Hamburger Beiträge zur Mathematik Nr. 858
ZMP–HH/20-14

Principal $\infty$-Bundles and Smooth String Group Models

Severin Bunk

###### Abstract

We provide a general, homotopy-theoretic definition of string group models
within an $\infty$-category of smooth spaces, and we present new smooth models
for the string group. Here, a smooth space is a presheaf of $\infty$-groupoids
on the category of cartesian spaces. The key to our definition and
construction of smooth string group models is a version of the singular
complex functor, which assigns to a smooth space an underlying ordinary space.
We provide new characterisations of principal $\infty$-bundles and group
extensions in $\infty$-topoi, building on work of Nikolaus, Schreiber, and
Stevenson. These insights allow us to transfer the definition of string group
extensions from the $\infty$-category of spaces to the $\infty$-category of
smooth spaces. Finally, we consider smooth higher-categorical group extensions
that arise as obstructions to the existence of equivariant structures on
gerbes. These extensions give rise to new smooth models for the string group,
as recently conjectured in joint work with Müller and Szabo.

###### Contents

  1. 1 Introduction and overview
  2. 2 Smooth spaces and $\infty$-topoi
    1. 2.1 Presheaves on cartesian spaces and the smooth singular complex
    2. 2.2 Background on $\infty$-topoi
  3. 3 Principal $\infty$-bundles and group extensions in $\infty$-topoi
    1. 3.1 Groups and group extensions
    2. 3.2 Group actions in $\infty$-categories
    3. 3.3 Principal $\infty$-bundles
  4. 4 Homotopy-theoretic smooth string group models
    1. 4.1 The definition of smooth string groups
    2. 4.2 Bundle gerbes and their symmetries
    3. 4.3 A smooth string group model
  5. A Actions and category objects

##  1 Introduction and overview

The most direct way to define the string group is via the Whitehead tower of
$\mathsf{O}(n)$,

$\cdots\longrightarrow{\mathrm{String}}(n)\longrightarrow{\mathrm{Spin}}(n)\longrightarrow\mathsf{S}\mathsf{O}(n)\longrightarrow\mathsf{O}(n)\,.$
(1.1)

By this approach, ${\mathrm{String}}(n)$ is defined as a 3-connected
topological space with a continuous map
${\mathrm{String}}(n)\to{\mathrm{Spin}}(n)$ which induces an isomorphism on
all homotopy groups except for in degree three. So far, this defines
${\mathrm{String}}(n)$ only as a space, but in [29] Stolz constructed
${\mathrm{String}}(n)$ as a topological group and the map
${\mathrm{String}}(n)\to{\mathrm{Spin}}(n)$ as a morphism of topological
groups. In fact, he presented a construction that produces, for any compact,
connected, and simply connected Lie group $H$, a morphism
${\mathrm{String}}(H)\to H$ of topological groups whose underlying continuous
map is a three-connected covering. A covering of this type is also called a
_string group extension of $H$_. In these conventions, we write
${\mathrm{String}}(n)\coloneqq{\mathrm{String}}({\mathrm{Spin}}(n))$.

The string group is important in geometry and topology in several ways.
Originally, Killingback [17] and Witten [33] investigated the two-dimensional
supersymmetric $\sigma$-model on background manifolds $M$ and found that this
is well-defined only if the free loop space $LM$ admits a spin structure.
Witten, moreover, computed the index of a hypothetical Dirac operator on $LM$
based on physical arguments, leading to the definition of the Witten genus. By
now, it has been understood that the Witten genus is related to the cohomology
theory of topological modular forms (TMF). The string group enters in this
story, for example by defining orientations in TMF [1, 12], analogously to how
the spin group underlies orientations in real K-theory.

Since the free loop space $LM$ is less tractable than the manifold $M$ itself,
it is an important question whether the condition that $LM$ admit a spin
structure can be recast as a condition on the manifold $M$ itself. This is
indeed the case: _spin_ structures on $LM$ correspond to _string_ structures
on $M$ [26, 27, 32]. Topologically, a string structure on $M$ is a lift of the
classifying map $M\to{\mathrm{B}}\mathsf{O}(n)$ of the tangent bundle $TM\to
M$ to a map $M\to{\mathrm{B}}{\mathrm{String}}(n)$. That is, a string
structure is a reduction of the structure group of $TM$ to
${\mathrm{String}}(n)$. From a geometric perspective, the interest ultimately
is in identifying consequences and constructions that are facilitated by a
string structure on a manifold. Concrete examples include the Höhn-Stolz
conjecture [16, 29] that the Witten genus is trivial for any Riemannian
$4k$-manifold with positive Ricci curvature which admits a string structure,
or the long-standing goal to define a Dirac operator on the loop space $LM$.

In order to study the differential geometric, rather than topological,
implications of string structures, it is paramount to have models for
${\mathrm{String}}(n)$ not just as a topological group, but as a group object
in some geometric category. For instance, given a Riemannian manifold $M$, the
construction of the Dirac operator associated with a spin structure on $M$
depends on the ability to glue the tangent bundle $TM$ from _smooth_
${\mathrm{Spin}}(n)$-valued functions. Technically, one also needs to find
local frames for $TM$ in which the Levi-Civita connection of $M$ is
represented by 1-forms valued in the Lie algebra $\mathfrak{spin}(n)$ rather
than $\mathfrak{so}(n)$; however, since the fibre of the map
${\mathrm{Spin}}(n)\to\mathsf{S}\mathsf{O}(n)$ is discrete, these Lie algebras
happen to be canonically isomorphic (for more background on spin geometry and
Dirac operators, we recommend [18]). Analogously to how spin structures on
$LM$ stem from string structures on $M$, a hypothetical Dirac operator on $LM$
may well stem from a geometric operator on $M$ itself (e.g. via some
transgression procedure), obtained from a further lift of the Levi-Civita
connection to the Lie algebra $\mathfrak{string}(n)$. However, for this to
make sense, one must work with a _smooth_ , rather than topological, model for
${\mathrm{String}}(n)$.

