\Conf^\scD$ describes smooth families of higher bundles with connection on $M$, $\Fix^\scD$ encodes smooth families of higher bundles without connection on $M$, and the morphism $\Conf^\scD \to \Fix^\scD$ forgets the connections.

For each section $\cG \in \Fix^\scD(\RR^0)$ we obtain an $\infty$-presheaf on $N\Cart$ which encodes smooth families of solutions on $\cG$ (such as connections on a fixed higher bundle which solve a certain set of field equations), together with the morphisms of such solutions which are intrinsic to the geometric set-up; we denote this $\infty$-presheaf by $\scSol(\cG)$.
Recall that, given a smooth higher group action $\Phi \colon \bbGamma \to \Diff_{[\cG]}(M)$, we obtain an associated smooth higher symmetry group $\scSym_\Phi(\cG)$.
We show that the $\infty$-group $\scSym_\Phi(\cG)$ naturally acts on $\scSol(\cG)$.

The moduli $\infty$-prestack of solutions on $\cG$ modulo the action of $\scSym_\Phi(\cG)$ is the $\infty$-categorical quotient

\begin{equation}
\scMdl_\Phi(\cG) \coloneqq \scSol(\cG) \dslash \scSym_\Phi(\cG)
\quad \in \scP(N\Cart)\,.
\end{equation}

We construct this quotient explicitly as an $\infty$-categorical left Kan extension and provide a presentation as a left fibration over $N\Cart^\opp$ (Definition <ref>).
Further, we show (Theorem <ref>):

Whenever $\bbGamma$ is a sheaf of ordinary groups on $\Cart$ (satisfying descent with respect to good open coverings) and the solution subpresheaf satisfies a certain descent condition, then $\scMdl_\Phi(\cG) \in \scP(N\Cart)$ is an $\infty$-stack (or $\infty$-sheaf), i.e. it satisfies descent with respect to good open covers.

A very important situation is the case where $(\bbGamma, \Phi) = (*, e)$ is the trivial group action on $M$, and thus $\scSym_e(\cG) = \scAut(\cG)$ is the smooth $\infty$-group of automorphisms of $\cG$ (if $\cG$ is a higher principal bundle on $M$, then $\scAut(\cG)$ is its smooth higher gauge group).
The following statement shows that this special case already encodes a large amount of information about the moduli stack for any choice of action $(\bbGamma, \Phi)$ (Theorem <ref>):

For each smooth $\infty$-group action $\Phi \colon \bbGamma \to \Diff_{[\cG]}(M)$ on $M$, there is a principal $\infty$-bundle

\begin{equation}
\scMdl_e(\cG) \longrightarrow \scMdl_\Phi(\cG)
\end{equation}

in $\scP(N\Cart)$ with structure group $\bbGamma$.

Let $S \colon \scP(N\Cart) \to \scS$ denote the functor which takes the underlying space of an $\infty$-presheaf on $\Cart$ [14, 13].
It also goes by the name of smooth singular complex, or concordance space [8] and is equivalent to the $\infty$-categorical colimit functor.
Since $S$ preserves group objects and principal $\infty$-bundles [13], we obtain insights into the topology of our moduli stacks (Corollaries <ref> and <ref>):

For each smooth $\infty$-group action $\Phi \colon \bbGamma \to \Diff_{[\cG]}(M)$ on $M$, there is a fibre sequence

\begin{equation}
\longrightarrow S \big( \scMdl_e(\cG) \big)
\longrightarrow S \big( \scMdl_\Phi(\cG) \big)
\end{equation}

of spaces.
In particular, this induces a long exact sequence of homotopy groups.

One of the main results in the first part of this paper is a simple sufficient criterion for when two higher moduli stacks are equivalent.
It applies to situations where we have an augmentation of the sequence (<ref>) by a further morphism, i.e. to a sequence

\begin{equation}
\label{eq:intro:Sol --> Conf --> Fix_0 --> Fix_1}
\begin{tikzcd}
\Sol^\scD \ar[r, hookrightarrow]
& \Conf^\scD \ar[r]
& \Fix_0^\scD \ar[r, "p"]
& \Fix_1^\scD
\end{tikzcd}
\end{equation}

of simplicial presheaves on $\rmB\scD$.
We say that $p$ is an 0-connected if, for each object $c \in \rmB\scD$, the morphism $p_{|c} \colon \Fix_0^\scD(c) \longrightarrow \Fix_1^\scD(c)$ induces a bijection of connected components.
Let $\cG_0 \in \Fix_0^\scD(\RR^0)$, and let $\cG_1 \coloneqq p(\cG_0)$ be its image in $\Fix_1^\scD(\RR^0)$.
In this case, the presheaves $\scSol^\scD(\cG_0)$ and $\scSol^\scD(\cG_1)$ of solutions with respect to the fixed data $\cG_0$ and $\cG_1$, respectively, are not equivalent, and neither are the associated smooth higher symmetry groups whose action we divide out.
A crucial example is where $\Fix_0^\scD$ describes $n$-gerbes on $M$ with $k$-connection, $\Fix_1^\scD$ describes $n$-gerbes with $l$-connection, for $l < k$, and the morphism $p$ forgets part of the connection data (see Section <ref> below for more detail on this example).
We show (Theorem <ref>):

In the situation (<ref>), suppose that $p$ is 0-connected.
Then, for each smooth $\infty$-group action $\Phi \colon \bbGamma \to \Diff_{[\cG]}(M)$, there is a canonical equivalence in $\scP(N\Cart)$ between moduli $\infty$-prestacks,

\begin{equation}
\scMdl_\Phi(\cG_0) \simeq \scMdl_\Phi(\cG_1)\,.
\end{equation}

This insight allows us to compare and reconcile different moduli problems which are a priori different (such as in NSNS supergravity; see below).
At the same time it allows us to treat certain moduli problems in several different, but equivalent ways, some of which may be particularly well adapted to the problems we wish to solve.

§.§ Applications to higher $\U(1)$-connections and supergravity

In Part <ref> of this paper we apply the general formalism and results from Part <ref> to the geometry of higher $\U(1)$-connections on higher gerbes, and thus of Deligne differential cocycles.

§.§.§ Properties of higher $\U(1)$-connections

We enhance the well-known construction of classifying $\infty$-stacks of $n$-gerbes with $k$-connection (see [29] for a review) to obtain simplicial presheaves

\begin{equation}
\Grb^{n,\scD}_{\nabla|k} \colon \rmB\scD^\opp \longrightarrow \sSet\,.
\end{equation}

These simplicial presheaves describe smooth families of $n$-gerbes with $k$-connections on a fixed manifold $M$, which are further acted on by the smooth diffeomorphism group of $M$.
They arise by means of the formalism in Part <ref> and the Dold-Kan correspondence, applied to a family-version of the Deligne complex.
Given an $n$-gerbe on $M$ without connection, $\cG \in \Grb^{n,M}(\RR^0)$, a $k$-connection on $\cG$ is a tuple $\cA^{(k)} = (A_1, \ldots, A_k)$ of locally defined forms of degrees $1, 2, \ldots, k$, satisfying certain compatibility conditions.
On the level of $\infty$-sheaves on $N\Cart$, similar objects were constructed already by Schreiber in [68] by means of the abstract formalism of (differential) concretification (see, for instance, <cit.> and <cit.>).
Here we employ explicit constructions of the $\infty$-sheaves describing smooth families of $n$-gerbes with $k$-connection
[Note that our use of the term moduli $\infty$-stacks differs from that in [68]:
there, it refers to the higher stacks describing smooth families of higher geometric structures on manifolds (or on other $\infty$-stacks), whereas here we reserve the term to describe quotients of solution stacks by the action of smooth higher symmetry groups.].

We make the following observations (Theorem <ref> and Section <ref>):

The $k$-connections on any $n$-gerbe on $M$ form a smooth $(k{-}1)$-groupoid.
Moreover, this $(k{-}1)$-groupoid carries a smooth affine action by a smooth simplicial vector space.

These are genuine features of higher connections.
The first result in particular is in stark contrast to classical gauge theory, where the connections on a given principal bundle simply form a set and do not come with an intrinsic notion of morphisms between them.
The second statement categorifies the fact that connections on principal bundles form affine spaces over the vector space of 1-forms valued in the adjoint bundle.

§.§.§ Symmetries of higher $\U(1)$-bundles and connections

Let $(\cG, \cA^{(k)})$ be a fixed $n$-gerbe with fixed $k$-connection $\cA^{(k)}$ on $M$, for $k \leq n$; we can think of this as a principal $\infty$-bundle for the smooth structure group $\rmB^{n-k}\rmB_\nabla^k \U(1)$, where $\rmB$ denotes the delooping, or classifying space, functor, and where $\rmB_\nabla^k \U(1)$ classifies principal $\infty$-bundles with connection for the group $\rmB^{k-1}\U(1)$.
We are interested in the higher groupoids of connections on $(\cG, \cA^{(k)})$, i.e. the ways of extending $\cA^{(k)}$ to a full connection on $\cG$, modulo the actions of $\scAut(\cG, \cA^{(k)})$ and $\scSym_\Phi(\cG, \cA^{(k)})$ (for any smooth action $\Phi \colon \bbGamma \to \Diff_{[\cG]}(M)$).

So far, we have written the action of the smooth higher automorphism group $\scAut(\cG, \cA^{(k)})$ only in terms of an $\infty$-functor, or a left fibration.
We devote a section to unravelling, or strictifying, this higher group action in terms of an action of a presheaf of simplicial groups.
We show that this smooth $\infty$-group action is presented precisely by a higher form of the standard action of $\U(1)$-gauge transformations, adapted to the structure group $\rmB^{n-k}\rmB_\nabla^k \U(1)$.
The strictification heavily relies on the Dold-Kan correspondence and is proven in Theorem <ref>.
The smooth higher gauge actions on higher groupoids of connections has not been considered before (though truncated versions appear in, for instance, [30, 32, 31, 72]); here we derive its full form from first principles.

There is a canonical equivalence

\begin{equation}
\scAut(\cG, \cA^{(k)})
\simeq \Grb^{n-1,M}_{\nabla,k}
\end{equation}

of smooth $\infty$-groups, where the right-hand side is the smooth higher group of $(n{-}1)$-gerbes with $k$-connection on $M$ and their canonical (abelian) tensor product.
In particular, these smooth higher groups categorify the $\infty$-dimensional Lie groups considered as the gauge groups in higher abelian gauge theories in [32, 31, 72], where only the zero-truncations of $\scAut(\cG, \cA^{(k)})$ were considered.

The smooth higher gauge and symmetry groups have interesting relations to higher-form symmetries in physics [34, 33].
In particular, they describe and formalise the higher symmetries in QFTs investigated in [69].
We point out that, for each fixed $n$-gerbe with $n$-connection on $M$ and each smooth map $\sigma \colon \Sigma^{(n-p)} \to M$ from a closed $(n{-}p)$-manifold $\Sigma^{(n-p)}$, higher gerbe holonomy provides a morphism of smooth groups (Remark <ref>, see also Remark <ref>)

\begin{equation}
U_{(-)}(\Sigma^{(n-p)}, \sigma) \colon \pi_p \scAut(\cG, \cA^{(n)}) \longrightarrow \U(1)\,,
\end{equation}

for $p \geq 0$, where the homotopy groups are formed in $\scP(N\Cart)$.
These morphisms only depend on the homotopy class of $\sigma$, for $p > 0$, and further descend to moduli stacks of connections on $(\cG, \cA^{(n)})$ modulo gauge transformations (Remark <ref>).
If we consider instead an $(n{+}1)$-connection $\cA$ on $\cG$, then homotopy-invariance also holds for $p = 0$.

Finally, we use the theory developed thus far to provide a new smooth model for the String group (as introduced in [70, 53]).
Let $H$ be a compact, simple, simply connected Lie group, and let $\Phi = L \colon H \to H$ be the action of $H$ on itself via left multiplication.
Further, let $\cG_\bas$ be a basic gerbe on $H$ (i.e. one whose class in $\rmH^3(H;\ZZ)$ is a generator).
This admits a distinguished connection $\cA_\bas$ [59].
It follows from [28] that the smooth 2-group $\scSym_L(\cG_\bas, \cA_\bas)$ of symmetries of $(\cG_\bas, \cA_\bas)$ which lift the action $L$ is a smooth model for the String group $\bbString(H)$ of $H$ (note that $\bbString(H)$ is defined only up to weak homotopy equivalence).
Further, it was shown in [10, 13] that also $\scSym_L(\cG_\bas)$ is a smooth higher group model for $\bbString(H)$.
Here, we prove the following intermediate case (Theorem <ref>):

Let $\cA^{(1)}$ be any 1-connection on the basic gerbe $\cG_\bas$ over a compact, simple, simply connected Lie group $H$.
Then, the extension of smooth $\infty$-groups (compare Theorem <ref>)

\begin{equation}
\scAut(\cG_\bas, \cA^{(1)}) \longrightarrow \scSym_L(\cG_\bas, \cA^{(1)}) \longrightarrow H
\end{equation}

is a smooth String group extension.

§.§.§ Gauge theory of higher $\U(1)$-connections

We present various important examples of higher gauge theoretic moduli stacks for higher $\U(1)$-connections, including higher Maxwell and Einstein-Maxwell theory, higher self-dual connections and higher $\U(1)$-BF theory, as recently explored in the context of Turaev-Viro models in [50] and references therein.
In particular, we extend arguments from [58] to show (Theorem <ref>):

For each Riemannian metric $g$ on a closed manifold $M$, and each $n$-gerbe with $k$-connection $(\cG, \cA^{(k)})$ on $M$ with $k \leq n$, there exists an extension of the $k$-connection $\cA^{(k)}$ on $\cG$ to a higher (vacuum) Maxwell solution with respect to $g$.

We then turn to the relation between different moduli stacks of connections on $n$-gerbes:
for $l, k \in \NN_0$ with $l < k$, let

\begin{equation}
\label{eq:intro:forgetting connections}
p^k_l \colon \Grb^{n,\scD}_{\nabla|k} \to \Grb^{n,\scD}_{\nabla|l}
\end{equation}

denote the morphism which forgets the highest form parts $(A_{l+1}, \ldots, A_k)$ of a $k$-connection on $\cG$.
Using the various properties of Čech-Deligne cohomology, we show the following technical result (Corollary <ref>), which allows us to apply Theorem <ref> to higher $\U(1)$-connections:

The morphism (<ref>) is a Kan fibration and 0-connected whenever $k \leq n$.

It follows that $\Diff_{[\cG, \cA^{(k)}]}(M) = \Diff_{[\cG]}(M)$ depends only on the equivalence class of $\cG$ whenever $k \leq n$.
As a direct application of Theorem <ref> and Proposition <ref>, we obtain that various pairs of moduli stacks of higher $\U(1)$-connections are equivalent.
For instance, we obtain:

Let $(\cG, \cA^{(k)})$ be an $n$-gerbe with $k$-connection on $M$, and let $\cA^{(l)}$ be the $l$-connection on $\cG$ obtained by forgetting part of the connection data (by means of the morphism (<ref>)).
For each smooth higher group action $\Phi \colon \bbGamma \to \Diff_{[\cG]}(M)$, there is a canonical equivalence of moduli $\infty$-(pre)stacks of Maxwell solutions on $(\cG, \cA^{(k)})$, and $(\cG, \cA^{(l)})$, respectively,

\begin{equation}
\scMdl_{MW, \Phi}(\cG, \cA^{(k)}) \simeq \scMdl_{MW, \Phi}(\cG, \cA^{(l)})\,.
\end{equation}

In other words, we can fix an arbitrary part of a higher $\U(1)$-connection on an $n$-gerbe in any moduli problem, as long as we use the correct notion of gauge transformation in each case.
We clarify which truncation of the full moduli stack of higher Maxwell theory for $n=1$ was investigated in [58] and compute the homotopy type of the higher stacks of $n$-gerbes with $k$-connection (Theorem <ref>) and of higher Maxwell solutions on $M$ (Theorem <ref>).

§.§.§ Moduli stacks of NSNS supergravity solutions

One motivation for the present paper is to achieve a comparison of two higher moduli stacks of solutions to NSNS supergravity on a fixed manifold $M$, as we now explain.
Through Gajer's insight, together with the relevance of differential cohomology theories in string theory [30, 24, 31, 72, 2, 68, 32], (higher) gerbes have come to play important roles in string theory and the various supergravity theories arising as its low-energy limits.
In particular, it has been argued that the B-field in string theory is described by a 2-connection on a 1-gerbe [52].
The investigation of solutions to supergravity equations and their moduli has long been an open problem in differential geometry (see, for instance, the introduction of [38] for a nice overview and further references), but only little is known in situations where the 1-gerbe underlying the B-field has non-trivial topology (i.e. the class $[\cG] \in \rmH^3(M;\ZZ)$ is non-zero).