Classical results on cohomology readily imply that it is impossible to
construct ${\mathrm{String}}(H)$ as a finite-dimensional Lie group (for any
compact, connected, simply connected Lie group $H$). Thus, to find geometric
models for ${\mathrm{String}}(H)$, one needs to look beyond the category of
smooth, finite-dimensional manifolds. Indeed, a number of models for
${\mathrm{String}}(H)$ have been found in (higher) categories of smooth spaces
that generalise the notion of a manifold in various ways [6, 15, 25, 31, 22,
13].

In each of these constructions, an extension

$A\longrightarrow{\mathrm{String}}(H)\longrightarrow H$ (1.2)

of a compact, simple, simply connected Lie group $H$ is constructed within the
chosen ambient category of smooth spaces. It is then argued that on the
underlying ordinary spaces (meaning topological spaces or simplicial sets) one
obtains a string group extension in the sense of (1.1). However, so far there
is no general definition of ${\mathrm{String}}(H)$ in a smooth context that
formalises this procedure. Consequently, in geometric models for
${\mathrm{String}}(H)$ the extending group $A$ currently has to be chosen ad
hoc as an explicit delooping of the Lie group $\mathsf{U}(1)$ in a rather
strict sense. This obscures the homotopy-theoretic nature of
${\mathrm{String}}(H)$, since from a homotopical point of view, not $A$ is
fixed, but only its homotopy type.

In [3], studying symmetries of gerbes, we came across extensions of Lie groups
$H$ not by a delooping of the Lie group $\mathsf{U}(1)$, but by the delooping
of the diffeological group $\mathsf{U}(1)^{H}$ of smooth maps from $H$ to
$\mathsf{U}(1)$. However, if $H$ is simply connected, then the smooth group
$\mathsf{U}(1)^{H}$ is homotopy equivalent to $\mathsf{U}(1)$. Therefore,
extensions of $H$ by the delooping ${\mathrm{B}}(\mathsf{U}(1)^{H})$
potentially have the correct homotopy type to produce smooth string group
extensions of $H$. Nevertheless, we could not make this rigorous due to the
lack of a homotopy-theoretic notion of smooth string group extensions that
does not fix the extending group, but only its homotopy type.

Here, we provide such a general definition of smooth string group extensions,
and we prove that the string group models proposed in [3] fit within this
definition. Let ${\mathscr{M}\mathrm{fd}}$ denote the category of manifolds
and smooth maps, and let
${\mathscr{C}\mathrm{art}}\subset{\mathscr{M}\mathrm{fd}}$ be the full
subcategory on those manifolds that are diffeomorphic to $\mathbb{R}^{n}$ for
any $n\in\mathbb{N}_{0}$. As our ambient category of smooth spaces, we choose
the $\infty$-category
${\mathbf{H}}_{\infty}\coloneqq{\mathscr{F}\mathrm{un}}({\mathscr{C}\mathrm{art}}^{\mathrm{op}},{\mathbf{S}})$
of presheaves of spaces on ${\mathscr{C}\mathrm{art}}$. This provides a very
general notion of smooth space: for instance, ${\mathbf{H}}_{\infty}$ contains
the categories of manifolds, diffeological spaces, and Lie groupoids. We write
$\underline{M}$ for the image of a manifold $M$ under the fully faithful
inclusion ${\mathscr{M}\mathrm{fd}}\hookrightarrow{\mathbf{H}}_{\infty}$.

The $\infty$-category ${\mathbf{H}}_{\infty}$ is even an $\infty$-topos, and
there exists an established theory of group objects in $\infty$-topoi [20].
Moreover, there exists a notion of principal $\infty$-bundles and extensions
of group objects in $\infty$-topoi due to [21]. A large part of this paper is
devoted to developing this theory further. In particular, we show that group
actions in $\infty$-topoi automatically form groupoid objects (Theorem 3.37)
and that principal $\infty$-bundles essentially consists of an effective
epimorphism and a principal group action (Theorem 3.58); this is analogous to
the definition of principal bundles of topological spaces as a locally trivial
map and a principal group action. Then, we provide the following
characterisation of extensions of group objects:

###### Theorem 1.3

Let ${\mathbf{H}}$ be an $\infty$-topos. Given a group object ${\widehat{A}}$
in ${\mathbf{H}}$, denote its underlying object in ${\mathbf{H}}$ by $A$. Let
${\widehat{A}}\xrightarrow{\widehat{\iota}}{\widehat{G}}\xrightarrow{\widehat{p}}{\widehat{H}}$
be a sequence of morphisms of group objects in ${\mathbf{H}}$. The following
are equivalent:

  1. (1)

${\widehat{A}}\xrightarrow{\widehat{\iota}}{\widehat{G}}\xrightarrow{\widehat{p}}{\widehat{H}}$
is an extension of group objects in ${\mathbf{H}}$, i.e. the sequence
${\mathrm{B}}A\to{\mathrm{B}}G\to{\mathrm{B}}H$ is a fibre sequence of pointed
connected objects in ${\mathbf{H}}$.