Recently, the introduction of generalised geometry has led to significant progress both in differential geometry in general and mathematical supergravity in particular [47, 43] (see also [37] for a textbook introduction).
In this approach, the B-field (in NSNS supergravity) is described as part of a generalised metric on an exact Courant algebroid on $M$.
Such a generalised metric can always be decomposed into a Riemannian metric on $M$ and an isotropic splitting of the exact Courant algebroid; the latter then corresponds to the B-field in NSNS supergravity.

The question which thus arises is how these two models for the geometry of NSNS supergravity are related and, even more importantly, how the solutions to NSNS supergravity in the two approaches compare.
As we explain below, there is no equivalence between the (higher) stacks of field configurations which the two models predict for NSNS supergravity.
It is therefore a highly important question whether and, if so, how the discrepancy between these two approaches to the B-field (Problem <ref>) and to moduli of NSNS supergravity solutions on $M$ (Problem <ref>) can be reconciled.

To answer these questions, we introduce models for exact Courant algebroids on $M$ (without and with isotropic splittings, respectively) in our formalism, i.e. as simplicial-presheaves,

\begin{equation}
\ECA^\scD,\, \ECA_\nabla^\scD \colon \rmB\scD^\opp \longrightarrow \sSet\,.
\end{equation}

Our description has the following pleasant feature:
Ševera's classification of exact Courant algebroids produces a canonical bijection between elements of $\rmH^3(M;\RR)$ and isomorphism classes of exact Courant algebroids on $M$.
Thus, the groupoid of exact Courant algebroids categorifies the degree-three de Rham cohomology of $M$.
A natural question is whether this also categorifies the abelian group structure on $\rmH^3(M;\RR)$.
For generic exact Courant algebroids, this appears to be unknown, but our model provides a groupoid which is canonically equivalent to that of exact Courant algebroids on $M$ and has a canonical symmetric monoidal structure which achieves this refined categorification (Theorem <ref>).

We enhance Hitchin's construction of the generalised tangent bundle to a morphism of simplicial presheaves

\begin{equation}
\AtCA_\nabla \colon \Grb^{1,\scD}_{\nabla|1} \longrightarrow \ECA^\scD
\end{equation}

on $\rmB\scD$.
It associates to a smooth family of gerbes with connective structure on $M$ a smooth family of exact Courant algebroids on $M$.
Then, as already observed by Hitchin [46], for each smooth family of 1-gerbes $(\cG, \cA^{(1)})$ with 1-connection on $M$, there is a canonical bijection between curvings on $(\cG, \cA^{(1)})$ (i.e. extensions of $\cA^{(1)}$ to a full connection on $\cG$) and smooth families of isotropic splittings of $\AtCA(\cG, \cA^{(1)})$.

In supergravity, (semi-classical) charge quantisation [24, 32, 31, 72] forces the Ševera class of the exact Courant algebroid describing the B-field to lie in the image of the rationalisation map $\rmH^3(M;\ZZ) \to \rmH^3(M;\RR)$, and thus the exact Courant algebroid to be a generalised tangent bundle of a 1-gerbe with connective structure $(\cG, \cA^{(1)})$ on $M$.
Viewing the latter as a principal 2-bundle on $M$ with structure 2-group $\rmB_\nabla \U(1)$, the associated exact Courant algebroid $\AtCA_\nabla(\cG, \cA^{(1)})$ is its higher Atiyah algebroid [23].
From this perspective, the correct notion of automorphisms acting on the B-field configurations are those of $(\cG, \cA^{(1)})$.
This is in analogy to how one can describe connections on a principal $G$-bundle $P$ as splittings of its Atiyah sequence, but the gauge transformations acting on the connections remain the automorphisms of $P$.
Describing connections on $P$ modulo automorphisms of the Atiyah algebroid of $P$ instead is a different interesting question.

We then still arrive at the following puzzle:
if the NSNS B-field is modelled as a connection on a fixed 1-gerbe $\cG$, its configurations form a smooth groupoid (see Theorem <ref> above), and the relevant smooth symmetry 2-group is $\scSym_\Phi(\cG)$.
In contrast, suppose the NSNS B-field is modelled as an isotropic splitting of an exact Courant algebroid (or as part of a generalised metric), which—by charge quantisation—we may assume to be of the form $\AtCA_\nabla(\cG, \cA^{(1)})$ for some 1-connection $\cA^{(1)}$ on $\cG$.
In that case, the configurations of the B-field form a smooth set, and the relevant smooth symmetry 2-group is $\scSym_\Phi(\cG, \cA^{(1)})$.
Neither the two objects describing configurations, nor the two symmetry 2-groups are equivalent.
Nevertheless, combining Theorem <ref> and Proposition <ref> allows us to prove (Theorems <ref> and <ref>):

Let $\cG$ be a 1-gerbe on $M$, and let $\cA^{(1)}$ be any 1-connection on $\cG$.
Further, let $\Phi \colon \bbGamma \to \Diff_{[\cG]}(M)$ be a smooth higher group action on $M$.

* There is a canonical equivalence of moduli $\infty$-stacks,

\begin{equation}
\scMdl_{B, \Phi}(\cG)
\simeq \scMdl_{B, \Phi}(\cG, \cA^{(1)})
\end{equation}

of NSNS B-fields described by $\cG$ and $\AtCA_\nabla(\cG, \cA^{(1)})$, respectively.

* There is a canonical equivalence of moduli $\infty$-prestacks of NSNS supergravity solutions

\begin{equation}
\scMdl_{NS, \Phi}(\cG)
\simeq \scMdl_{NS, \Phi}(\cG, \cA^{(1)})\,,
\end{equation}

modelled on $\cG$ and $\AtCA_\nabla(\cG, \cA^{(1)})$, respectively.

Solutions of NSNS supergravity are in particular generalised Ricci solitons [37]. In fact, most of our later statement regarding the moduli space of NSNS supergravity solutions apply mutatis mutandis to the moduli space of generalised Ricci solitons.

This resolves the apparent discrepancy between the two approaches to NSNS supergravity moduli:
while the two models are not equivalent at the level of configurations, and also do not have equivalent symmetry 2-groups, these differences cancel each other in the passage to the moduli $\infty$-prestacks.
It will be highly interesting to explore similar results in the setting of heterotic supergravity in the future, where we expect analogous links between connections on Chern-Simons 2-gerbes [74, 19] and structures on String algebroids [39, 40].

§.§ Notation

* $\Mfd$ denotes the category of smooth manifolds and smooth maps.

* $\Cart \subset \Mfd$ is the full subcategory of cartesian spaces.

* $\Cartfam$ denotes the category whose objects are families of cartesian spaces, parameterised by objects $c \in \Cart$ (Definition <ref>).

* $\Gpd$ denotes the 2-category of (small) groupoids, and $\Cat$ the 2-category of (small) categories.

* $\scS$ is the $\infty$-category of spaces (or, equivalently, $\infty$-groupoids).

* By an $\infty$-category, we mean a simplicial set satisfying the inner horn-lifting conditions in the sense of, for instance, [57, 20].

* We write $\Fun(-,-)$ to denote a category of functors between ordinary categories and $\scFun(-,-)$ to denote $\infty$-categories of functors between $\infty$-categories.

* We write $\sSet = \Fun(\bbDelta^\opp, \Set)$ for the category of simplicial sets.

* If $\scC$ is a simplicially enriched category, we write $\ul{\scC}(-,-) \colon \scC^\opp \times \scC \to \sSet$ for the simplicially enriched hom functor.

* For an $\infty$-category $A$, we abbreviate the $\infty$-category of $\infty$-presheaves on $A$ as $\scP(A) \coloneqq \scFun(A^\opp, \scS)$.

* For a category $\scC$ and object $c \in \scC$, we write $\scC_{/c}$ and $\scC_{c/}$ for the slice categories over and under $c$, respectively.
The same applies if $\scC$ is an $\infty$-category.

* $\Diff(M)$ is the diffeomorphism group of a manifold $M$.
We write $\Diff_{[\cG]}(M)$ for the subgroup of $\Diff(M)$ which preserves the equivalence class of a given geometric structure $\cG$ on $M$ (Definition <ref>).

* Given a group $H$, we let $\rmB H \in \Gpd$ denote its delooping:
this is the groupoid with a single object, whose group of automorphisms is $H$.

* We write $\rmB\scD$ for the Grothendieck construction of the functor $\Cart^\opp \to \Gpd$, $c \mapsto \rmB \Diff(M)$.
Analogously, we write $\rmB\scD[\cG]$ for the Grothendieck construction of $\Cart^\opp \to \Gpd$, $c \mapsto \rmB \Diff_{[\cG]}(M)$.

* We introduce a category $\GCov^\scD$ whose objects are smooth families of good open coverings of $c \times M$, where $c \in \Cart$ is a cartesian space.
Its morphisms are a combination of pulling coverings back along fibrewise diffeomorphisms and refining the result by a good open covering (Definition <ref>).

* We write $\Ch_{\geq 0}$ for the category of non-negatively graded chain complexes of abelian groups.

* $\Ab_\Delta$ denotes the category of simplicial abelian groups.

* We write $\varGamma \colon \Ch_{\geq 0} \to \Ab_\Delta$ for (one of the two functors in the) Dold-Kan correspondence.

* We use bold-face font to denote $\infty$-presheaves and roman font to denote ordinary or simplicial presheaves.

* We write $\textint F \to N\scC$ to denote the left fibration associated to a (1-categorical) functor $F \colon \scC \to \sSet$.

§.§ Acknowledgements

The authors would like to thank André Henriques, Lukas Müller, Dmitri Pavlov, Sakura Schäfer-Nameki, and Richard Szabo for helpful discussions. SB's research was funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under the project number 468806966.
The work of C.S.S. is funded by the Germany Excellence Strategy Quantum Universe - 390833306 and the 2022 Leonardo Grant for Researchers and Cultural Creators of the BBVA Foundation.

Moduli $\infty$-stacks of higher geometric structures on manifolds

This first part of the paper develops a general formalism for constructing smooth families of higher geometric structures on manifolds, taking into account the smooth action of the diffeomorphisms of the manifold.
Both of these aspects are key to obtaining a full picture of the global moduli $\infty$-stacks of higher geometric structures.

Higher geometric structures are, in full generality, described in terms of $\infty$-sheaves
[We use the term $\infty$-sheaf and $\infty$-stack interchangeably.]
with respect to (good) open coverings.
As we work on 1-categorical sites, these $\infty$-sheaves can always be presented in terms of simplicial presheaves which satisfy homotopy descent (see, for instance, <cit.>).
Since such presentations provide additional computational tools, we start our set-up from simplicial homotopy sheaves on an appropriate category with Grothendieck coverage (Sections <ref>, <ref> and <ref>).

We then construct the smooth higher automorphism and symmetry groups of higher geometric structures and prove their universal property (Sections <ref> and <ref>).
In order to encode the smooth and homotopy coherent actions of these groups which underlie the definition of moduli $\infty$-stacks, we pass to the language of left-fibrations (another model for $\infty$-presheaves <cit.>).

Finally, using the formalism developed so far, we construct moduli $\infty$-prestacks of higher geometric data (Section <ref>) and prove a useful criterion for when two different moduli $\infty$-prestacks are equivalent.
The set-up we present is motivated by field-theoretic problems, but applies in more general situations.
In Section <ref> we show that under certain conditions our moduli $\infty$-prestacks satisfy descent with respect to good open coverings of parameter spaces, i.e. form $\infty$-stacks.

§ SITES FOR SMOOTH FAMILIES OF MANIFOLDS AND DIFFEOMORPHISMS

In this section we set the stage for our study of higher geometric structures on manifolds.
We introduce a particular category with a Grothendieck coverage, whose sheaves encode smooth families of geometric data on a fixed manifold $M$, parameterised by cartesian spaces, and the smooth action of the diffeomorphism group $\Diff(M)$.

§.§ Smooth families of cartesian spaces

We begin by constructing a (Grothendieck) site which we use to parameterise smooth families of higher geometric structures.

Let $\Mfd$ be the category of smooth manifolds and smooth maps.
Let $\Cart \subset \Mfd$ be the full subcategory on those manifolds $c$ which are diffeomorphic to $\RR^n$, where $n$ ranges over all non-negative integers.

Let $\Cartfam$ denote the following category:
its objects are smooth, locally trivial fibre bundles $\hat{c} \to c$, where $c \in \Cart$, and whose fibre is a cartesian space
[Note that the bundles $(\hat{c} \to c)$ are also globally trivial by the fact that $c$ is smoothly contractible and <cit.>.
It follows that $\hat{c}$ is itself a cartesian space.].
The morphisms $(\hat{c} \to c) \longrightarrow (\hat{d} \to d)$ are commutative squares

\begin{equation}
\begin{tikzcd}
\hat{c} \ar[r, "\hat{f}"] \ar[d]
& \hat{d} \ar[d]
\\
c \ar[r, "f"']
& d
\end{tikzcd}
\end{equation}

of morphisms in $\Cart$.

In particular, for each object $(\hat{c} \to c)$ in $\Cartfam$ there is a cartesian space $d$ and some isomorphism in $\Cartfam$ of the form

\begin{equation}
\begin{tikzcd}
\hat{c} \ar[r, "\cong"] \ar[d]
&c \times d \ar[d, "\pr_c"]
\\
c \ar[r, equal]
& c
\end{tikzcd}
\end{equation}

We say an open covering $\cU = \{U_a\}_{a \in \Lambda}$ of a manifold $M$ is good if each finite intersection $U_{a_0 \cdots a_n} \coloneqq U_{a_0} \cap \cdots, \cap U_{a_n}$, for $a_0, \ldots, a_n \in \Lambda$, is either empty or a cartesian space.

Every open covering of each manifold $M$ admits a refinement by a good open covering <cit.>.

A covering of an object $(\hat{c} \to c) \in \Cartfam$ is a family $(\hat{\cU} \to \cU) = \{(\hat{U}_a \to U_a)\}_{a \in \Lambda}$ of subsets

\begin{equation}
\begin{tikzcd}
\hat{U}_a \ar[r, hookrightarrow] \ar[d]
& \hat{c} \ar[d]
\\
U_a \ar[r, hookrightarrow]
& c
\end{tikzcd}
\end{equation}

such that

* $\hat{\cU}$ and $\cU$ are good open coverings of $\hat{c}$ and $c$, respectively, and

* Every finite intersection $(\hat{U}_{a_0 \cdots a_n} \to U_{a_0 \cdots a_n})$ is either empty or an object in $\Cartfam$.

Let $c,d \in \Cart$ with good open coverings $\cU = \{U_a\}_{a \in \Lambda}$ of $c$ and $\cV = \{V_b\}_{b \in \Xi}$ of $d$.
Then, the product covering $\{U_a \times V_b\}_{a \in \Lambda, b \in \Xi}$ is a covering of $(\pr_c \colon c \times d \to c)$ in $\Cartfam$.

The second condition in Definition <ref> is not implied by the first:
for instance, let x$c = d = \RR$ and consider the object $(\pr_c \colon c \times d \to c) \in \Cartfam$.
The covering of $c \times d$ and $c$ depicted here:

satisfies condition (1), but violates condition (2) as the projection $\hat{U}_{abc} \to U_{abc}$ is not surjective.

Consider a covering $(\hat{\cU} \to \cU)$ of an object $(\hat{c} \to c) \cong (c \times c' \to c) \in \Cartfam$.
For each $x \in c$, we obtain an induced open covering $\hat{\cU}_{|x}$ of the fibre $d$.
This covering can vary from point to point, as illustrated in the following example for $c = c' = \RR$:

Here each rectangle of single boundary colour in the top space $c \times c' \cong \RR^2$ is a patch in the good open covering $\hat{\cU} = \{\hat{U}_a, \hat{U}_b, \hat{U}_c, \hat{U}_d\}$ of $c \times c'$, and we have displayed in each rectangle which arises as an intersection of the patches all indices of the patches which enter in the intersection.
In particular, this will allow us to describe smooth families of geometric data on $M$ without having to fix a good open covering of $c \times M$ for each family first.

Let $\scC$ be a category.
A Grothendieck coverage on $\scC$ is a collection $\tau$ of coverings, for each object $c \in \scC$; a covering is a family $\cU = \{f_a \colon c_a \to c\}_{a \in \Lambda}$ of morphisms with codomain $c$.
These coverings satisfy the following property:
if $\cU$ is a covering of $c \in \scC$ and $h \in \scC(d,c)$ is any morphism, then there exists a covering $\cV = \{g_b \colon d_b \to d\}_{b \in \Xi} \in \tau$ of $d$ and a map $\lambda \colon \Xi \to \Lambda$ such that, for each $b \in \Xi$, there is a commutative diagram in $\scC$:

\begin{equation}
\begin{tikzcd}
d_b \ar[r, "h_b"] \ar[d, "g_b"']
& c_{\lambda(b)} \ar[d, "f_{\lambda(b)}"]
\\
d \ar[r, "h"']
& c
\end{tikzcd}
\end{equation}

The good open coverings endow the category $\Cart$ with a Grothendieck coverage (see <cit.>).

The coverings on $\Cartfam$ define a Grothendieck coverage on $\Cartfam$, which we denote by $\tau_\rmfam$.

Consider a covering $(\hat{\cU} \to \cU)$ of $(\hat{d} \to d)$ and a morphism $(\hat{f},f) \colon (\hat{c} \to c) \longrightarrow (\hat{d} \to d)$.
Since $\hat{c} \to c$ is a trivialisable bundle, we find an isomorphism $(\varphi, 1_c) \colon (c \times \tilde{c} \to c) \longrightarrow (\hat{c} \to c)$, where the domain object is the trivial $\tilde{c}$-bundle over $c$.
The open covering $\hat{\cU}$ of $\hat{d}$ pulls back to an open covering $(\hat{f} \varphi)^{-1}(\hat{\cU})$ of $c \times \tilde{c}$.
By the properties of the product topology, this admits a refinement of the form $\hat{\cV}' = \{V'_b \times \tilde{V}'_b\}_{b \in \Xi}$, where $V'_b$ and $\tilde{V}'_b$ are open subsets of $c$ and $\tilde{c}$, respectively.
Projecting onto each factor, we obtain open coverings $\cV' = \{V'_b\}_{b \in \Xi}$ of $c$ and $\tilde{\cV}' = \{ \tilde{V}'_b\}_{b \in \Xi}$ of $\tilde{c}$.
We find good refinements $\cW = \{W_i\}_{i \in I}$ and $\tilde{\cW} = \{W_j\}_{j \in J}$ of these coverings (see the proof of <cit.>).
Then, the families $\hat{\cW} = \{\varphi(W_i \times \tilde{W}_j)\}_{i \in I, j \in J}$ and $\cW$ fit together to give a covering $(\hat{\cW} \to \cW)$ of $(\hat{c} \to c)$ with the desired properties.

§.§ Smooth families of diffeomorphisms of $M$

Next, we include the action of the diffeomorphisms group $\Diff(M)$ of $M$.
We view $\Diff(M)$ as a group object in $\Fun(\Cart^\opp, \Set)$:
it assigns to each $c \in \Cart$ the set of all smooth maps $\varphi^\dashv \colon c \times M \to c \times M$ which commute with the projection to $c$, restrict to a diffeomorphism of $M$ at each $x \in c$, and which admit an inverse map which also has these properties.
Equivalently, we can describe the data $\varphi^\dashv$ as a smooth map $\varphi \colon c \to \Diff(M)$.
Let $\Gpd$ denote the 2-category of groupoids.
Given a group object $H \in \Fun(\Cart^\opp, \Set)$, we let $\rmB H \colon \Cart^\opp \to \Gpd$ denote the strict functor which assigns to $c \in \Cart$ the groupoid $\rmB(H(c))$ consisting of a single object, which has the group $H(c)$ as its automorphisms.

Fix a manifold $M$.
We introduce the shorthand notation

\begin{equation}
\begin{tikzcd}
\rmB \scD \coloneqq \smallint \big( \rmB \Diff(M) \big) \ar[r, "\pi_M"]
& \Cart
\end{tikzcd}
\end{equation}

for the the Grothendieck construction of $\rmB \Diff(M)$.
Concretely, the objects of $\rmB \scD$ are the same as those of $\Cart$, but a morphism $c \to d$ in $\rmB \scD$ is a pair $(f, \varphi)$ of a morphism $f \in \Cart(c,d)$ and a smooth map $\varphi \colon c \to \Diff(M)$; these data induce a commutative square in $\Mfd$,

\begin{equation}
\begin{tikzcd}
c \times M \ar[r, "f \wr \varphi"] \ar[d]
& d \times M \ar[d]
\\
c \ar[r, "f"']
& d
\end{tikzcd}
\end{equation}

Here, for $x \in c$ and $z \in M$, we set

\begin{equation}
\label{eq:wr notation}
(f \wr \varphi)(x,z) = \big( f(x), \varphi^\dashv(x,z) \big)\,.
\end{equation}

Applying the Grothendieck construction to the contravariant functor $\rmB \Diff(M) \colon \Cart^\opp \to \Gpd$ and taking nerves produces a right fibration

\begin{equation}
N\pi_M \colon N \rmB \scD \longrightarrow N \Cart\,.
\end{equation}

In this article the opposite category $\rmB\scD^\opp$ plays an important role.
Its nerve is the domain of the left fibration $N\pi_M \colon N\rmB\scD^\opp \longrightarrow N\Cart^\opp$ (where we suppress the $(-)^\opp$ on the morphism for ease of notation; this will be clear from context).
We remark that, for each $c \in \Cart$, the fibre $(\rmB\scD^\opp)_{|c}$ is the delooping of the opposite group $\Diff(M)^\rev$—i.e. $\Diff(M)(c)$ with its multiplication reversed—rather than $\Diff(M)(c)$.
That is, there is a canonical isomorphism

\begin{equation}
\cong r_{\Cart^\opp}^* \rmB \big( \Diff(M)^\rev \big)
\end{equation}

of simplicial sets over $N\Cart^\opp$, where $r_{\Cart^\opp}^* \colon \Fun(\Cart^\opp, \sSet) \longrightarrow \sSet_{/N \Cart^\opp}$ is the rectification functor for presheaves of simplicial sets from Definition <ref>.
This reversal is indeed natural for us to consider since the $\Diff(M)$-actions we will encounter are via pullback of geometric structures; therefore, they naturally arise as right actions, but we will often encounter them as left actions of $\Diff(M)^\rev$.

Observe that there is a canonical functor

\begin{equation}
e_M \colon \Cart \longrightarrow \rmB \scD\,,
\end{equation}

which is the identity on objects and sends a morphism $f \colon c \to d$ in $\Cart$ to the morphism $(f, 1_M)$ in $\rmB \scD$.
It further satisfies the identity

\begin{equation}
\pi_M \circ e_M = 1_{\Cart}\,,
\end{equation}

i.e. $e_M$ is a section of $\pi_M$.

Next, we associate to $M$ a category which encodes smooth families of good open coverings of $M$ and the action of $\Diff(M)$ on these coverings.

We define a category $\GCov^\scD$, whose objects are pairs $(c, \hat{\cU} \to \cU)$ of the following data: $c \in \Cart$ is a cartesian space, and $(\hat{\cU} \to \cU)$ is a family

\begin{equation}
(\hat{\cU} \to \cU) = \{(\hat{U}_a \to U_a) \in \Cartfam\}_{a \in \Lambda}
\end{equation}

of objects $(\hat{U}_a \to U_a) \in \Cartfam$ satisfying the following properties:

* $\hat{U}_a \subset c \times M$ is an open subset,

* $U_a \subset c$ is an open subset,

* for each $a \in \Lambda$, the canonical diagram

\begin{equation}
\begin{tikzcd}
\hat{U}_a \ar[r, hookrightarrow] \ar[d]
& c \times M \ar[d]
\\
U_a \ar[r, hookrightarrow]
& c
\end{tikzcd}
\end{equation}

of manifolds and smooth maps commutes.

* $\{\hat{U}_a\}_{a \in \Lambda}$ and $\{U_a\}_{a \in \Lambda}$ form good open coverings of $c \times M$ and $c$, respectively, and

* each pair $(\hat{U}_{a_0 \cdots a_l} \to U_{a_0 \cdots a_l})$ of finite intersections is again an object in $\Cartfam$.

A morphism in $\GCov^\scD$ is defined as follows:
consider two objects $(c, \hat{\cU} \to \cU)$ and $(d, \hat{\cV} \to \cV)$ in $\GCov^\scD$, whose coverings read as $(\hat{\cU} \to \cU) = \{\hat{U}_a \to U_a\}_{a \in \Lambda}$ and $(\hat{\cV} \to \cV) = \{\hat{V}_b \to V_b\}_{b \in \Xi}$.
A morphism $(c, \hat{\cU} \to \cU) \longrightarrow (d, \hat{\cV} \to \cV)$ in $\GCov^\scD$ consists of a morphism $(f, \varphi) \colon c \to d$ in $\rmB \scD$ and a map $\lambda \colon \Lambda \to \Xi$ of indexing sets such that the restriction of $(f \wr \varphi \to f) \colon (c {\times} M \to c) \longrightarrow (d {\times} M \to d)$ to $(\hat{U}_a \to U_a)$ factors through $(\hat{V}_{\lambda(a)} \to V_{\lambda(a)})$, for each $a \in \Lambda$.

Let $\cW = \{W_i \hookrightarrow M\}_{i \in \Xi}$ be a good open covering of $M$, and let $\cU = \{U_a \hookrightarrow c\}_{a \in \Lambda}$ be a good open covering of $c \in \Cart$.
We obtain the product covering

\begin{equation}
\hat{\cU} = \cW {\times} \cU = \{U_a {\times} W_i \hookrightarrow c \times M\}_{a \in \Lambda, i \in \Xi}\,.
\end{equation}

Then, the canonical projection $(\hat{\cU} \to \cU)$ is an object in $\GCov^\scD$.
Let $\varphi \colon c \to \Diff(M)$ be a smooth family of diffeomorphisms and $f \colon c \to c$ a diffeomorphism.
Set $\hat{U}'_{ai} \coloneqq (f \wr \varphi)^{-1}(U_a \times W_i) \subset c \times M$ and $U'_a \coloneqq f^{-1}(U_a) \subset c$.
We obtain a new object $(\hat{\cU}' \to \cU') \in \GCov^\scD$, where $\hat{\cU}' = \{\hat{U}'_{ai}\}_{a \in \Lambda, i \in \Xi}$ and $\cU' = \{U'_a\}_{a \in \Lambda}$.
This comes with a canonical isomorphism

\begin{equation}
(\lambda = \id, f \wr \varphi \to f) \colon (\hat{\cU}' \to \cU') \longrightarrow (\hat{\cU} \to \cU)
\end{equation}

in $\GCov^\scD$.
Note that the new covering $(\hat{\cU}' \to \cU')$ is, in general, no longer of product form; rather, it has been twisted by the diffeomorphism $f \wr \varphi \colon c {\times} M \to c {\times} M$.

We may depict morphisms in $\GCov^\scD$ as commutative cubes

\begin{equation}
\begin{tikzcd}[column sep={1.5cm,between origins}, row sep={1.cm,between origins}]
\hat{\cU} \ar[dr] \ar[rr] \ar[dd]
& & \hat{\cV} \ar[dr] \ar[dd]
\\
& c \times M \ar[rr, "f \wr \varphi" {pos=0.25}, crossing over]
& & d \times M \ar[dd]
\\
\cU \ar[rr] \ar[dr]
& & \cV \ar[dr]
\\
& c \ar[rr, "f"'] \ar[from=uu, crossing over]
& & d
\end{tikzcd}
\end{equation}

leaving the map $\lambda$ implicit.

There is a canonical projection functor

\begin{equation}
\label{eq:varpi^scD}
\varpi^\scD \colon \GCov^\scD \longrightarrow \rmB \scD\,,
\end{equation}

whose fibre $(\varpi^\scD)^{-1}(c) = \GCov^\scD_{|c}$ over $c \in \rmB\scD$ is the cofiltered category of good open coverings of $(c {\times} M \to c)$ (the cofilteredness is seen analogously to the proof of Lemma <ref>).

Let $c \in \rmB \scD$.
The inclusion $\jmath^\scD_c \colon \GCov^\scD_{|c} \hookrightarrow \GCov^\scD_{c/}$ of the fibre into the under-category is homotopy cofinal.

An object in $X \in \GCov^\scD_{/c}$ consists of a morphism $(f, \varphi) \colon c \to d$ in $\rmB\scD$, for some object $d \in \rmB \scD$, together with an object $(\hat{\cV} \to \cV)$ in $\GCov^\scD_{|d}$.
We have to show that, for each $X \in \GCov_{/c}$, the nerve

\begin{equation}
\cong N \big( \GCov^\scD_{|c} \big) \underset{N (\GCov^\scD_{c/})}{\times} N \big( (\GCov^\scD_{|c})_{/X} \big)
\end{equation}

is a weakly contractible simplicial set.
However, the comma category $\jmath_c/X$ can be described as the category of all those good open coverings of $(c {\times} M \to c)$ which refine the open covering $((f \wr \varphi)^{-1}(\hat{\cV}) \to f^{-1}(\cV))$ of $(c {\times} M \to c)$ (the latter is, in general, not differentiably good).
This category is cofiltered, so that its nerve is indeed contractible.

The $\infty$-functor $N\varpi^\scD \colon N \GCov^\scD \longrightarrow N \rmB \scD$ satisfies the following:

* It is an isofibration.

* It is smooth (in the sense of <cit.>).

The nerve of any functor between (1-)categories is an inner fibration <cit.>; thus, $N\varpi^\scD$ is an inner fibration.
To see that it is even an isofibration, let $(c_1, \hat{\cU} \to \cU)$ be an object in $\GCov^\scD$, where $\cU = \{U_i\}_{i \in \Lambda}$ and $\hat{\cU} = \{\hat{U}_i\}_{i \in \Lambda}$ (compare Definition <ref>).
Let $(f, \varphi) \colon c_0 \to c_1$ be an isomorphism in $\rmB\scD$; that is, $f \colon c_0 \to c_1$ is a diffeomorphism of cartesian spaces.
Let $V_i \coloneqq f^{-1}(U_i)$ and $\hat{V}_i \coloneqq (f \wr \varphi)^{-1}(\hat{U}_i)$, for each $i \in \Lambda$, and set $\cV = \{V_i\}_{i \in \Lambda}$ and $\hat{\cV} = \{\hat{V}_i\}_{i \in \Lambda}$.
Then, the triple $(f, \varphi, \lambda = 1_\Lambda)$ provides an isomorphism $(c_0, \hat{\cV} \to \cV) \to (c_0, \hat{\cU} \to \cU)$ in $\GCov^\scD$ which maps to $(f, \varphi)$ under $\varpi^\scD$.
This shows that $N\varpi^\scD$ is and isofibration.
The second claim now follows from Lemma <ref> and <cit.>.

Let $\GCov \subset \GCov^\scD$ be the wide
[A subcategory is called wide if it contains all objects of its ambient category.]
subcategory whose morphisms are only those morphisms in $\GCov^\scD$ which are of the form $(f, \varphi) = (f,1_M)$.
Thus, $\GCov$ comes with a canonical projection functor $\varpi \colon \GCov \to \Cart$.

The $\infty$-functor $N\varpi \colon N\GCov \to N\Cart$ is a smooth isofibration by the same arguments as in Lemma <ref>.

We obtain a commutative diagram of categories

\begin{equation}
\begin{tikzcd}[column sep=1.25cm, row sep=1cm]
\GCov \ar[r, hookrightarrow, "\hat{e}_M"] \ar[d, "\varpi"']
& \GCov^\scD \ar[d, "\varpi^\scD"]
\\
\Cart \ar[r, shift left=0.1cm, "e_M"]
& \rmB \scD \ar[l, shift left=0.1cm, "\pi_M"]
\end{tikzcd}
\end{equation}

By definition of the subcategory $\GCov \subset \GCov^\scD$, we have the following lemma:

The commutative square of simplicial sets

\begin{equation}
\begin{tikzcd}[column sep=1.25cm, row sep=1cm]
N\GCov \ar[r, hookrightarrow, "N\hat{e}_M"] \ar[d, "N\varpi"']
& N\GCov^\scD \ar[d, "N\varpi^\scD"]
\\
N\Cart \ar[r, "Ne_M"]
& N\rmB \scD
\end{tikzcd}
\end{equation}

is cartesian.

For each $c \in \Cart$, the following statements hold true:

* The inclusion $\jmath^\scD_c \colon \GCov_{|c} \hookrightarrow \GCov_{c/}$ is homotopy cofinal.

* The canonical inclusion $\imath_c \colon \GCov_{/c} \hookrightarrow \GCov^\scD_{c/}$ is homotopy cofinal

Claim 1 follows analogously to Lemma <ref>, noting that there is a coincidence of fibres

\begin{equation}
\GCov_{|c} = \GCov^\scD_{|c}\,,
\end{equation}

for each $c \in \Cart$.
Claim 2 is then a combination of Claim 1, Lemma <ref> and <cit.>.