  2. (2)

The sequence
${\widehat{A}}\xrightarrow{\widehat{\iota}}{\widehat{G}}\xrightarrow{\widehat{p}}{\widehat{H}}$
is a fibre sequence of group objects in ${\mathbf{H}}$.

  3. (3)

The sequence $A\xrightarrow{\iota}G\xrightarrow{p}H$ is a fibre sequence in
${\mathbf{H}}$.

  4. (4)

The map $p\colon G\to H$ together with the action of $A$ on $G$ induced by
$\iota$ define a principal $A$-bundle over $H$.

Point (1) in Theorem 1.3 is the definition of extensions of group objects in
$\infty$-topoi from [21]. In order to give a general homotopy-theoretic
definition of string group extensions within ${\mathbf{H}}_{\infty}$, we need
to associate an underlying space to an object in ${\mathbf{H}}_{\infty}$. In
[9] we investigated (a model categorical presentation of) a functor
$\mathrm{S}_{e}\colon{\mathbf{H}}_{\infty}\to{\mathbf{S}}$ from
${\mathbf{H}}_{\infty}$ to the $\infty$-category ${\mathbf{S}}$ of spaces. It
evaluates a smooth space $X\in{\mathbf{H}}_{\infty}$ on the extended affine
simplices $\Delta_{e}^{k}\in{\mathscr{C}\mathrm{art}}$ and then takes the
geometric realisation of the resulting simplicial object in
${\mathbf{S}}$—that is, $\mathrm{S}_{e}$ is a smooth version of the singular
complex functor. Here, we give further interpretation and context to this
functor: the adjunction ${\widetilde{\mathsf{c}}}\dashv\Gamma$, where
$\Gamma\colon{\mathbf{H}}_{\infty}\to{\mathbf{S}}$ is the global-section
functor, fits into a triple adjunction
$\Pi\dashv{\widetilde{\mathsf{c}}}\dashv\Gamma\dashv codisc$, where $codisc$
is fully faithful and where $\Pi$ preserves finite products. That is, the
$\infty$-topos ${\mathbf{H}}_{\infty}$ is cohesive.

###### Theorem 1.4

The functor $\mathrm{S}_{e}\colon{\mathbf{H}}_{\infty}\to{\mathbf{S}}$ is part
of the cohesion of ${\mathbf{H}}_{\infty}$: there is a canonical equivalence

$\Pi\simeq\mathrm{S}_{e}\,.$ (1.5)

This has already been indicated in [2] and proven on the level of model
categories of simplicial presheaves in [9]; here we provide an
$\infty$-categorical proof based on findings from [9]. Let
${\mathrm{L}}\colon{\mathbf{H}}\to{\mathbf{H}}^{\prime}$ be a functor between
$\infty$-topoi which preserves finite products and geometric realisations of
simplicial objects. We show that ${\mathrm{L}}$ maps principal
$\infty$-bundles in ${\mathbf{H}}$ to principal $\infty$-bundles in
${\mathbf{H}}^{\prime}$ and group extensions in ${\mathbf{H}}$ to group
extensions in ${\mathbf{H}}^{\prime}$. (This relies on Theorem 3.37.) In
particular, the functor
$\mathrm{S}_{e}\colon{\mathbf{H}}_{\infty}\to{\mathbf{S}}$ has these
properties. In ${\mathbf{S}}$, a string group extension of a compact,
connected, simply connected Lie group $H$ can be defined as usual: it is an
extension $A\to{\mathrm{String}}(H)\to H$ of group objects in ${\mathbf{S}}$
such that ${\mathrm{String}}(H)$ is 3-connected and such that the morphism
${\mathrm{String}}(H)\to H$ induces an isomorphism on all homotopy groups
except for in degree three. Using that $\mathrm{S}_{e}\underline{M}\simeq M$
for any manifold $M$ [9] and that $\mathrm{S}_{e}$ preserves principal
$\infty$-bundles and group extensions, we are now able to transfer this
definition to ${\mathbf{H}}_{\infty}$:

###### Definition 1.6

Let $H$ be a compact, simple, and simply connected Lie group, and let
$\underline{{\widehat{H}}}$ denote the induced group object in
${\mathbf{H}}_{\infty}$. An extension
${\widehat{A}}\longrightarrow\widehat{{\mathrm{String}}(\underline{H})}\longrightarrow\underline{{\widehat{H}}}$
of group objects in ${\mathbf{H}}_{\infty}$ is called a _smooth string group
extension of $H$_ if its image under $\mathrm{S}_{e}$ is a string group
extension in ${\mathbf{S}}$.

Finally, we show that the string group models conjectured in [3] fit within
Definition 1.6. Let $M$ be a manifold endowed with a bundle gerbe
$\mathcal{G}$ (a categorified hermitean line bundle). In [3], we addressed the
question of when an action of a Lie group $H$ on $M$ lifts to an equivariant
structure on $\mathcal{G}$. We found that the obstruction to such a lift is an
extension

${\mathrm{HLB}}^{M}\overset{i}{\longrightarrow}{\mathrm{Sym}}(\mathcal{G})\overset{p}{\longrightarrow}H$
(1.7)

of $H$ by the smooth 2-group ${\mathrm{HLB}}^{M}$ of hermitean line bundles on
$M$. Each of the above objects can be interpreted as a group object in
${\mathbf{H}}_{\infty}$ via the nerve functor, and so (1.7) provides an
extension of $H$ as a group object in ${\mathbf{H}}_{\infty}$. The case
relevant for string group extensions is $M=H$, where $H$ is a compact,
connected and simply connected Lie group, acting on itself via left
multiplication. Since $H$ is 2-connected, there is an objectwise equivalence
${\mathrm{HLB}}^{H}\simeq{\mathrm{B}}(\mathsf{U}(1)^{H})$, and since $H$ is
1-connected, there is a smooth homotopy equivalence
$\mathsf{U}(1)^{H}\simeq\mathsf{U}(1)$. Therefore, the extending group in
(1.7) has the correct homotopy type for a string group extension. We prove:

###### Theorem 1.8

Let $H$ be a compact, simple, simply connected Lie group, and let $N$ be the
nerve functor. We consider the left-action of $H$ on itself via left
multiplication. Let $\mathcal{G}\in{\mathscr{G}\mathrm{rb}}(H)$ be a gerbe on
$H$ whose class in $\mathrm{H}^{3}(H;\mathbb{Z})\cong\mathbb{Z}$ is a
generator. The sequence

${\widehat{N{\mathrm{HLB}}^{H}}}$${\widehat{N{\mathrm{Sym}}(\mathcal{G})}}$${\underline{{\widehat{H}}}}$$\scriptstyle{\widehat{Ni}}$$\scriptstyle{\widehat{Np}}$
(1.9)

is a smooth string group extension of $H$.

This string group model is somewhat similar to the model in [13], which is
obtained by studying symmetries of gerbes with connection. However, the
presence of connections forces the extending group to be the delooping
${\mathrm{B}}\mathsf{U}(1)$ in this case. It is interesting that the
connection does not change the homotopy type of the extension. A similar
observation has been made in [3], where a second extension of $H$, equivalent
to (1.7), was constructed with a connection on the gerbe $\mathcal{G}$ acting
as crucial auxiliary data. Since that second extension is equivalent to the
one in (1.7) [3], it gives rise to a second smooth string group extension of
$H$. Finally, we expect that most (or possibly all) of the aforementioned
smooth string group models fit within Definition 1.6.

#### Organisation.

In Section 2 we investigate the functor
$\mathrm{S}_{e}\colon{\mathbf{H}}_{\infty}\to{\mathbf{S}}$. Further, we recall
some basic notions and facts about $\infty$-topoi and prove Theorem 1.4.

Section 3 is devoted to the theory of group objects, group extensions, and
principal $\infty$-bundles in $\infty$-topoi. We recall the definitions of
these notions from [21] and provide new characterisations of principal
$\infty$-bundles and group extensions. In particular, we prove Theorem 1.3.

In Section 4, we use the results obtained thus far to transfer the definition
of string group extensions in ${\mathbf{S}}$ to the $\infty$-topos
${\mathbf{H}}_{\infty}$. After recalling from [3] the smooth 2-group
extensions which obstruct the existence of equivariant structures on gerbes,
we show that these extensions give rise to new smooth models for string group,
thus proving Theorem 1.8.

Finally, in Appendix A we prove Theorem 3.37: we show that group actions in an
$\infty$-topos give rise to groupoid objects.

#### Notation.

We usually make no notational distinction between ordinary categories and
$\infty$-categories; the nerve functor will be used implicitly where
necessary.

We write ${\mathbbe{\Delta}}$ for the simplex category, and
${\mathscr{S}\mathrm{et}_{\Delta}}$ for the category of simplicial sets. In a
simplicial category $\mathscr{C}$, we denote the simplicially enriched hom-
functor by
$\underline{\mathscr{C}}(-,-)\colon\mathscr{C}^{\mathrm{op}}\times\mathscr{C}\to{\mathscr{S}\mathrm{et}_{\Delta}}$.

We write
$|{-}|={\mathrm{colim}}^{\mathscr{C}}_{{\mathbbe{\Delta}}^{\mathrm{op}}}$ for
the colimit of simplicial objects in an $\infty$-category $\mathscr{C}$.
Moreover, we also refer to $|X|$ (if it exists) as the geometric realisation
of a simplicial object $X$ in $\mathscr{C}$.

Usually, we denote $\infty$-categories by letters
$\mathscr{C},\mathscr{D},\dots$, but for $\infty$-topoi we use bold-face
letters ${\mathbf{H}}$. In particular, the $\infty$-topos of spaces is denoted
by ${\mathbf{S}}$. We write
$\underline{\mathscr{D}}(-,-)\colon\mathscr{D}^{\mathrm{op}}\times\mathscr{D}\to{\mathbf{S}}$
for the mapping spaces in an $\infty$-category $\mathscr{D}$.

We model $\infty$-categories by quasi-categories. Given an $\infty$-category
$\mathscr{C}$ and a simplicial set $K\in{\mathscr{S}\mathrm{et}_{\Delta}}$, we
write
${\mathscr{F}\mathrm{un}}(K,\mathscr{C})=\underline{{\mathscr{S}\mathrm{et}_{\Delta}}}(K,\mathscr{C})=\mathscr{C}^{K}$
for the $\infty$-category of functors from $K$ to $\mathscr{C}$.

We let ${\mathbbe{\Delta}}_{+}$ denote the augmented simplex category, i.e.
the category ${\mathbbe{\Delta}}$ with an initial object adjoined. We usually
do not distinguish notationally between augmented simplicial objects
$X\in{\mathscr{F}\mathrm{un}}({\mathbbe{\Delta}}_{+}^{\mathrm{op}},\mathscr{C})$
in an $\infty$-category $\mathscr{C}$ and their underlying simplicial objects.
If we wish to make this distinction explicit for clarity, we will denote the
latter by the restriction $X_{|{\mathbbe{\Delta}}^{\mathrm{op}}}$.