§ SMOOTH FAMILIES OF HIGHER GEOMETRIC STRUCTURES

In this section we present a functorial construction which produces simplicial presheaves on $\rmB\scD$ from simplicial homotopy sheaves on $\Cartfam$.
This allows us to build smooth families of global geometric data on $M$ from local models.

We make the following definitions:

* Let $\scH \coloneqq \Fun(\Cart^\opp, \sSet)$ denote the category of simplicial presheaves on $\Cart$.
We view this as endowed with the projective model structure.

* Let $\scH^{loc}$ denote the left Bousfield localisation of $\scH$ at the Čech nerves of good open coverings of cartesian spaces.

* Let $\scH_\rmfam \coloneqq \Fun(\Cartfam^\opp, \sSet)$ denote the category of simplicial presheaves on $\Cartfam$, endowed with the projective model structure.

* By $\scH_\rmfam^{loc}$ we denote the left Bousfield localisation of $\scH_\rmfam$ at the Čech nerves of $\tau_\rmfam$-coverings (see Definition <ref>).

Each of the above model structures is simplicial.
We denote the simplicially enriched hom functor in a simplicial category $\scC$ by $\ul{\scC}(-,-) \colon \scC^\opp \times \scC \to \sSet$.

There is a functor $\v{C} \colon \GCov^\scD \longrightarrow \scH_\rmfam$ defined as follows:
consider a good open covering $(\hat{\cU} \to \cU)$ of $(c \times M \to c)$, consisting of patches $(\hat{\cU} \to \cU) = \{(\hat{U}_a \to U_a)\}_{a \in \Lambda}$.
The functor $\v{C}$ sends this to the simplicial presheaf on $\Cartfam$ whose $l$-th level is

\begin{equation}
\v{C}(c, \hat{\cU} \to \cU) = \coprod_{a_0 \cdots a_l \in \Lambda} h_{(\hat{U}_{a_0 \cdots a_l} \to U_{a_0 \cdots a_l})}\,,
\end{equation}

where $h$ denotes the Yoneda embedding of $\Cartfam$.
The action of $\v{C}$ on morphisms is canonically induced by the Yoneda embedding of $\Cartfam$.
Since $\v{C}(c, \hat{\cU} \to \cU)$ is levelwise a coproduct of representables, it is cofibrant in $\scH_\rmfam$.

Consider the presheaf $\ul{(c \times M \to c)}$ on $\Cartfam$ which sends an object $(\hat{d} \to d)$ to the set of commutative square of smooth maps

\begin{equation}
\begin{tikzcd}
\hat{d} \ar[d] \ar[r]
& c \times M \ar[d, "\pr_c"]
\\
d \ar[r]
& c
\end{tikzcd}
\end{equation}

The above Čech nerve is even a cofibrant approximation in $\scH_\rmfam^{loc}$ of this presheaf:
this follows, for instance, from the fact that the canonical morphism

\begin{equation}
\v{C}(c, \hat{\cU} \to \cU)_0 = \coprod_{a \in \Lambda} h_{(\hat{U}_a \to U_a)}
\longrightarrow \ul{(c \times M \to c)}
\end{equation}

is a local epimorphism with respect to the coverage $\tau_\rmfam$ on $\Cartfam$ (see Section <ref>).
Its Čech nerve is then a weak equivalence in the local model structure by <cit.>).

There exists a functor

\begin{align}
\label{eq:sPShs on GCov^D from sPShs on Cart_fam}
&\check{C}^{\scD*} \colon \scH_\rmfam^{loc} \longrightarrow \Fun \big( (\GCov^\scD)^\opp, \sSet \big)\,,
\qquad
G \longmapsto \check{C}^{\scD*} G\,,
\\
&\check{C}^{\scD*} G (c, \hat{\cU} \to \cU)
\coloneqq \ul{\scH_\rmfam} \big( \v{C}(c, \hat{\cU} \to \cU), G \big)
\end{align}

Endowing $\Fun ((\GCov^\scD)^\opp, \sSet)$ with the projective model structure, this functor preserves fibrations, trivial fibrations, limits, as well as weak equivalences between fibrant objects.
In particular, it maps fibrant objects in $\scH_\rmfam^{loc}$ to fibrant objects of $\Fun ((\GCov^\scD)^\opp, \sSet)$.

Analogously, we define the functor

\begin{align}
&\check{C}^{M*} \colon \scH_\rmfam^{loc} \longrightarrow \Fun \big( \GCov^\opp, \sSet \big)\,,
\qquad
G \longmapsto \check{C}^{M*} G\,,
\\
&\check{C}^{M*} G (c, \hat{\cU} \to \cU)
\coloneqq \ul{\scH_\rmfam} \big( \v{C}(c, \hat{\cU} \to \cU), G \big)
\end{align}

This functor has the same properties as $\check{C}^{\scD*}$ above.

Let $\check{C}^{\scD*} G$ be a simplicial presheaf on $\GCov^\scD$ obtained via Construction <ref> from a fibrant object $G \in \scH_\rmfam^{loc}$.
For each $c \in \rmB\scD$, the functor $\check{C}^{\scD*} G$ restricts to an essentially constant functor $(\GCov^\scD)_{|c}^\opp \longrightarrow \sSet$ on the fibre over $c \in \Cart$, i.e. $\check{C}^{\scD*} G$ sends each morphism in $\GCov^\scD_{|c}$ to a weak equivalence.
The same applies to the functor $\check{C}^{M*}$.

This is a direct consequence of the fact that $G$ satisfies descent with respect to $\tau_\rmfam$.

Our goal now is to rid ourselves of the dependence on choices of good open coverings.
We achieve this by averaging over all such coverings by means of the following left Kan extensions:

The (1-categorical) left Kan extensions

\begin{align}
\Lan_{\varpi^\scD} \colon \Fun \big( (\GCov^\scD)^\opp, \sSet \big) &\longrightarrow \Fun \big( \rmB\scD^\opp, \sSet \big)
\quad \text{and}
\\
\Lan_\varpi \colon \Fun \big( \GCov^\opp, \sSet \big) &\longrightarrow \Fun \big( \Cart^\opp, \sSet \big)
\end{align}

preserve finite limits, as well as projective weak equivalences and fibrations (with respect to the Kan-Quillen model structure on $\sSet$).
In particular, they preserves projectively fibrant objects.

We present the proof of the statement for $\Lan_{\varpi^\scD}$; the proof for $\Lan_\varpi$ is entirely analogous.
Let $\wtG^\scD \colon (\GCov^\scD)^\opp \to \sSet$ be a functor.
By Lemmas <ref> and <ref>, for each $c \in \Cart$, the canonical morphism of simplicial sets

\begin{equation}
\label{eq:Lan comparison mp}
\colim \big( \wtG^\scD \colon (\GCov^\scD_{/c})^\opp \to \sSet \big)
\longrightarrow \colim \big( \wtG^\scD \colon (\GCov^\scD_{|c})^\opp \to \sSet \big)
\end{equation}

is an isomorphism.
The isomorphisms (<ref>) are not natural in $c$ since the fibres $\GCov^\scD_{|c}$ do not depend on $c$ functorially (the assignment $c \mapsto \GCov^\scD_{|c}$ is not a functor $\rmB \scD^\opp \to \Cat$).
For fixed $c \in \Cart$, they are, however, natural in the diagram $\wtG^\scD$.
We thus obtain, for each $c \in \Cart$, a canonical isomorphism

\begin{align}
\big( \Lan_{\varpi^\scD} \wtG^\scD \big) (c) &\cong \colim \Big( \wtG^\scD \colon (\GCov^\scD)_{|c}^\opp \to \sSet \Big)\,,
\end{align}

natural in $\wtG^\scD$.
Since all the properties we need to check are objectwise and stable under isomorphism, it suffices to show that, for each $c \in \Cart$, the functor

\begin{equation}
\Fun \big( (\GCov^\scD)^\opp, \sSet \big) \longrightarrow \sSet\,,
\qquad
\wtG^\scD \longmapsto \colim \big( \wtG^\scD \colon (\GCov^\scD_{|c})^\opp \to \sSet \big)
\end{equation}

preserves finite limits, fibrations and weak equivalences.

The indexing category $\GCov^\scD_{|c}$ is cofiltered (see Remark <ref>), so that the above colimit commutes with finite limits.
In particular, it follows that the colimit of a constant diagram $(\GCov^\scD)_{|c}^\opp \to \sSet$ with value $K \in \sSet$ is canonically isomorphic to $K$.
Thus, $\Lan_{\varpi^\scD}$ preserves final objects.

Next, since Kan fibrations are stable under filtered colimits <cit.> and closed under isomorphism, it follows that $\Lan_{\varpi^\scD}$ preserves Kan fibrations.
As $\Lan_{\varpi^\scD}$ also preserves final objects, we obtain that it preserves projectively fibrant objects.

Finally, again because $(\GCov^\scD)_{|c}^\opp$ is filtered, it follows from <cit.> that the above colimit is, in fact, a homotopy colimit.
Consequently, the left Kan extension $\Lan_{\varpi^\scD}$ is even a homotopy left Kan extension and thus preserves objectwise weak equivalences.

The two left Kan extensions in Proposition <ref> are related by a base change formula:

There is a canonical natural isomorphism

\begin{equation}
e_M^* \circ \Lan_{\varpi^\scD}
\cong
\Lan_\varpi \circ \hat{e}_M^*
\end{equation}

of functors $\Fun((\GCov^\scD)^\opp, \sSet) \longrightarrow \Fun(\Cart^\opp, \sSet)$.

For each $c \in \Cart$, the functor $e_M$ induces a functor $\GCov_{c/} \to (\GCov^\scD)_{c/}$; this is precisely the functor $\jmath^\scD_c$ considered in Lemma <ref>.
By that lemma, the functor $\jmath^\scD_c$ is homotopy cofinal and hence, in particular, cofinal.

We also have the following $\infty$-categorical version of Lemma <ref>.
Let $\scS$ denote the $\infty$-category of spaces.
For $\scC$ an $\infty$-category, we write $\scP(\scC) \coloneqq \scFun(\scC^\opp, \scS)$ for the $\infty$-category of $\infty$-presheaves on $\scC$.

Let $\widetilde{\bbG}^\scD \in \scP(N\GCov^\scD)$ be any $\infty$-presheaf on $N\GCov^\scD$.
The canonical base change morphism

\begin{equation}
N \varpi_! \circ N\hat{e}_M^* \widetilde{\bbG}^\scD \longrightarrow
N e_M^* \circ N\varpi^\scD_! \widetilde{\bbG}^\scD
\end{equation}

is an equivalence in $\scP(N\GCov^\scD)$.

This follows from Lemmas <ref> and <ref>, together with the Proper Base Change Theorem <cit.>.

We finish this section by summarising our main constructions so far:

We have the following functors:

* The functor

\begin{equation}
\check{C}^{\scD*} \colon \scH_\rmfam^{loc} \longrightarrow \Fun \big( (\GCov^\scD)^\opp, \sSet \big)\,,
\qquad
G \longmapsto \check{C}^{\scD*} G = \ul{\scH}_\rmfam \big( \v{C}(-), G \big)\,,
\end{equation}

which preserves limits, fibrations, trivial fibrations and weak equivalences between fibrant objects.
Further, it maps fibrant objects to functors which send refinements of coverings to weak equivalences.

* The functor

\begin{equation}
\Lan_{\varpi^\scD} \colon \Fun \big( (\GCov^\scD)^\opp, \sSet \big) \longrightarrow \Fun(\rmB\scD^\opp, \sSet)\,,
\qquad
\wtG^\scD \longmapsto \Lan_{\varpi^\scD} \wtG^\scD\,,
\end{equation}

preserves colimits, finite limits, fibrations and weak equivalences between fibrant objects.

* The composition of the above functors,

\begin{equation}
\Lan_{\varpi^\scD} \circ \check{C}^{\scD*} \colon \scH_\rmfam^{loc} \longrightarrow \Fun(\rmB\scD^\opp, \sSet)\,,
\qquad
G \longmapsto \Lan_{\varpi^\scD} \check{C}^{\scD*} G\,,
\end{equation}

preserves finite limits and fibrations (hence also fibrant objects), as well as weak equivalences between fibrant objects.

Analogous statements hold true for the functors $\check{C}^{M*}$ and $\Lan_\varpi$.

§ AVERAGING OVER OPEN COVERINGS AS AN $\INFTY$-CATEGORICAL LOCALISATION

In Proposition <ref> we saw that averaging over open coverings in $\GCov^\scD$ has nice technical properties.
Here we provide a deeper interpretation of this left Kan extension:
in Theorem <ref> we show that on associated left fibrations it amounts precisely to an $\infty$-categorical localisation at the morphisms arising from refinements of good open coverings.
This shows that the construction in Theorem <ref> is indeed a presentation of the correct way to pass from local presentations of geometric data to families of global objects on $M$.
While that is conceptually important and gives credence to our formalism, the reader interested mainly in applications may skip this section.

Let $q \colon C \to D$ be a morphism of $\infty$-categories.
Suppose that $q$ has the following properties:

* it is an inner fibration,

* it is smooth or proper, and

* for each vertex $d \in D$, the fibre $C_{|d} = q^{-1}(d) \subset C$ is a weakly contractible simplicial set.

Then, for each cartesian square

\begin{equation}
\begin{tikzcd}
A \ar[r, "u"] \ar[d, "q'"']
& C \ar[d, "q"]
\\
B \ar[r, "v"']
& D
\end{tikzcd}
\end{equation}

of $\infty$-categories, the $\infty$-functor $q'$ exhibits $\scB$ as the $\infty$-categorical localisation of $\scA$ at those morphisms in $\scA$ which become identities under $q'$.

Let $q \colon C \to D$ be a morphism of $\infty$-categories.
Suppose $q$ is an inner fibration which is also smooth or proper  and all of whose fibres are weakly contractible.
Then, $q$ exhibits $D$ as the $\infty$-categorical localisation of $C$ at those morphisms in $C$ whose image under $q$ is an identity morphism in $D$.

This is the Proposition <ref> in the case where $u$ and $v$ are identities.

In the situation of Corollary <ref>, let $W \subset C_1$ be the class of edges which become identity morphisms under $q$.
Let $\overline{W} \subset C_1$ denote the saturation of $W$ in the sense of <cit.>, i.e. the class of edges $f$ in $C$ such that $q(f)$ is an equivalence in the $\infty$-category $D$.
Then, $q$ equivalently exhibits $D$ as the localisation of $C$ at $\overline{W}$.

The following statements hold true:

* The morphism of $\infty$-categories

\begin{equation}
N \varpi^\scD \colon N \GCov^\scD \longrightarrow N\rmB \scD
\end{equation}

is an $\infty$-categorical localisation at the class $W^\scD$ of refinements of $\tau_\rmfam$-coverings, or, equivalently at its saturation $\overline{W}{}^\scD$; this is the class of morphisms $((\hat{f}, f), \varphi)$ in $\GCov^\scD$ whose image in $\rmB\scD$ is an isomorphism.
This is equivalent to $f$ being a diffeomorphism of cartesian spaces.

* The morphism of $\infty$-categories

\begin{equation}
N \varpi \colon N \GCov \longrightarrow N\Cart
\end{equation}

is an $\infty$-categorical localisation at the class $W$ of refinements of $\tau_\rmfam$-coverings, or, equivalently at its saturation $\overline{W}$; this is the class of morphisms in $\GCov$ whose image in $\Cart$ is a diffeomorphism of cartesian spaces.

The functor $N\varpi^\scD$ is a smooth inner fibration by Lemma <ref>.
Remark <ref> implies that it also has weakly contractible fibres.

Consider a cartesian diagram of $\infty$-categories

\begin{equation}
\begin{tikzcd}
A \ar[r, "u"] \ar[d, "p"']
& C \ar[d, "q"]
\\
B \ar[r, "v"']
& D
\end{tikzcd}
\end{equation}

Suppose that $q$ has the properties in Proposition <ref> and that, additionally, $v$ is conservative (i.e. reflects equivalences).
Let $\overline{W} \subset C_1$ denote the saturation of the class of morphisms at which $q$ localises $C$ (compare Corollary <ref>).
Then, $p$ exhibits $B$ as the $\infty$-categorical localisation of $A$ at those edges in $A$ whose image under $u$ lies in $\overline{W}$.
That is,

\begin{equation}
B \simeq L_{u^{-1}(\overline{W})} A\,.
\end{equation}

By Proposition <ref> we have that $p$ exhibits $B$ as the localisation of $A$ at the class $V$ of morphisms whose image under $p$ are identities.
The saturation $\overline{V} \subset A_1$ of this class consists of those morphisms in $A$ which are mapped to equivalences in $B$ under $p$.

\begin{equation}
B \simeq L_{\overline{V}} A\,.
\end{equation}

We claim that an edge $f \in A_1$ lies in $\overline{V}$ precisely if $u(f) \in \overline{W}$.
Since $v$ is conservative by assumption, $\overline{V}$ is the class of edges in $A$ whose image under $v \circ p$ is an equivalence in $D \simeq L_{\overline{W}}C$.
That is, $f \in \overline{V}$ if and only if $vp(f) \in D_1$ is an equivalence.
By the commutativity of the diagram, $vp(f) = qu(f)$, and $qu(f)$ is an equivalence if and only if $u(f) \in \overline{W}$.

Let $S \in \sSet$ be a simplicial set, and consider a commutative triangle

\begin{equation}
\begin{tikzcd}
A \ar[rr, "f"] \ar[dr, "p"']
& & B \ar[dl, "q"]
\\
& S &
\end{tikzcd}
\end{equation}

Suppose that $p$ and $q$ are left fibrations.
Then, $f$ is a covariant weak equivalence in $\sSet_{/S}$ if and only if it is a categorical weak equivalence, i.e. a weak equivalence in the Joyal model structure on $\sSet$.

Let $T$ denote a class of morphisms in a model category $\scC$ such that the left Bousfield localisation $L_T \scC$ exists.
Then, a morphism between fibrant objects in $L_T \scC$ is a weak equivalence precisely if it is a weak equivalence in the original model category $\scC$.
We apply this to the following setting:
the Joyal model structure on $\sSet$ induces a model structure on the slice category $\sSet_{/S}$ (see, for instance, <cit.>).
The covariant model structure on $\sSet_{/S}$ is the left Bousfield localisation of this induced model structure at all morphisms <cit.>

\begin{equation}
\begin{tikzcd}
\Lambda^n_0 \ar[rr, hookrightarrow] \ar[dr]
& & \Delta^n \ar[dl]
\\
& S &
\end{tikzcd}
\end{equation}

where $n$ ranges over $\NN_0$ and, for each $n$, we consider all possible morphisms $\Delta^n \to S$.
This shows the claim.

We now apply the above statements to the left fibrations associated to the simplicial presheaves from Section <ref>.
These left fibrations are computed using the rectification functor (see Appendix <ref>, as well as [45, 12] for more background):
given a category $\scC$, rectification is a functor

\begin{equation}
r_\scC^* \colon \Fun(\scC, \sSet) \longrightarrow \sSet_{/N\scC}\,.
\end{equation}

Endowing $\Fun(\scC, \sSet)$ with the projective model structure and $\sSet_{/N\scC}$ with the covariant model structure, $r_\scC^*$ is even a right Quillen equivalence (see Theorem <ref>).
Given a simplicial presheaf $F^\scD \colon \rmB \scD^\opp \to \sSet$ or $F^M \colon \Cart^\opp \to \sSet$, we write

\begin{equation}
\label{eq:base change on left fibs tint F^scD}
\textint F^\scD \coloneqq r_{\rmB\scD^\opp}^* F^\scD
\qquad \text{and} \qquad
\textint F^\scD \coloneqq r_{\Cart^\opp}^* F^M\,,
\end{equation}

By Lemma <ref> there is a canonical isomorphism

\begin{equation}
Ne_M^* \textint F^\scD
\cong \textint e_M^*F^\scD
\end{equation}

in $\sSet_{/N \Cart^\opp}$.

For a category $\scC$, we denote the $\infty$-categorical localisation of $\Fun(\scC, \sSet)$ at the objectwise weak homotopy equivalences by

\begin{equation}
\gamma_\scC \colon N\Fun(\scC, \sSet) \longrightarrow \scFun(N\scC, \scS)\,.
\end{equation}

Recall further that $r_\scC^*$ produces the left fibrations classified by the $\infty$-presheaves obtained via the localisation functor $\gamma_\scC$ (Theorem <ref>).
In other words, for $F \colon \scC \to \sSet$, the $\infty$-functor $\gamma_\scC F \colon N\scC \to \scS$ classifies the left fibration $r_\scC^*F \to N\scC$.
Given this insight, we see that the isomorphism (<ref>) is a manifestation on the level of left fibrations of the base change equivalence in Proposition <ref>.

Recall the projection functor $\varpi^\scD \colon \GCov^\scD \to \rmB\scD$ from (<ref>).

Let $\wtG^\scD \colon (\GCov^\scD)^\opp \to \sSet$ be a simplicial presheaf such that, for each $c \in \rmB\scD$ the restriction of $\wtG^\scD$ to the fibre $\GCov^\scD_{|c} = (\varpi^\scD){}^{-1}(c)$ to the fibre at $c$ is an essentially constant functor (i.e. it sends each morphism in the domain to a weak equivalence in its codomain).
Then, there is a canonical covariant equivalence

\begin{equation}
\textint \wtG^\scD
\longrightarrow (N\varpi^\scD)^* \textint \Lan_{\varpi^\scD} \wtG^\scD
\end{equation}

of left fibrations over $(N\GCov^\scD)^\opp$.

Given an object $X = (c, \hat{\cU} \to \cU) \in \GCov^\scD$, consider the canonical cocone morphism

\begin{align}
\eta_{|X} \colon \wtG^\scD(X)
\longrightarrow &\colim \Big( \wtG^\scD \colon \big( \GCov^\scD_{/c} \big)^\opp \to \sSet \Big)
\\
&\cong \colim \Big( i_c^*\wtG^\scD \colon \big( \GCov^\scD_{|c} \big)^\opp \to \sSet \Big)\,.
\end{align}

The morphisms $\eta_{|X}$ are the components of a natural transformation:
this is the unit of the adjunction $(\varpi^\scD)^* \dashv \Lan_{\varpi^\scD}$.
The isomorphism in the second line stems from Proposition <ref>; it is, in general not natural in $c$.
However, by Remark <ref> the colimit in the second line is filtered; together with the assumption that $i_c^*\wtG^\scD$ is essentially constant, this shows that the composition of $\eta_{|X}$ with the above isomorphism is a weak equivalence, for each $X \in \GCov^\scD$.
It follows by the two-out-of-three property that $\eta$ is a weak equivalence between fibrant objects in the projective model structure on simplicial presheaves on $\GCov^\scD$.
By Theorem <ref>, we thus obtain a covariant weak equivalence

\begin{equation}
\textint \wtG^\scD
\longrightarrow r_{(\GCov^\scD)^\opp}^* \circ (\varpi^\scD)^* \circ \Lan_{\varpi^\scD} \wtG^\scD
\end{equation}

between fibrant objects in $\sSet_{/N(\GCov^\scD)^\opp}$.
Finally, on the right-hand side we have a further canonical isomorphism

\begin{equation}
r_{(\GCov^\scD)^\opp}^* \circ (\varpi^\scD)^* \circ \Lan_{\varpi^\scD} \wtG^\scD
\cong N(\varpi^\scD)^* \circ r_{\rmB\scD^\opp}^* \circ \Lan_{\varpi^\scD} \wtG^\scD
= N(\varpi^\scD)^* \textint \Lan_{\varpi^\scD} \wtG^\scD\,,
\end{equation}

induced by Lemma <ref>.

Let $\wtG^\scD \colon (\GCov^\scD)^\opp \to \sSet$ be a simplicial presheaf.
Suppose that $\wtG^\scD$ is projectively fibrant and essentially constant along the fibres of $\varpi^\scD \colon \GCov^\scD \to \rmB\scD$ (as in the assumptions of Lemma <ref>).
There is a commutative diagram

\begin{equation}
\begin{tikzcd}[row sep=0.75cm, column sep=1cm]
\textint \wtG^\scD \ar[r, "v"] \ar[d, "q"']
& N(\GCov^\scD)^\opp \ar[d, "N\varpi^\scD"]
\\
\textint \Lan_{\varpi^\scD} \wtG^\scD \ar[r, "u"']
& N\rmB\scD^\opp
\end{tikzcd}
\end{equation}

in $\sSet$, and the following statements hold true:

* This diagram is homotopy cartesian in the Joyal model structure on $\sSet$.

* The horizontal morphisms are left fibrations.

* The morphism $q$ exhibits $\textint \Lan_{\varpi^\scD} \wtG^\scD$ as the $\infty$-categorical localisation of $\textint \wtG^\scD$ at those morphisms whose image under $u$ is in the class $\overline{W}{}^\scD$ from Corollary <ref>(1).

An analogous statement holds true for the map $\textint \wtG^M \to \textint \Lan_\varpi \wtG^M$ for any projectively fibrant simplicial presheaf $\wtG^M \colon \GCov^\opp \to \sSet$ which is essentially constant along the fibres of the functor $\varpi \colon \GCov \to \Cart$ and the class $\overline{W}$ from Corollary <ref>(2).

Note that any projectively fibrant simplicial presheaf $\wtG^\scD$ on $\GCov^\scD$ gives rise to a projectively fibrant simplicial presheaf $\wtG^M \coloneqq \iota^*\wtG^\scD$ on $\GCov$.
Moreover, if the former is essentially constant on the fibres of $\varpi^\scD$, then the latter is essentially constant along the fibres of $\varpi$.

Theorem <ref> shows that the $\infty$-category $\textint \Lan_{\varpi^\scD} \wtG^\scD$ arises by starting from the $\infty$-category $\textint \wtG^\scD$ and inverting (in the $\infty$-categorical sense) those of its morphisms which stem from reparameterisations (i.e. isomorphisms in $\Cart$), smooth families of diffeomorphisms of $M$ and refinements of $\tau_\rmfam$-coverings.
In the case of $\wtG^M$ we only invert morphisms coming from reparameterisations and refinements of $\tau_\rmfam$-coverings.

The morphisms $q$ and $v$ are induced by the morphism in Lemma <ref> and the universal property of pullbacks; this produces the commutative diagram.
Lemmas <ref> and <ref> imply that, in order to show Claim (1), it suffices to show that the strict pullback

\begin{equation}
\label{eq:Nvarpi^* tint wtG^D-square}
\begin{tikzcd}[row sep=1cm, column sep=1.25cm]
(N\varpi^\scD)^* \big( \textint \Lan_{\varpi^\scD} \wtG^\scD \big) \ar[r, "v'"] \ar[d, "q'"']
& N(\GCov^\scD)^\opp \ar[d, "N\varpi^\scD"]
\\
\textint \Lan_{\varpi^\scD} \wtG^\scD \ar[r, "u"']
& N\rmB\scD^\opp
\end{tikzcd}
\end{equation}

is homotopy cartesian in the Joyal model structure.
This follows because its top right, bottom left and bottom right vertices are $\infty$-categories and $N\omega^\scD$ is an isofibration between $\infty$-categories (see Lemma <ref>); therefore, it is a fibration in the Joyal model structure <cit.>.
Claim (2) holds by Theorem <ref>.
Finally, we show Claim (3):
first, note that by Claim (2) and <cit.> it follows that $v'$ and $u$ are conservative isofibrations.
Thus, we can apply Lemma <ref> to obtain that the morphism $q'$ in (<ref>) is an $\infty$-categorical localisation functor at those edges in $(N\varpi^\scD)^* \big( \textint \Lan_{\varpi^\scD} \wtG^\scD \big)$ whose image under $v'$ is in $\overline{W}{}^\scD$.
The claim now follows because Lemmas <ref> and <ref> provide a weak categorical equivalence $\eta'$ which fits into the commutative diagram

\begin{equation}
\begin{tikzcd}
\textint \wtG^\scD \ar[dr, "\eta'" description] \ar[drr, bend left=15, "v" description] \ar[ddr, bend right=30, "q" description]
& &
\\
& (N \varpi^\scD)^* \textint \Lan_{\varpi^\scD} \wtG^\scD \ar[r, "v'"] \ar[d, "q'"']
& N(\GCov^\scD)^\opp \ar[d, "N\varpi^\scD"]
\\
& \textint \Lan_{\varpi^\scD} \wtG^\scD \ar[r, "u"']
& N\Cart^\opp
\end{tikzcd}
\end{equation}

The claim then follows since localisations are compatible with weak categorical equivalences <cit.>.
The proof for $\textint \wtG^M \to \textint \Lan_\varpi \wtG^M$ and the class $\overline{W}$ is analogous.

§ DIFFEOMORPHISMS PRESERVING THE EQUIVALENCE CLASS OF A GEOMETRIC STRUCTURE

We now specialise to diffeomorphisms which preserve the equivalence class of a given geometric structure $\cG$ on $M$.
They are precisely those diffeomorphisms of $M$ which admit a lift to $\cG$, and therefore may appear in symmetries of $\cG$.

Let $G^\scD$ be a projectively fibrant simplicial presheaf on $\rmB\scD$, and let $\cG \in G^M(\RR^0)$ be a section of $G^\scD = e_M^*G^\scD$ over $\RR^0$.
Equivalently, since $\RR^0 \in \Cart$ is a final object, a vertex in $G^M(\RR^0)$ is a morphism of simplicial presheaves

\begin{equation}
\cG \colon \Delta^0 \longrightarrow G^M = e_M^* G^\scD\,,
\end{equation}

i.e. a global section of $\G^M$ over $\Cart^\opp$.
Explicitly, this section reads as

\begin{equation}
c \mapsto \coll_c^*\cG\,,
\end{equation}

where $\coll_c \colon c \to \RR^0$ is the collapse morphism in $\Cart$.
However, we will usually suppress the collapse morphisms and simply denote the section by $\cG$, as above.

Given a section $\cG \in G^M(\RR^0)$, let

\begin{equation}
\Diff_{[\cG]}(M) \subset \Diff(M)
\end{equation}

denote the subpresheaf of groups on $\Cart$ defined by

\begin{equation}
\Diff_{[\cG]}(M)(c)
= \big\{ \varphi \in \Diff(M)(c)\, \big| \, (1_c, \varphi)^*\cG \simeq \cG \text{ in } G^\scD(c) \big\}\,.
\end{equation}

Equivalently, the morphism

\begin{equation}
G^\scD(1_c, \varphi) \colon G^\scD(c) \longrightarrow G^\scD(c)
\end{equation}

of simplicial sets preserves the connected component of $\cG \in (e_M^*G^\scD)(c) = G^\scD(c)$.
We let

\begin{equation}
\rmB \scD[\cG] \subset \rmB \scD
\end{equation}

denote the Grothendieck construction of the delooping of $\Diff_{[\cG]}(M)$, and we denote the associated inclusion map by

\begin{equation}
\jmath_{[\cG]} \colon \rmB \scD[\cG] \hookrightarrow \rmB \scD\,.
\end{equation}

As it will always be clear whether we are using $\rmB\scD$ or its subcategory $\rmB\scD[\cG]$, we will still denote the canonical functors between $\rmB\scD[\cG]$ and $\Cart$ by

\begin{equation}
\begin{tikzcd}
e_M : \Cart \ar[r, shift left=0.05cm]
& \rmB\scD[\cG] : \pi_M\,. \ar[l, shift left=0.05cm]
\end{tikzcd}
\end{equation}

Observe that each of these functors acts as the identity at the level of objects.