If ${\mathscr{M}}$ is a simplicial model category, then
${\mathscr{M}}^{\circ}$ is the full simplicial subcategory on the cofibrant-
fibrant objects of ${\mathscr{M}}$. Recall from [20] that the coherent nerve
$N({\mathscr{M}}^{\circ})$ is an $\infty$-category.

If $\mathscr{C}$ is a (small) $\infty$-category, we write
$\mathscr{P}(\mathscr{C})={\mathscr{F}\mathrm{un}}(\mathscr{C}^{\mathrm{op}},{\mathbf{S}})$
for the $\infty$-category of presheaves of spaces on $\mathscr{C}$.

#### Acknowledgements.

The author would like to thank Lukas Müller, Birgit Richter, Christoph
Schweigert, Walker Stern, and Konrad Waldorf for helpful discussions. The
author acknowledges partial support by the Deutsche Forschungsgemeinschaft
(DFG, German Research Foundation) under Germany’s Excellence Strategy—EXC 2121
“Quantum Universe”—390833306.

##  2 Smooth spaces and $\infty$-topoi

In this section we recall and develop some background on the
$\infty$-categories most relevant in this paper. Most importantly, we consider
a presheaf $\infty$-category ${\mathbf{H}}_{\infty}$, whose objects can be
interpreted as a general notion of smooth spaces. We study an
$\infty$-categorical version
$\mathrm{S}_{e}\colon{\mathbf{H}}_{\infty}\to{\mathbf{S}}$ of a Quillen
functor considered in [9], which provides a version of the singular complex
functor for smooth spaces. Subsequently, we briefly recall the definition of
an $\infty$-topos and of cohesion of $\infty$-topoi, and we show that
$\mathrm{S}_{e}$ is part of the cohesion of ${\mathbf{H}}_{\infty}$.

###  2.1 Presheaves on cartesian spaces and the smooth singular complex

We let ${\mathscr{C}\mathrm{art}}$ denote the (small) category whose objects
are submanifolds of $\mathbb{R}^{\infty}$ that are diffeomorphic to
$\mathbb{R}^{n}$ for any $n\in\mathbb{N}_{0}$, and whose morphisms are the
smooth maps between these manifolds. We let

${\mathbf{H}}_{\infty}\coloneqq\mathscr{P}({\mathscr{C}\mathrm{art}})={\mathscr{F}\mathrm{un}}({\mathscr{C}\mathrm{art}}^{\mathrm{op}},{\mathbf{S}})$
(2.1)

denote the $\infty$-category of presheaves of spaces on
${\mathscr{C}\mathrm{art}}$. The $\infty$-category ${\mathbf{H}}_{\infty}$ is
presented by several model categories of simplicial presheaves on
${\mathscr{C}\mathrm{art}}$—for example, there is a canonical equivalence [20]

${\mathbf{H}}_{\infty}\simeq
N\big{(}(\mathscr{H}_{\infty}^{i})^{\circ}\big{)}\,,$ (2.2)

where $\mathscr{H}_{\infty}^{i}$ is the category of simplicial presheaves on
${\mathscr{C}\mathrm{art}}$, endowed with the injective model structure.

Let $I\coloneqq\\{c\times\mathbb{R}\to c\,|\,c\in{\mathscr{C}\mathrm{art}}\\}$
denote the set of morphisms in ${\mathscr{C}\mathrm{art}}$ of the form
$1_{c}\times c_{\mathbb{R}}$, where $c_{\mathbb{R}}\colon\mathbb{R}\to*$ is
the map that collapses the real line to the point. We can localise both
$\mathscr{H}_{\infty}^{i}$ and ${\mathbf{H}}_{\infty}$ at this set of
morphisms (or rather at its image under the Yoneda embedding), and there is
still a canonical equivalence between the localisations [20],

$N\big{(}(L_{I}\mathscr{H}_{\infty}^{i})^{\circ}\big{)}\simeq
L_{I}{\mathbf{H}}_{\infty}\,.$ (2.3)

The simplicial model categories $\mathscr{H}_{\infty}^{i}$ and
$L_{I}\mathscr{H}_{\infty}^{i}$ were the subject of [9]. On the level of their
underlying $\infty$-categories, one of the main results of that paper can be
phrased as follows. For $k\in\mathbb{N}_{0}$, we let
$\Delta_{e}^{k}\coloneqq\\{t\in\mathbb{R}^{k+1}\,|\,\sum_{i=0}^{k}t^{i}=1\\}$
denote the extended (affine) $k$-simplex. This is a $k$-dimensional affine
subspace of $\mathbb{R}^{k+1}$, and hence forms a cartesian space. The face
and degeneracy maps of the standard topological simplices $|\Delta^{k}|$
extend to the extended affine simplices, turning them into a functor

$\Delta_{e}\colon{\mathbbe{\Delta}}\to{\mathscr{C}\mathrm{art}}\,,\quad[k]\mapsto\Delta_{e}^{k}\,.$
(2.4)

We let $\mathrm{S}_{e}\colon{\mathbf{H}}_{\infty}\to{\mathbf{S}}$ denote the
composition of functors

${\mathrm{S}_{e}\colon{\mathbf{H}}_{\infty}}$${{\mathscr{F}\mathrm{un}}({\mathbbe{\Delta}}^{\mathrm{op}},{\mathbf{S}})}$${{\mathbf{S}}\,.}$$\scriptstyle{\Delta_{e}^{*}}$$\scriptstyle{{\mathrm{colim}}}$
(2.5)

We refer to this functor as the _smooth singular complex functor_ ; viewing
the $\infty$-category ${\mathbf{H}}_{\infty}$ as an $\infty$-category of
smooth spaces, $\mathrm{S}_{e}$ thus assigns an underlying ordinary space to a
smooth space.