Analogously, we obtain an inclusion of a wide
[A subcategory is wide if it contains all objects of its ambient category.]

\begin{equation}
\iota_{[\cG]} \colon \GCov^{\scD[\cG]} \hookrightarrow \GCov^\scD\,,
\end{equation}

where $\GCov^{\scD[\cG]}$ is defined as the strict pullback of categories

\begin{equation}
\label{eq:GCov^D_[G] defining pb square}
\begin{tikzcd}
\GCov^{\scD[\cG]} \ar[r, "\iota_{[\cG]}"] \ar[d, "\varpi^\scD_{[\cG]}"']
& \GCov^\scD \ar[d, "\varpi^\scD"]
\\
\rmB \scD[\cG] \ar[r, "\jmath_{[\cG]}"']
& \rmB \scD
\end{tikzcd}
\end{equation}

Given a simplicial presheaf $G^\scD \in \Fun(\rmB\scD^\opp, \sSet)$, we denote its restriction to $\rmB\scD[\cG]$ by $G^{\scD[\cG]} \in \Fun(\rmB\scD[\cG]^\opp, \sSet)$.
We denote its associated left fibration by

\begin{equation}
\textint G^{\scD[\cG]} \coloneqq r_{\rmB\scD[\cG]^\opp}^* G^{\scD[\cG]}
\longrightarrow N\rmB\scD[\cG]^\opp\,.
\end{equation}

Note that by Lemma <ref> there is a canonical isomorphism

\begin{equation}
\textint G^{\scD[\cG]} \cong \jmath_{[\cG]}^* \textint G^\scD\,.
\end{equation}

Observe that $\cG$ defines a section of $G^M$, but not of $G^{\scD[\cG]}$.
Nevertheless, since $\Cart$, $\rmB\scD$ and $\rmB\scD[\cG]$ each have the same objects, for each $c \in \rmB\scD[\cG]$ there is a canonical identification $G^M(c) = G^{\scD[\cG]}(c)$.
Even though this is not a natural transformation, it specifies a connected component of $G^\scD(c)$.
Restricting to $\rmB\scD[\cG]$, these connected components define a simplicial subpresheaf of $G^{\scD[\cG]}$:

Let $G^\scD$ be a projectively fibrant simplicial presheaf on $\rmB\scD$, and let $\cG \in G^M(\RR^0)$.
We make the following definitions:

* We let $G^{\scD[\cG]}_{[\cG]} \subset G^{\scD[\cG]}$ denote the simplicial subpresheaf on $\rmB \scD[\cG]$ such that, for each $c \in \rmB\scD[\cG]$,

\begin{equation}
G^{\scD[\cG]}_{[\cG]}(c) \subset G^{\scD[\cG]}(c) = G^M(c)
\end{equation}

is the full connected component of the vertex $\cG(c) \in G^M(c) = G^{\scD[\cG]}(c)$.
In particular, the object $G^{\scD[\cG]}_{[\cG]} \in \Fun(\rmB\scD[\cG]^\opp, \sSet)$ is again projectively fibrant if $G^\scD$ was so in $\Fun(\rmB\scD^\opp, \sSet)$.

* We also define the associated left fibration over $N\rmB\scD[\cG]^\opp$,

\begin{equation}
\big( \textint G^{\scD[\cG]}_{[\cG]} \big)
\coloneqq r_{\rmB\scD[\cG]^\opp}^* G^{\scD[\cG]}_{[\cG]}
\subset \textint G^{\scD[\cG]}\,.
\end{equation}

* Recall that $G^M \coloneqq e_M^* G^\scD$.
Observing that $G^\scD(\RR^0) = G^M(\RR^0)$, we also define a simplicial subpresheaf $G^M_{[\cG]} \subset G^M$ on $\Cart$ by requiring that

\begin{equation}
G^M_{[\cG]}(c) \subset G^M(c)
\end{equation}

is the connected component of $\cG \in G^M(c)$, for each $c \in \Cart$.

* We again also define the associated left fibration over $N\Cart^\opp$,

\begin{equation}
\big( \textint G^M_{[\cG]} \big)
\coloneqq r_{\Cart^\opp}^* G^M_{[\cG]}
\subset \textint G^M\,.
\end{equation}

With these definitions, we have the following immediate observation:

In the setting of Definition <ref> there is a canonical isomorphism

\begin{equation}
(N e_M)^* \textint G^{\scD[\cG]}_{[\cG]}
\cong \textint G^M_{[\cG]}
\end{equation}

of left fibrations over $N\Cart^\opp$.

We have canonical isomorphisms

\begin{align}
(N e_M)^* \textint G^{\scD[\cG]}_{[\cG]}
= (N e_M)^* r_{\rmB\scD[\cG]^\opp}^* G^{\scD[\cG]}_{[\cG]}
\cong r_{\Cart^\opp}^* e_M^* G^{\scD[\cG]}_{[\cG]}
\cong r_{\Cart^\opp}^* G^M_{[\cG]}\,,
\end{align}

where we have used Lemma <ref>.

We could have equivalently defined $\textint G^{\scD[\cG]}_{[\cG]}$ as follows:
the section $\cG \in \G^\scD(\RR^0)$ defines a section $\tilde{\sigma}\cG$ of the left fibration $\textint G^M \to N\Cart^\opp$.
From this, we obtain a section $\sigma \cG$ of the left fibration $(N \pi_M)_! \textint G^\scD \longrightarrow N\Cart^\opp$ (note that $N \pi_M$ is a left fibration), i.e. a commutative diagram of $\infty$-categories

\begin{equation}
\begin{tikzcd}[column sep=1.25cm, row sep=0.75cm]
N \Cart^\opp \ar[r, "\tilde{\sigma} \cG" description] \ar[rr, bend left=25, "\sigma \cG" description] \ar[dr, equal]
& \textint G^M \ar[r, hookrightarrow] \ar[d]
& \textint G^{\scD[\cG]} \ar[dl]
\\
& N\Cart^\opp &
\end{tikzcd}
\end{equation}

Then, the simplicial subset $\textint G^M_{[\cG]} \subset \textint G^M$ is the essential image of the $\infty$-functor $\tilde{\sigma}\cG$.
Analogously, we could have defined the simplicial subset

\begin{equation}
\big( \textint G^{\scD[\cG]}_{[\cG]} \big)
\subset \textint G^\scD
\end{equation}

as the essential image of the $\infty$-functor $\sigma$.
Note that here it is crucial that we restrict to the subcategory $\rmB\scD[\cG] \subset \rmB\scD$.

We point out that while the canonical morphism $\textint G^{\scD[\cG]}_{[\cG]} \longrightarrow N\rmB\scD[\cG]^\opp$ is a left fibration, the composite morphism $\textint G^{\scD[\cG]}_{[\cG]} \longrightarrow N\rmB\scD[\cG]^\opp \hookrightarrow N\rmB\scD[\cG]^\opp$ does, in general, no longer have this property.

Let $G^\scD$ be a projectively fibrant simplicial presheaf on $\rmB\scD$, and let $\cG \in G^\scD(\RR^0)$.
The morphism

\begin{equation}
(N\iota_{[\cG]})_{c/} \colon \big( N\GCov^{\scD[\cG]} \big)_{c/}
\hookrightarrow \big( N\GCov^\scD \big)_{c/}
\end{equation}

is a cofinal morphism of simplicial sets, for each $c \in \Cart$.

For each $c \in \Cart$, there is an identity of fibres

\begin{equation}
\big( \GCov^{\scD[\cG]} \big)_{|c}
= \big( \GCov^\scD \big)_{|c}\,.
\end{equation}

Moreover, for each $c \in \Cart$, there is a commutative diagram of simplicial sets

\begin{equation}
\begin{tikzcd}
\big( N\GCov^\scD \big)_{|c} \ar[r] \ar[d]
& \big( N\GCov^{\scD[\cG]} \big)_{c/} \ar[r, "{(N\iota_{[\cG]})_{c/}}"] \ar[d, "{(N\varpi^\scD_{[\cG]})_{c/}}"']
& \big( N\GCov^\scD \big)_{c/} \ar[d, "{(N\varpi^\scD)_{c/}}"]
\\
\Delta^0 \ar[r, "\{c\}"']
& \big( N\rmB \scD[\cG] \big)_{c/} \ar[r, "{(N\jmath_{[\cG]})_{c/}}"']
& \big( N\rmB \scD \big)_{c/}
\end{tikzcd}
\end{equation}

Each square in this diagram is a pullback diagram.
Since smooth morphisms of simplicial sets are stable under pullback, each vertical morphism in this diagram is a smooth morphism in $\sSet$ by Lemma <ref>.
Then, it follows (for instance from <cit.>) that the top left horizontal morphism is cofinal.
Since we already know (see Lemma <ref>) that the composition of the top horizontal arrows is cofinal, we infer that the morphism $(N\iota_{[\cG]})_{c/}$ is cofinal as well <cit.>.

Let $G^\scD$ be a projectively fibrant simplicial presheaf on $\rmB\scD$, and let $\cG$ be an element of $G^\scD(\RR^0)$.
The following statements hold true:

* The morphism $N\varpi^\scD_{[\cG]} \colon N\GCov^{\scD[\cG]} \longrightarrow N\rmB\scD[\cG]$ is smooth.

* For each simplicial presheaf $\widetilde{F}^\scD$ on $\GCov^\scD$, there is a canonical isomorphism of simplicial presheaves on $\rmB \scD[\cG]$

\begin{equation}
\jmath_{[\cG]}^* \Lan_{\varpi^\scD} \widetilde{F}^\scD
\cong \Lan_{\varpi^\scD_{[\cG]}} \imath_{[\cG]}^* \widetilde{F}^\scD\,.
\end{equation}

* Let $\wtF^\scD$ be a projectively fibrant simplicial presheaf on $\GCov^\scD$.
Set $F^\scD \coloneqq \Lan_{\varpi^\scD} \wtF^\scD$ and $F^M \coloneqq \Lan_\varpi \imath^* \wtF^\scD$.
Suppose that $\cG \in F^\scD(\RR^0)$.
There are canonical isomorphisms of simplicial presheaves on $\Cart$

\begin{equation}
N e_M^* \big( F^{\scD[\cG]}_{[\cG]} \big)
\cong N e_M^* \Big( \big( \Lan_{\varpi^\scD_{[\cG]}} \imath_{[\cG]}^*\wtF^\scD \big)_{[\cG]} \Big)
\cong \big( \Lan_\varpi \imath^* \wtF^\scD \big)_{[\cG]}\,.
\end{equation}

Claim (1) holds true because smooth maps of simplicial sets are stable under pullbacks and the definition of $\GCov^{\scD[\cG]}$ as a pullback (see Equation (<ref>)).
Claim (2) follows from Lemma <ref> (essentially, this is a presentation of the smooth base change for $\infty$-presheaves associated to the square (<ref>)).

To see Claim (3), first recall that $e_M \colon \Cart \to \rmB\scD[\cG]$ is the identity on objects.
Consequently, taking the connected components of $\cG$ over each $c \in \Cart$ commutes with restricting along $e_M$; that is, there is a canonical isomorphism

\begin{equation}
N e_M^* \Big( \big( \Lan_{\varpi^\scD_{[\cG]}} \imath_{[\cG]}^*\wtF^\scD \big)_{[\cG]} \Big)
\cong \Big( N e_M^* \Big( \big( \Lan_{\varpi^\scD_{[\cG]}} \imath_{[\cG]}^*\wtF^\scD \big) \Big)_{[\cG]}\,.
\end{equation}

Next, consider the commutative diagram

\begin{equation}
\begin{tikzcd}[column sep=1.25cm, row sep=0.75cm]
\GCov \ar[r, hookrightarrow, "\hat{e}_M"] \ar[d, "\varpi"']
& \GCov^{\scD[\cG]} \ar[d, "\varpi^\scD_{[\cG]}"]
\\
\Cart \ar[r, "e_M"']
& \rmB\scD[\cG]
\end{tikzcd}
\end{equation}

This is a strict pullback square of categories.
The associated base change morphism

\begin{equation}
\Lan_\varpi \circ \hat{e}_M^*
\longrightarrow	e_M^* \circ \Lan_{\varpi^\scD_{[\cG]}}
\end{equation}

is an isomorphism; this follows by the same argument as Lemma <ref>, using that Lemmas <ref> and <ref> still apply with $\GCov^{\scD[\cG]}$ in place of $\GCov^\scD$.
Thus, we obtain a canonical isomorphism

\begin{equation}
N e_M^* \big( \Lan_{\varpi^\scD_{[\cG]}} (\imath_{[\cG]}^*\wtF^\scD) \big)
\cong \Lan_\varpi \imath'{}^* (\imath_{[\cG]}^* \wtF^\scD)\,.
\end{equation}

That completes the proof.

Let $G^\scD$ be a projectively fibrant simplicial presheaf on $\rmB\scD$.
We emphasise that all definitions in this section only depend on the connected component of $\cG$ in $\G^\scD(\RR^0)$, and not on the actual choice of $\cG$ itself.
That is, if $\cG \simeq \cG'$ in $G^M(\RR^0)$, then we have

= _['](M) ,
= ['] ,
= ^['] ,

= G^['] ,
= G^[']_['] ,

= G^['] ,
= G^[']_['] ,

and similarly for the pullbacks of the above simplicial presheaves and left fibrations to the categories $\GCov$ and $\Cart$.

§ HIGHER SYMMETRY GROUPS OF HIGHER GEOMETRIC STRUCTURES

In this section we use the left fibrations associated to the simplicial presheaves considered so far to construct the smooth higher automorphism and symmetry groups of a fixed higher geometric structure $\cG$ on $M$.
More precisely, for each smooth action $\Phi \colon \bbGamma \to \Diff_{[\cG]}(M)$ of a smooth higher group $\bbGamma$ on $M$ we construct a smooth $\infty$-group of symmetries which lift the action of elements of $\bbGamma$ to $\cG$.
We show that these higher symmetry groups are extensions of $\bbGamma$ by the smooth higher automorphism group of $\cG$ (Theorem <ref> and Corollary <ref>) and that the smooth higher symmetry groups completely classify smooth equivariant structures on $\cG$ (Theorem <ref> and Corollary <ref>).

§.§ The smooth higher automorphism and symmetry groups

First, given a smooth higher group action $\Phi \colon \bbGamma \to \Diff_{[\cG]}(M)$, we define the smooth higher automorphism and symmetry groups of a fixed higher geometric structure $\cG$ on $M$.
We show that there is a short exact sequence of smooth $\infty$-groups, exhibiting the smooth higher symmetry group of $\cG$ as an extension of $\bbGamma$ by the smooth higher automorphism group of $\cG$.

Let $G^\scD$ be a projectively fibrant simplicial presheaf on $\rmB\scD$, and let $\cG \in G^\scD(\RR^0)$.
Recall the sections $\tilde{\sigma} \cG$ and $\sigma \cG$ from Remark <ref>.
We define the following simplicial sets:

* We denote by

\begin{equation}
\rmB \AUT^\rev(\cG) \subset \textint G^M
\end{equation}

the full simplicial subset (i.e. the full $\infty$-subcategory) on the vertices in the image of the section $\tilde{\sigma} \cG \colon N\Cart^\opp \to \textint G^M$ from Remark <ref>.

* We denote by

\begin{equation}
\rmB \SYM^\rev(\cG) \subset \textint G^{\scD[\cG]}
\end{equation}

the full simplicial subset (i.e. the full $\infty$-subcategory) on the vertices in the image of the composition

\begin{equation}
\label{eq:section from (cG,cA^(k))}
\begin{tikzcd}
\sigma \cG \colon N\Cart^\opp \ar[r, "\tilde{\sigma} \cG"]
& \textint G^M \ar[r]
& \textint G^{\scD[\cG]}\,.
\end{tikzcd}
\end{equation}

The simplicial set $\rmB\SYM^\rev(\cG)$ has a unique vertex over each object $c \in \Cart$; it consists of the constant family $\tilde{\sigma}\cG(c) \in G^{\scD[\cG]}(c) = (\textint G^{\scD[\cG]})_{|c}$ (recall that $\tilde{\sigma}\cG(c) = \coll_c^*\cG$, where $\cG \in G^M(\RR^0)$ is the fixed chosen vertex and $\coll_c \colon c \to \RR^0$ is the canonical collapse morphism).
A 1-simplex in $\rmB\SYM^\rev(\cG)$ is a pair $((f, \varphi), \psi)$, consisting of a morphism $(f,\varphi) \colon c_0 \to c_1$ in $\rmB\scD[\cG]$ and an equivalence

\begin{equation}
\psi \colon (f, \varphi)^* \big( \tilde{\sigma} \cG(c_1) \big) \longrightarrow \tilde{\sigma}\cG(c_0)
\end{equation}

in $G^\scD(c_0) = G^M(c_0)$.
Note that, in general, $(f, \varphi)^* (\tilde{\sigma} \cG(c_1)) \neq \tilde{\sigma} \cG(c_0)$, but as long as $(f,\varphi)$ is a morphism in $\rmB\scD[\cG] \subset \rmB\scD$, there is an equivalence in $G^\scD(c_0)$ as above, by definition of $\rmB\scD[\cG]$.
In particular, $\rmB\SYM^\rev(\cG)$ is not the rectification of a simplicial subpresheaf of $G^{\scD[\cG]}$.

Finally, the vertices in $\rmB\AUT^\rev(\cG)$ are the same as those in $\rmB\SYM^\rev(\cG)$, and morphisms are those morphisms in $\rmB\SYM^\rev(\cG)$ where $\varphi = \id_M$.

The superscript $\rev$ indicates that $\rmB\SYM^\rev(\cG)$ is naturally related to the opposite of the group of symmetries of $\cG$; this is consistent with the fact that the codomain of the left fibration $\rmB\SYM^\rev(\cG) \to N\rmB\scD[\cG]^\opp$ describes the opposite of the diffeomorphism group $\Diff_{[\cG]}(M)$.