###### Theorem 2.6

_[ 9]_ There exist adjunctions of $\infty$-categories

${{\mathbf{H}}_{\infty}}$${L_{I}{\mathbf{H}}_{\infty}}$${{\mathbf{S}}}$${{\mathbf{S}}}$$\scriptstyle{\operatorname{Loc}}$$\scriptstyle{\perp}$$\scriptstyle{\mathrm{S}_{e}}$$\scriptstyle{\dashv}$$\scriptstyle{\iota}$$\scriptstyle{\mathrm{S}_{e}^{I}}$$\scriptstyle{{\mathrm{R}}_{e}}$$\scriptstyle{{\mathrm{L}}_{e}^{I}}$$\scriptstyle{\dashv}$$\scriptstyle{{\mathrm{R}}_{e}^{I}}$$\scriptstyle{\dashv}$
(2.7)

where $\mathrm{S}_{e}^{I}$ is the restriction of $\mathrm{S}_{e}$ to
$L_{I}{\mathbf{H}}_{\infty}\subset{\mathbf{H}}_{\infty}$. Furthermore, the
following statements hold true:

  1. (1)

The functor $\mathrm{S}_{e}\colon{\mathbf{H}}_{\infty}\to{\mathbf{S}}$
preserves and reflects $I$-local equivalences.

  2. (2)

The morphism $\iota$ is fully faithful, i.e. $\operatorname{Loc}$ is a
reflective localisation.

  3. (3)

The three right-hand vertical functors are equivalences of
$\infty$-categories.

  4. (4)

The diagram obtained by omitting the morphism ${\mathrm{L}}^{I}_{e}$ is
(weakly) commutative.

###### Proof.

The first claim follows readily from Proposition 3.6, Corollary 3.12, and
Corollary 3.37 of [9]. (Note that model categorical presentations of
${\mathbf{H}}_{\infty}$, $L_{I}{\mathbf{H}}_{\infty}$, and ${\mathbf{S}}$ are
used in [9], and the functors in the statement are presented by Quillen
functors.)

Further, claim (1) follows readily from [9, Cor. 3.15]. Claim (2) follows from
general properties of $\infty$-categories underlying simplicial model
categories and their Bousfield localisations [20]. Claim (3) is the version on
the underlying $\infty$-categories of Theorems 3.14 and 3.40 of [9]. Claim (4)
holds true because the diagram of the right-adjoints clearly commutes ($\iota$
is an inclusion, and ${\mathrm{R}}_{e}$ simply factors through
$L_{I}{\mathbf{H}}_{\infty}\subset{\mathbf{H}}_{\infty}$ [9]). ∎

###### Remark 2.8

There is a fully faithful embedding
${\mathscr{M}\mathrm{fd}}\hookrightarrow{\mathbf{H}}_{\infty}$ from the
category of manifolds into ${\mathbf{H}}_{\infty}$: it sends a manifold $M$ to
the presheaf $\underline{M}$ of discrete spaces that maps a cartesian space
$c$ to the set ${\mathscr{M}\mathrm{fd}}(c,M)$ of smooth maps from $c$ to $M$.
By [9, Thm. 5.1] there is a canonical equivalence of spaces
$M\simeq\mathrm{S}_{e}\underline{M}$ for any $M\in{\mathscr{M}\mathrm{fd}}$,
which is natural in $M$. $\triangleleft$

###### Proposition 2.9

The localisation functor $\operatorname{Loc}\colon{\mathbf{H}}_{\infty}\to
L_{I}{\mathbf{H}}_{\infty}$ preserves finite products. The class $W_{I}$ of
$I$-local equivalences in ${\mathbf{H}}_{\infty}$ is closed under finite
products.

###### Proof.

By [9, Prop. 2.13], the localisation $L_{I}{\mathbf{H}}_{\infty}$ agrees with
the localisation $L_{W}{\mathbf{H}}_{\infty}$ of ${\mathbf{H}}_{\infty}$ at
all collapse morphisms $c\to*$, for $c\in{\mathscr{C}\mathrm{art}}$. The class
$W$ is stable under finite products in ${\mathbf{H}}_{\infty}$, since
${\mathscr{C}\mathrm{art}}$ has finite products. Therefore, the first claim
follows from [11, Cor. 7.1.16]. The second claim then follows since a morphism
in ${\mathbf{H}}_{\infty}$ is in $W_{I}$ precisely if its image under
$\operatorname{Loc}$ is an equivalence [20, Prop. 5.5.4.15]. ∎

###### Proposition 2.10

For $X,Y\in{\mathbf{H}}_{\infty}$, let $Y^{X}\in{\mathbf{H}}_{\infty}$ denote
their internal hom object in ${\mathbf{H}}_{\infty}$. The localisation functor
$\operatorname{Loc}\colon{\mathbf{H}}_{\infty}\to L_{I}{\mathbf{H}}_{\infty}$
is given (up to equivalence) by

$\operatorname{Loc}\simeq\underset{{\mathbbe{\Delta}}^{\mathrm{op}}}{{\mathrm{colim}}}^{{\mathbf{H}}_{\infty}}\big{(}(-)^{\Delta_{e}}\big{)}\,.$
(2.11)

###### Proof.

By Theorem 2.6(4), there is a canonical equivalence
$S_{e}^{I}\circ\operatorname{Loc}\simeq\mathrm{S}_{e}$. Combining this with
Theorem 2.6(3), we obtain canonical equivalences