Let $\scC$ be an $\infty$-category, let $c$ be an object of $\scC$, and let $F \colon K \to \scC_{/c}$ be a diagram in $\scC_{/c}$, indexed by some simplicial set $K$.
If $K$ is weakly contractible (i.e. weakly equivalent to $\Delta^0$ in the Kan-Quillen model structure on $\sSet$), then the projection functor $\pr \colon \scC_{/c} \to \scC$ preserves and reflects limits.
That is, $\overline{F} \colon K^\triangleleft \to \scC_{/c}$ is a limit diagram if and only if $\pr \circ \overline{F} \colon K^\triangleleft \to \scC$ is so.

The limit of $F$ in $\scC_{/c}$ agrees with the limit of

\begin{equation}
F * c \colon K^\triangleright = K * \Delta^0 \longrightarrow \scC\,,
\end{equation}

together with its canonical morphism to $c$, where $(-)*(-)$ denotes the join construction of simplicial sets.
The canonical inclusion $K \hookrightarrow K * \Delta^0$ is a cofinal morphism of simplicial sets if and only if $K$ is weakly contractible.

In the case where $\scC$, $\scD$ and $K$ are 1-categories and we are interested in 1-categorical limits it even suffices to have the indexing category $K$ connected; this is enough to ensure that the inclusion $K \hookrightarrow K * [0]$ is a cofinal functor of 1-categories (though, in general, it will not be homotopy cofinal).

Consider the setting of Definition <ref>.
For the commutative diagram

\begin{equation}
\begin{tikzcd}
\rmB \AUT^\rev(\cG) \ar[r, hookrightarrow] \ar[d]
& \rmB \SYM^\rev(\cG) \ar[d]
\\
N\Cart^\opp \ar[r, hookrightarrow, "Ne_M"']
& N \rmB \scD[\cG]^\opp
\end{tikzcd}
\end{equation}

of simplicial sets, the following statements hold true:

* Both vertical morphisms are left fibrations.

* The square is a pullback diagram, and hence (by Claim (1)) even a homotopy pullback square in the Joyal model structure.

* Augmenting by the map $N \rmB \scD[\cG] \longrightarrow N\Cart^\opp$ produces a cartesian square in $\sSet_{/N \Cart^\opp}$, which is even homotopy cartesian in the covariant model structure.

For Claim (1) it suffices to show that the morphism

\begin{equation}
\rmB \SYM^\rev(\cG) \longrightarrow \rmB \scD[\cG]^\opp
\end{equation}

is a left fibration.
The statement for the left-hand vertical morphism will then follow from Claim (2).
We have a sequence

\begin{equation}
\begin{tikzcd}
\rmB \SYM^\rev(\cG) \ar[r, hookrightarrow]
&  \textint G^{\scD[\cG]}_{[\cG]} \ar[r]
& N \rmB \scD[\cG]^\opp\,.
\end{tikzcd}
\end{equation}

The first morphism is an inclusion of a full $\infty$-subcategory, and the second morphism is a left fibration.
We need to show that the composition has the right lifting property with respect to all left horn inclusions $\Lambda^r_s \hookrightarrow \Delta^r$, for $0 \leq s < r \in \NN_0$.
Since the first morphism is an inclusion of a full $\infty$-subcategory, it automatically has the right lifting property with respect to all left horn inclusions except possibly the case $\Lambda^1_0 \hookrightarrow \Delta^1$.
As the second morphism is a left fibration, the only non-trivial lifting problems for the composition arise for the horn inclusion $\Lambda^1_0 \hookrightarrow \Delta^1$.
We check the lifting property explicitly:
a commutative diagram

\begin{equation}
\begin{tikzcd}
\Lambda^1_0 \ar[r] \ar[d, hookrightarrow]
& \rmB \SYM^\rev(\cG) \ar[d]
\\
\Delta^1 \ar[r]
& N \rmB \scD[\cG]^\opp
\end{tikzcd}
\end{equation}

is equivalent to specifying a 1-simplex in $N\rmB\scD[\cG]^\opp$ and a lift of its initial vertex to $\rmB \SYM^\rev(\cG)$.
The 1-simplex is a pair $(f,\varphi)$ of a smooth map $f \colon c_1 \to c_0$ of cartesian spaces and a smooth map $\varphi \colon c_1 \to \Diff_{[\cG]}(M)$.
Let $\coll_c \colon c \to \RR^0$ denote the collapse map in $\Cart$.
A solution to the lifting problem amounts to specifying an equivalence

\begin{equation}
(\coll_{c_1}, 1_M)^* (\cG)
\simeq (f, \varphi)^* (\coll_{c_0}, 1_M)^* (\cG)
= (1_{c_1}, \varphi)^* (\coll_{c_1}, 1_M)^* (\cG)
\end{equation}

in in $G^\scD(c_1)$ (see also the construction of the rectification functor $r_\scC^*$ in <cit.> and <cit.>).
Such an equivalence exists since $f$ takes values in $\Diff_{[\cG]}(M)$ (see Definition <ref>).

Claim (2) follows by inspection of the diagram:
the inclusion $\rmB\AUT^\rev(\cG) \hookrightarrow \rmB\SYM^\rev(\cG)$ is a bijection on vertices, and a higher simplex in $\rmB\SYM^\rev(\cG)$ lies in $\rmB\AUT^\rev(\cG)$ precisely if its image in $N\rmB\scD[\cG]$ consists of a composable chain $(f_0, \varphi_0), \ldots, (f_n, \varphi_n)$ of morphisms in $\rmB\scD[\cG]$ with $\varphi_i = 1_M$, for each $i = 1, \ldots, n$.
Claim (3) follows readily from Lemma <ref> (see also Remark <ref>) and Claim (2), since the covariant fibrations between left fibrant objects are precisely the left fibrations (see the opposite of <cit.>).

In the setting of Definition <ref> the canonical inclusion morphisms

\begin{equation}
\rmB \SYM^\rev(\cG) \hookrightarrow \textint G^{\scD[\cG]}_{[\cG]}\,,
\qquad
\rmB \AUT^\rev(\cG) \hookrightarrow \textint G^M_{[\cG]}
\end{equation}

are covariant weak equivalences in $\sSet_{/N \rmB \scD[\cG]^\opp}$ and $\sSet_{/N \Cart^\opp}$, respectively.

We show the claim for $\rmB\SYM^\rev(\cG)$; the proof for $\rmB\AUT^\rev(\cG)$ is analogous.
First, note that both $\rmB\SYM^\rev(\cG)$ and $\textint G^{\scD[\cG]}_{[\cG]}$ are fibrant in the covariant model structure on $\sSet_{/N \rmB \scD[\cG]^\opp}$ (by Lemma <ref> and Definition <ref>, respectively).
It follows that both simplicial sets are $\infty$-categories.
Recall that the simplicial subset

\begin{equation}
\textint G^{\scD[\cG]}_{[\cG]} \subset \textint G^{\scD[\cG]}
\end{equation}

is the essential image of the functor (<ref>) (by Remark <ref>).
The further simplicial subset

\begin{equation}
\rmB\SYM^\rev(\cG) \subset \textint G^{\scD[\cG]}_{[\cG]}
\end{equation}

is the full simplicial subset on all vertices in the image of the functor  (<ref>); it follows readily that the inclusion

\begin{equation}
\rmB\SYM^\rev(\cG) \subset \textint G^{\scD[\cG]}_{[\cG]}
\end{equation}

is both fully faithful and essentially surjective as a functor between $\infty$-categories.
That is, it is a weak equivalence in the Joyal model structure <cit.>.
Lemma <ref> then implies that it is also a covariant weak equivalence in $\sSet_{/N \rmB \scD[\cG]^\opp}$.

Note that, for each $c \in \Cart$, the fibres $\rmB\AUT^\rev(\cG)_{|c}$ and $\rmB\SYM^\rev(\cG)_{|c}$ are reduced simplicial sets, i.e. they have a unique vertex.
The looping-delooping Quillen equivalence between reduced simplicial sets and simplicial groups <cit.> shows that the left fibrations $\rmB\AUT^\rev(\cG) \to N\Cart^\opp$ and $\rmB\SYM^\rev(\cG) \to N\Cart^\opp$ present presheaves of $\infty$-groups on $N\Cart$ by <cit.> (see also the following remark and Definition <ref>).

We recall the equivalence between pointed connected objects in an $\infty$-topos $\scX$ and group objects in $\scX$ <cit.>.
Concretely, there are inverse equivalences of $\infty$-categories

\begin{equation}
\label{eq:Omega -| B equivalence}
\begin{tikzcd}
\Omega : \scX^{*/}_{\geq 1} \ar[r, shift left=0.1cm]
& \Grp(\scX) : \rmB\,, \ar[l, shift left=0.1cm]
\end{tikzcd}
\end{equation}

where $\scX^{*/}_{\geq 1} \subset \scX$ denotes the $\infty$-subcategory of pointed connected objects in $\scX$, and $\Grp(\scX)$ denotes the $\infty$-category of group objects in $\scX$ (see, for instance, <cit.> and [60, 13]).
The functor $\rmB$ takes the classifying object of a group object, while $\Omega$ takes the loop object (with its homotopy coherent multiplication) of a pointed connected object in $\scX$.
Here we will mostly be interested in $\infty$-topoi which arise as $\infty$-categories of presheaves of spaces.

A smooth $\infty$-group is a group object in the $\infty$-topos $\scP(N\Cart)$.
The $\infty$-category of smooth $\infty$-groups is $\Grp(\scP(N\Cart))$.

Let $\rho \colon \bbDelta \to \bbDelta$ be the functor defined by

\begin{equation}
\rho[n] = [n]\,,
\qquad
\rho(\lambda)(i) = n - \lambda(m-i)
\end{equation}

for each $\lambda \in \bbDelta([m], [n])$ and $i \in \{0, \ldots, m\}$ (compare, for instance, <cit.>).
This functor is a strict involution, i.e. $\rho \circ \rho = 1_\bbDelta$.
In particular, it is an auto-equivalence of $\bbDelta$.

Let $\scC$ be an $\infty$-category.
We define the involution

\begin{equation}
\rev \coloneqq (N\rho^\opp)^* \colon \scFun(N\bbDelta^\opp, \scC) \longrightarrow \scFun(N\bbDelta^\opp, \scC)
\end{equation}

on the $\infty$-category of simplicial objects in $\scC$.

An $\infty$-category $\scC$ is said to have geometric realisations if all $\bbDelta^\opp$-shaped diagrams in $\scC$ have colimits.

If $\scC$ is an $\infty$-category which admits geometric realisations and $X \in \scFun(N\bbDelta^\opp, \scC)$ is a simplicial object in $\scC$, then there is a canonical equivalence

\begin{equation}
|\rev X| \simeq |X|
\end{equation}

in $\scC$, natural in $X$.

Since $\rho$ is an equivalence of categories, the induced functor $N\rho^\opp$ on $N \bbDelta^\opp$ is an equivalence.
Thus, it is final <cit.>.

In the setting of Definition <ref>, we make the following definitions:

* We denote the $\infty$-presheaf which classifies the left fibration $\pi_\SYM \colon \rmB \SYM^\rev(\cG) \longrightarrow N \Cart^\opp$ by

\begin{equation}
\rmB \scSym^\rev(\cG) \colon N \Cart^\opp \longrightarrow \scS\,.
\end{equation}

Since the fibres of $\pi_\SYM$ are reduced simplicial sets, the values of $\rmB \scSym^\rev(\cG)$ are pointed, connected spaces.
By Remark <ref>, there is thus a unique group object associated to $\rmB\scSym^\rev(\cG)$, which we denote by

\begin{equation}
\scSym^\rev(\cG) \in \Grp \big( \scFun(N \Cart^\opp, \scS) \big)\,.
\end{equation}

* The reverse simplicial object

\begin{equation}
\scSym(\cG) \coloneqq \big( \scSym^\rev(\cG) \big)^\rev
\quad \in \Grp \big( \scFun (N \Cart^\opp, \scS) \big)
\end{equation}

is the smooth higher group of symmetries of $\cG \in \G^\scD(\RR^0)$.

* Similarly, we let

\begin{equation}
\rmB\scAut^\rev(\cG) \colon N\Cart^\opp \longrightarrow \scS
\end{equation}

denote the $\infty$-presheaf which classifies the left fibration $\pi_\AUT \colon \rmB\AUT^\rev(\cG) \longrightarrow N\Cart^\opp$.
We denote its associated group object by

\begin{equation}
\scAut^\rev(\cG) \in \Grp \big( \scFun (N \Cart^\opp, \scS) \big)\,.
\end{equation}

* The reverse simplicial object

\begin{equation}
\scAut(\cG) \coloneqq \big( \scAut^\rev(\cG) \big)^\rev
\quad \in \Grp \big( \scFun (N \Cart^\opp, \scS) \big)
\end{equation}

is the smooth higher group of automorphisms of $\cG \in G^\scD(\RR^0)$.

We denote by

\begin{equation}
\bbG^{\scD[\cG]}_{[\cG]} = \gamma_{\rmB\scD[\cG]^\opp}^* G^{\scD[\cG]}_{[\cG]}
\end{equation}

the $\infty$-presheaf associated to $G^{\scD[\cG]}_{[\cG]}$.

There is a canonical commutative triangle

\begin{equation}
\begin{tikzcd}[column sep={2cm,between origins}, row sep=0.75cm]
\rmB\scSym^\rev(\cG) \ar[rr] \ar[dr]
& & (N\pi_M)_! \bbG^{\scD[\cG]}_{[\cG]} \ar[dl]
\\
& \rmB \Diff_{[\cG]}^\rev(M) &
\end{tikzcd}
\end{equation}

The top morphism is an objectwise equivalence.

For each $\infty$-category $A$, there is a canonical equivalence between the $\infty$-categorical localisation of $\sSet_{/A}$ at the covariant weak equivalences and the $\infty$-category $\scFun(A, \scS)$ <cit.>.
By construction, the triangle in the statement is presented under this equivalence by the commutative triangle

\begin{equation}
\begin{tikzcd}[column sep={2cm,between origins}, row sep=0.75cm]
\rmB\SYM^\rev(\cG) \ar[rr, hookrightarrow] \ar[dr]
& & \textint G^{\scD[\cG]}_{[\cG]} \ar[dl]
\\
& N \rmB\scD[\cG] &
\end{tikzcd}
\end{equation}

of simplicial sets over $N\Cart^\opp$.
Lemma <ref> shows that the top morphism is a covariant weak equivalence over $N\rmB\scD[\cG]^\opp$.
Since each object in the covariant model structure is cofibrant, under the left Quillen functor

\begin{equation}
(N\pi_M)_! \colon \sSet_{/N \rmB \scD[\cG]^\opp} \longrightarrow \sSet_{/N \Cart^\opp}
\end{equation}

this becomes a covariant weak equivalence over $N\Cart^\opp$.

Let $\scX$ be an $\infty$-topos.
An extension of group objects in $\scX$ is a sequence of morphisms $\bbA \to \bbH \to \bbGamma$ in the $\infty$-category $\Grp(\scX)$ of group objects in $\scX$ such that the induced sequence $\rmB\bbA \to \rmB\bbH \to \rmB\bbGamma$ is a fibre sequence in $\scX$.

Proposition <ref>(3) has the following direct consequence:

We have an extension of group objects in $\scFun(N\Cart^\opp, \scS)$:

\begin{equation}
\begin{tikzcd}
\scAut(\cG) \ar[r]
& \scSym(\cG) \ar[r]
& \Diff_{[\cG]}(M)\,.
\end{tikzcd}
\end{equation}

By Proposition <ref>(3) and the presentation of the $\infty$-category $\scP(N\Cart)$ by the covariant model category $\sSet_{/N \Cart^\opp}$ (see <cit.>) the square

\begin{equation}
\begin{tikzcd}
\rmB \scAut^\rev(\cG) \ar[r] \ar[d]
& \rmB \scSym^\rev(\cG) \ar[d]
\\
* \ar[r]
& \rmB\Diff_{[\cG]}^\rev(M)
\end{tikzcd}
\end{equation}

is a fibre sequence in $\scFun(N\Cart^\opp, \scS)$.
Lemma <ref> shows that this is equivalent to fact that the fibre square for the classifying objects of the non-reversed group objects is a fibre sequence.

The smooth $\infty$-group $\scSym(\cG)$ can be interpreted as consisting of all possible lifts of elements $\varphi \in \Diff_{[\cG]}(M)$ to $\cG$ (see, in particular, Theorem <ref> and Corollary <ref> below).
We now associate similar universal groups to any smooth group action on $M$.

Consider a covariantly fibrant object $Q \to N\Cart^\opp$ in $\sSet_{/N \Cart^\opp}$ and suppose that the fibre $Q_{|c}$ is a reduced simplicial set, for each $c \in \Cart$.
Suppose $\phi \colon Q \to N\rmB\scD[\cG]^\opp$ is a morphism in $\sSet_{/N \Cart^\opp}$.
We define a left fibration over $Q$ as the pullback (this indeed produces a left fibration by Proposition <ref>(1))

\begin{equation}
\begin{tikzcd}
\rmB\SYM^\rev_\phi(\cG) \ar[r] \ar[d, "\psi_\phi"']
& \rmB\SYM^\rev(\cG) \ar[d]
\\
Q \ar[r, "\phi"']
& N\rmB\scD[\cG]^\opp
\end{tikzcd}
\end{equation}

Note that there is a canonical factorisation

\begin{equation}
\begin{tikzcd}[column sep={1.5cm,between origins}, row sep=0.75cm]
N\Cart^\opp \ar[rr, "\eta_M"] \ar[dr, "Ne_M"']
& & Q \ar[dl, "\phi"]
\\
& N\rmB\scD[\cG]^\opp
\end{tikzcd}
\end{equation}

of the morphism $e_M \colon N\Cart^\opp \longrightarrow N\rmB\scD[\cG]^\opp$ through $\phi$.

Let $H$ be a presheaf of groups on $\Cart$, and let $\psi \colon H \to \Diff_{[\cG]}(M)$ be a morphism of group objects in $\scH$.
Recall the isomorphism

\begin{equation}
N\rmB\scD[\cG]^\opp \cong r_{\Cart^\opp}^* \big( N \circ \rmB \Diff_{[\cG]}(M)^\rev \big)
\end{equation}

from Remark <ref>.
Consider the simplicial set

\begin{equation}
Q = \rmB H^\rev
\coloneqq r_{\Cart^\opp}^* \rmB H^\rev\,,
\end{equation}

where $\rmB$ is the delooping functor, together with the canonical morphism $\phi \coloneqq r_{\Cart^\opp}^* \rmB(\phi^\rev) \colon Q \to N\rmB\scD[\cG]$.
Then, the left fibration $\rmB\SYM^\rev_\phi(\cG)$ encodes those symmetries of $\cG$ which cover the action of elements of $H$.

We have the following relative version of Proposition <ref>:

In the situation of Definition <ref>, we have a commutative square

\begin{equation}
\begin{tikzcd}
\rmB \AUT^\rev(\cG) \ar[r, hookrightarrow] \ar[d, "\psi_\Aut"']
& \rmB \SYM^\rev_\phi(\cG) \ar[d]
\\
N\Cart^\opp \ar[r, hookrightarrow, "\eta_M"']
& Q
\end{tikzcd}
\end{equation}

of simplicial sets.
Moreover, the following statements hold true:

* Both vertical morphisms are left fibrations.

* The square is a pullback diagram, and hence (by Claim (1)) even a homotopy pullback square in the Joyal model structure.

* Augmenting by the map $Q \to N\Cart^\opp$, this produces a cartesian square in $\sSet_{/N \Cart^\opp}$, which is even homotopy cartesian in the covariant model structure.

Claim (1) follows from Proposition <ref>(1) (for the left-hand vertical morphism) and Definition <ref> (for the right-hand vertical morphism).
Claim (2) follows by the pasting law for pullbacks, and Claim (3) follows from Claim (2) and Lemma <ref> (see also Remark <ref>).

The left fibration $Q \to N\Cart^\opp$ is classified by an $\infty$-presheaf $\rmB\bbGamma^\rev \colon N\Cart^\opp \to \scS$ which takes values in pointed, connected spaces.
It thus presents an essentially unique group object $\bbGamma = (\Omega \rmB \bbGamma^\rev)^\rev$ in $\scP(N\Cart)$ (recall the equivalence (<ref>)).
Then, the morphism $\phi \colon Q \to N\rmB\scD[\cG]^\opp$ presents a morphism $\Phi \colon \bbGamma \to \Diff_{[\cG]}(M)$ of group objects in $\scP(N\Cart)$.

Let $\bbGamma$ be a smooth $\infty$-group, i.e. a group object in $\scP(N\Cart)$, and let $\Phi \colon \bbGamma \to \Diff_{[\cG]}(M)$ be a morphism of smooth $\infty$-groups.
We define a smooth $\infty$-group as the pullback in $\Grp(\scP(N\Cart))$,

\begin{equation}
\begin{tikzcd}
\scSym_\Phi(\cG) \ar[r] \ar[d]
& \scSym(\cG) \ar[d]
\\
\bbGamma \ar[r, "\Phi"']
& \Diff_{[\cG]}(M)
\end{tikzcd}
\end{equation}

By construction, this is an extension of $\bbGamma$ by $\scAut(\cG)$, i.e. we have a fibre sequence in $\Grp(\scP(N\Cart))$:

\begin{equation}
\begin{tikzcd}
\scAut(\cG) \ar[r]
& \scSym_\Phi(\cG) \ar[r]
& \bbGamma\,.
\end{tikzcd}
\end{equation}

This follows readily from Corollary <ref> and the pasting law for pullbacks.

In the preceding discussion the left fibration $\rmB\SYM_\phi^\rev(\cG) \to Q$ presents the morphism

\begin{equation}
\scSym_\Phi(\cG)^\rev \to \bbGamma^\rev
\end{equation}

of smooth $\infty$-groups.

§.§ The universal property of the group of symmetries

In this subsection we prove the relation between the smooth higher symmetry group of a higher geometric structure $\cG$ and smooth equivariant structures on $\cG$.

Let $G^\scD$ be a projectively fibrant simplicial presheaf on $\rmB\scD$, and let $\cG \in \G^\scD(\RR^0)$ be a section.
Let $G^{\scD[\cG]}_{[\cG]}$ be the associated projectively fibrant simplicial presheaf on $\rmB\scD[\cG]$ (see Definition <ref>).
We denote by

\begin{equation}
\bbG^{\scD[\cG]}_{[\cG]} \coloneqq \gamma_{\rmB\scD[\cG]^\opp}^*G^{\scD[\cG]}_{[\cG]}
\end{equation}

the associated $\infty$-presheaf on $N\rmB\scD[\cG]$.
By construction, there is a canonical morphism of $\infty$-presheaves $\bbG^{\scD[\cG]}_{[\cG]} \to \rmB\Diff_{[\cG]}^\rev(M)$.
In this section we exhibit the geometric and structural significance of the smooth $\infty$-group $\scSym(\cG)$ associated to any higher geometric structure $\cG \in G^\scD$ on $M$:
it arises from a universal property (it represents a certain $\infty$-functor) and completely controls smooth families of equivariant structures on $\cG$.
As before, let $\phi \colon Q \to N\rmB\scD[\cG]^\opp$ be a morphism of left fibrations over $N\Cart^\opp$ which presents the $\infty$-group action $\Phi \colon \bbGamma \to \Diff_{[\cG]}(M)$.
We denote the left fibration $Q \to N\Cart^\opp$ by $q$.

Let $\Phi \colon \bbGamma \to \Diff(M)$ be a morphism of group objects in $\scP(N\Cart)$.
The space of $(\bbGamma, \Phi)$-equivariant structures on $\cG$ is the pullback of $\infty$-groupoids

\begin{equation}
\begin{tikzcd}[column sep=1.5cm]
\Equivar_\Phi(\cG) \ar[r] \ar[d]
& \scP(N\Cart)_{/\rmB\Diff_{[\cG]}^\rev(M)} \big( \rmB \bbGamma^\rev, (N\pi_M)_! \bbG^{\scD[\cG]}_{[\cG]} \big) \ar[d]
\\
\Delta^0 \ar[r, "\{\cG\}"']
& \big( (N\pi_M)_! \bbG^{\scD[\cG]}_{[\cG]} \big)(\RR^0)
\end{tikzcd}
\end{equation}

The right-hand vertical morphism in this square arises from the fact that $\rmB \bbGamma^\rev$ is a pointed object of $\scP(N\Cart)$, i.e. it has a distinguished global section.
This induces a morphism

\begin{equation}
\scP(N\Cart)_{/\rmB\Diff_{[\cG]}^\rev(M)} \big( \rmB \bbGamma^\rev, (N\pi_M)_! \bbG^{\scD[\cG]}_{[\cG]} \big)
\longrightarrow \big( (N\pi_M)_! \bbG^{\scD[\cG]}_{[\cG]} \big) (\RR^0)\,.
\end{equation}

The morphism $e_M \colon \Cart^\opp \to \rmB\scD[\cG]$ induces a canonical morphism

\begin{equation}
\bbG^M_{[\cG]}
= Ne_M^* \bbG^{\scD[\cG]}_{[\cG]} \longrightarrow
\big( (N\pi_M)_! \bbG^{\scD[\cG]}_{[\cG]} \big)\,.
\end{equation}

The bottom horizontal morphism is the composition of this morphism by $\cG \colon \Delta^0 \to Ne_M^* \bbG^{\scD[\cG]}$.

Let us justify Definition <ref>.
The $\infty$-functor $\bbG^{\scD[\cG]}_{[\cG]} \colon N\rmB\scD[\cG]^\opp \to \scS$ encodes the smooth right action of $\Diff_{[\cG]}(M)$ on $\bbG^{M}_{[\cG]} \in \scP(N\Cart)$.
Analogously, the $\infty$-functor $\phi^*\bbG^{\scD[\cG]}_{[\cG]} \colon Q \to \scS$ encodes the smooth right action of $\bbGamma$ on $\bbG^{M}_{[\cG]} \in \scP(N\Cart)$ via the morphism $\Phi \colon \bbGamma \to \Diff_{[\cG]}(M)$.
The $\infty$-groupoid of sections of $\bbG^{M}_{[\cG]}$ endowed with smooth homotopy fixed-point data is given by the mapping space $\scFun(Q, \scS) (\sfe_Q, \phi^* \bbG^{\scD[\cG]}_{[\cG]})$, where $\sfe_Q \in \scFun(Q,\scS)$ is the constant functor with value the final object in $\scS$.
Therefore, the space of $(\bbGamma, \Phi)$-equivariant structures on $\cG$ is the full $\infty$-subgroupoid of $\scFun(Q, \scS) (\sfe_Q, \bbG^{M}_{[\cG]})$ on those vertices whose underlying section of $\bbG^{\scD[\cG]}_{[\cG]}$ is the constant section $\cG$ (we could instead have defined $\Equivar_\Phi(\cG)$ in this way).

Consider a commutative triangle of $\infty$-categories

\begin{equation}
\begin{tikzcd}
A \ar[rr, "\phi"] \ar[dr, "q"']
& & B \ar[dl, "\pi"]
\\
& C &
\end{tikzcd}
\end{equation}

where $q$ and $\pi$ are left fibrations.
Let $\sfe_A \in \scFun(A, \scS)$ be the $\infty$-functor with constant value the final object in $\scS$, and analogously for $\sfe_B \in \scFun(B, \scS)$.
For each $F \in \scFun(B, \scS)$, there is a canonical equivalence

\begin{equation}
\phi^*F
\simeq \sfe_A \underset{q^* \pi_! \sfe_B}{\times} q^* \pi_! F\,,
\end{equation}

in $\scFun(A,\scS)$, natural in $F$.

Let $\textint F \to B$ denote the left fibration classified by $F$.
Since $\pi$ is a left fibration, the left Kan extension $\pi_!F \colon C \to \scS$ classifies the composed left fibration $\textint F \to B \to C$ (see <cit.>).
The left fibration classified by $e_B$ is the identity morphism $B \to B$.
We have a commutative diagram of simplicial sets

\begin{equation}
\begin{tikzcd}
\phi^* \textint F \ar[r] \ar[d]
& q^* \textint F \ar[r] \ar[d]
& \textint F \ar[d]
\\
A \ar[r] \ar[dr, equal]
& q^*B \ar[r] \ar[d]
& B \ar[d, "\pi"]
\\
& A \ar[r, "q"']
& C
\end{tikzcd}
\end{equation}

Its middle horizontal composition is $\phi$, and by the pasting law each square is a pullback square.
In particular, the left upper square is homotopy cartesian in the covariant model structure on $\sSet_{/A}$; it presents the equivalence in the claim.

We obtain a pullback square of $\infty$-groupoids

\begin{equation}
\begin{tikzcd}
\scFun(Q, \scS) \big( \sfe_Q, \phi^* \bbG^{\scD[\cG]}_{[\cG]} \big) \ar[r] \ar[d]
& \scFun(Q, \scS) \big( \sfe_Q, q^* (N\pi_M)_! \bbG^{\scD[\cG]}_{[\cG]} \big) \ar[d]
\\
\Delta^0 \simeq \scFun(Q, \scS) \big( e_Q, e_Q \big) \ar[r]
& \scFun(Q, \scS) \big( \sfe_Q, q^* (N\pi_M)_! \sfe_{N\rmB\scD[\cG]^\opp} \big)
\end{tikzcd}
\end{equation}

where the bottom morphism is induced by the action $\Phi$ (which corresponds to a morphism $\sfe_Q \to \phi^* \sfe_{N\rmB\scD[\cG]^\opp}$ in $\scFun(Q, \scS)$) and the unit of the adjunction $(N\pi_M)_! \dashv (N\pi_M)^*$.
Finally, recall that $q_! \sfe_Q$ is the $\infty$-functor presented by the left fibration $q$.
Using this, we see that the Kan extension adjunctions induce canonical equivalences

\begin{align}
\scFun(Q, \scS) \big( \sfe_Q, q^* (N\pi_M)_! \sfe_{N\rmB\scD[\cG]^\opp} \big)
&\simeq \scFun(N\Cart^\opp, \scS) \big( \rmB\bbGamma^\rev, \rmB\Diff_{[\cG]}^\rev(M) \big)\,,
\\
\scFun(Q, \scS) \big( \sfe_Q, q^* (N\pi_M)_! \bbG^{\scD[\cG]}_{[\cG]} \big)
&\simeq \scFun(N\Cart^\opp, \scS) \big( \rmB\bbGamma^\rev, (N\pi_M)_! \bbG^{\scD[\cG]}_{[\cG]} \big)\,.
\end{align}

Combining this with the above pullback square, we obtain a canonical equivalence

\begin{equation}
\scFun(Q, \scS) \big( \sfe_Q, \phi^* \bbG^{\scD[\cG]}_{[\cG]} \big)
\simeq \scP(N\Cart)_{/\rmB\Diff_{[\cG]}^\rev(M)} \big( \rmB \bbGamma^\rev, (N\pi_M)_! \bbG^{\scD[\cG]}_{[\cG]} \big)\,,
\end{equation}

where on the left-hand side we have the $\infty$-groupoid of smooth homotopy fixed points of the $\bbGamma$-action on $\bbG^M_{[\cG]}$.
Then, the further pullback in Definition <ref> restricts the $\infty$-subgroupoid of such homotopy fixed points whose underlying section is $\cG$, i.e. the smooth equivariant structures on $\cG$.

Varying the pair $(\bbGamma, \Phi)$, the above construction canonically assembles into a functor

\begin{equation}
\Equivar(\cG) \colon \big( \Grp \big( \scP(N\Cart) \big)_{/\Diff_{[\cG]}(M)} \big)^\opp \longrightarrow \scS
\end{equation}

which assigns to a $\bbGamma$-action $(\bbGamma, \Phi)$ on $M$ the space of $(\bbGamma, \Phi)$-equivariant structures on $\cG$ as in Definition <ref>.
We can further extend this to a functor on the $\infty$-category $\Grp \big( \scP(N\Cart) \big)_{/\Diff(M)}$ of all smooth $\infty$-group actions on $M$ by setting $\Equivar_\Phi(\cG) = \emptyset$ whenever $\Phi \colon \bbGamma \to \Diff(M)$ does not factor through $\Diff_{[\cG]}(M)$ (in that case, no $(\bbGamma, \Phi)$-equivariant structure can exist on $\cG$).

The $\infty$-functor

\begin{equation}
\Equivar(\cG) \colon \big( \Grp \big( \scP(N\Cart) \big)_{/\Diff(M)} \big)^\opp \longrightarrow \scS
\end{equation}

is representable by the morphism $\scSym(\cG) \to \Diff_{[\cG]}(M) \hookrightarrow \Diff(M)$.
That is, there is a canonical natural equivalence

\begin{equation}
\label{eq:Equivar equivalence}
\Equivar_\Phi(\cG)
\simeq \Grp \big( \scP(N\Cart) \big)_{/\Diff(M)} \big( \bbGamma, \scSym(\cG) \big)
\end{equation}

between $(\bbGamma,\Phi)$-equivariant structures on $\cG$ and lifts of the morphism $\Phi \colon \bbGamma \to \Diff(M)$ of smooth $\infty$-groups along the morphism $\scSym(\cG) \to \Diff_{[\cG]}(M)$.