$\operatorname{Loc}\simeq{\mathrm{L}}_{e}^{I}\circ\mathrm{S}_{e}^{I}\circ\operatorname{Loc}\simeq{\mathrm{L}}_{e}^{I}\circ\mathrm{S}_{e}\,.$
(2.12)

Consider the adjunction
$\tilde{\mathsf{c}}:{\mathbf{S}}\rightleftarrows{\mathbf{H}}_{\infty}:\operatorname{ev}_{*}$,
where $\tilde{\mathsf{c}}$ assigns to a space $K$ the constant presheaf with
value $K$, and where $\operatorname{ev}_{*}$ evaluates a presheaf on the final
object $*\in{\mathscr{C}\mathrm{art}}$. These functors induce an equivalence
$\tilde{\mathsf{c}}:{\mathbf{S}}\rightleftarrows
L_{I}{\mathbf{H}}_{\infty}:\operatorname{ev}_{*}$ [9, Thm. 2.17], and there is
a canonical equivalence $\operatorname{ev}_{*}\simeq\mathrm{S}_{e}^{I}$ of
functors $L_{I}{\mathbf{H}}_{\infty}\to{\mathbf{S}}$ by [9, Prop. 2.7, Cor.
3.15]. By adjointness, we also obtain a canonical equivalence
$\tilde{\mathsf{c}}\simeq{\mathrm{L}}_{e}^{I}$. Consequently, there is a
canonical equivalence

$\operatorname{Loc}\simeq\tilde{\mathsf{c}}\circ\mathrm{S}_{e}\,.$ (2.13)

We observe that there exists a canonical equivalence

$\mathrm{S}_{e}=\underset{{\mathbbe{\Delta}}^{\mathrm{op}}}{{\mathrm{colim}}}^{\mathbf{S}}\big{(}\Delta_{e}^{*}(-)\big{)}\simeq\operatorname{ev}_{*}\circ\,\underset{{\mathbbe{\Delta}}^{\mathrm{op}}}{{\mathrm{colim}}}^{{\mathbf{H}}_{\infty}}\big{(}(-)^{\Delta_{e}}\big{)}\,.$
(2.14)

By [9, Prop. 6.2], we have that
${\mathrm{colim}}_{{\mathbbe{\Delta}}^{\mathrm{op}}}^{{\mathbf{H}}_{\infty}}((-)^{\Delta_{e}})$
is a functor ${\mathbf{H}}_{\infty}\to L_{I}{\mathbf{H}}_{\infty}$; that is,
it takes values in $I$-local objects. It follows that there are canonical
equivalences

$\operatorname{Loc}\simeq\tilde{\mathsf{c}}\circ\mathrm{S}_{e}\simeq\tilde{\mathsf{c}}\circ\operatorname{ev}_{*}\circ\underset{{\mathbbe{\Delta}}^{\mathrm{op}}}{{\mathrm{colim}}}^{{\mathbf{H}}_{\infty}}\big{(}(-)^{\Delta_{e}}\big{)}\simeq\underset{{\mathbbe{\Delta}}^{\mathrm{op}}}{{\mathrm{colim}}}^{{\mathbf{H}}_{\infty}}\big{(}(-)^{\Delta_{e}}\big{)}\,.$
(2.15)

This completes the proof. ∎

###  2.2 Background on $\infty$-topoi

In this section, we briefly recall some background on $\infty$-topoi. Most of
the material in this section can be found in [20, 24, 21]. For
$n\in\mathbb{N}_{0}$ and a subset $S\subset[n]$, let
$\Delta^{S}\subset\Delta^{n}$ be the full $\infty$-subcategory on the vertices
that lie in $S$. There is a canonical isomorphism
$\Delta^{S}\cong\Delta^{|S|}$ as simplicial sets, where $|S|$ is the
cardinality of $S$. The simplicial set $\Delta^{S}$ can equivalently be seen
as the image of an inclusion $\Delta^{|S|}\hookrightarrow\Delta^{n}$ that
sends the $i$-th vertex of $\Delta^{|S|}$ to the vertex of $\Delta^{n}$ which
corresponds to the $i$-th element of $S$ (with the order induced from the
inclusion $S\subset[n]$). Given a simplicial object in an $\infty$-category
$\mathscr{C}$, i.e.
$\widehat{X}\in{\mathscr{F}\mathrm{un}}({\mathbbe{\Delta}}^{\mathrm{op}},\mathscr{C})$,
we set $\widehat{X}(S)\coloneqq\widehat{X}(\Delta^{|S|})$. This comes with a
canonical morphism $\widehat{X}_{n}\to\widehat{X}(S)$, induced by the
inclusion $S\subset[n]$.

###### Definition 2.16

Let $\mathscr{C}$ be an $\infty$-category. A _groupoid object_ in
$\mathscr{C}$ is a simplicial object
$\widehat{X}\in{\mathscr{F}\mathrm{un}}({\mathbbe{\Delta}}^{\mathrm{op}},\mathscr{C})$
such that, for every $n\in\mathbb{N}_{0}$ and every partition $[n]=S\cup
S^{\prime}$ (as finite sets) with $S\cap S^{\prime}\cong\\{*\\}$ consisting of
a single element, the diagram

${\widehat{X}_{n}}$${\widehat{X}(S^{\prime})}$${\widehat{X}(S)}$${\widehat{X}_{0}}$
(2.17)

is a pullback diagram in $\mathscr{C}$.

We denote the full subcategory of
${\mathscr{F}\mathrm{un}}({\mathbbe{\Delta}}^{\mathrm{op}},\mathscr{C})$ on
the groupoid objects by

${\mathscr{G}\mathrm{pd}}(\mathscr{C})\subset{\mathscr{F}\mathrm{un}}({\mathbbe{\Delta}}^{\mathrm{op}},\mathscr{C})\,.$
(2.18)

Let ${\mathbbe{\Delta}}_{+}$ denote the simplex category with an initial
object $[-1]$ adjoined. For $n\in\mathbb{N}_{0}$, let
${\mathbbe{\Delta}}_{+,\leq n}\subset{\mathbbe{\Delta}}_{+}$ be the full
subcategory on the objects $[-1],\ldots,[n]$. In particular,
${\mathbbe{\Delta}}_{+,\leq 0}^{\mathrm{op}}$ is the category with two objects
and one non-trivial morphism $[0]\to[-1]$. Therefore, any morphism $p\colon
P\to X$ in an $\infty$-category $\mathscr{C}$ defines an object
$\\{p\\}\in{\mathscr{F}\mathrm{un}}({\mathbbe{\Delta}}_{+,\leq
0}^{\mathrm{op}},\mathscr{C})$.

###### Definition 2.19

Given a morphism $p\colon P\to X$ in an $\infty$-category $\mathscr{C}$, its
_Čech nerve ${\check{C}}p$_ (if it exists) is the augmented simplicial object
obtained as the right Kan extension

${{\mathbbe{\Delta}}_{+,\leq
0}^{\mathrm{op}}}$${\mathscr{C}}$${{\mathbbe{\Delta}}_{+}^{\mathrm{op}}}$$\scriptstyle{\\{p\\}}$$\scriptstyle{\imath}$$\scriptstyle{{\check{C}}p}$
(2.20)

That is, ${\check{C}}p=\mathrm{Ran}_{\imath}\\{p\\}$, where $\imath$ is the
inclusion ${\mathbbe{\Delta}}_{+,\leq
0}^{\mathrm{op}}\hookrightarrow{\mathbbe{\Delta}}_{+}^{\mathrm{op}}$.

For later use, we record:

###### Proposition 2.21

_[ 20, Prop. 6.1.2.11]_ Let $\mathscr{C}$ be an $\infty$-category, and let
$\widehat{X}\colon{\mathbbe{\Delta}}_{+}^{\mathrm{op}}\to\mathscr{C}$ be an
augmented simplicial object. The following are equivalent:

  1. (1)

$\widehat{X}$ is a right Kan extension of
$\widehat{X}_{|{\mathbbe{\Delta}}_{+,\leq 0}^{\mathrm{op}}}$.

  2. (2)

The underlying simplicial object
$\widehat{X}_{|{\mathbbe{\Delta}}^{\mathrm{op}}}$ is a groupoid object in
$\mathscr{C}$ and the diagram

$\widehat{X}_{|{\mathbbe{\Delta}}_{+,\leq
1}^{\mathrm{op}}}=\quad\leavevmode\hbox to61.47pt{\vbox
to50.79pt{\pgfpicture\makeatletter\hbox{\hskip
31.56558pt\lower-25.44595pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{}{}{}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{\offinterlineskip{}{}{{{}}{{}}{{}}{{}}}{{{}}}{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-29.89996pt}{-25.34612pt}\pgfsys@invoke{
}\hbox{\vbox{\halign{\pgf@matrix@init@row\pgf@matrix@step@column{\pgf@matrix@startcell#\pgf@matrix@endcell}&#\pgf@matrix@padding&&\pgf@matrix@step@column{\pgf@matrix@startcell#\pgf@matrix@endcell}&#\pgf@matrix@padding\cr\hfil\hskip
8.48332pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-4.17778pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{${\widehat{X}_{1}}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}}}&\hskip
8.48332pt\hfil&\hfil\hskip 32.48329pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-4.17778pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{${\widehat{X}_{0}}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}&\hskip
8.48332pt\hfil\cr\vskip 18.00005pt\cr\hfil\hskip
8.48332pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-4.17778pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{${\widehat{X}_{0}}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}&\hskip
8.48332pt\hfil&\hfil\hskip 33.41663pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-5.11111pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{${\widehat{X}_{-1}}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}&\hskip
9.41666pt\hfil\cr}}}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}}{{{{}}}{{}}{{}}{{}}{{}}}} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} {}{ {}{}{}}{}{ {}{}{}} {{{{{}}{
{}{}}{}{}{{}{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}{}{}{{{}{}}}{}{{}}{}{}{}{{{}{}}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.39998pt}\pgfsys@invoke{
}{}{}{}{}{{}}{}{}{{}}\pgfsys@moveto{-12.73332pt}{11.50003pt}\pgfsys@lineto{11.40002pt}{11.50003pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{11.6pt}{11.50003pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-3.28838pt}{15.14168pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{$\scriptstyle{d_{0}}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{ {}{}{}}{}{ {}{}{}}
{{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}{}{}{{{}{}}}{}{{}}{}{}{}{{{}{}}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.39998pt}\pgfsys@invoke{
}{}{}{}{}{{}}{}{}{{}}\pgfsys@moveto{-21.41664pt}{3.33588pt}\pgfsys@lineto{-21.41664pt}{-13.86421pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{0.0}{-1.0}{1.0}{0.0}{-21.41664pt}{-14.0642pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{{}{}}}{{}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-29.41281pt}{-7.25024pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{$\scriptstyle{d_{1}}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{ {}{}{}}{}{ {}{}{}}
{{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.39998pt}\pgfsys@invoke{
}{}{}{}{}{{}}{}{}{{}}\pgfsys@moveto{20.4833pt}{3.33588pt}\pgfsys@lineto{20.4833pt}{-13.86421pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{0.0}{-1.0}{1.0}{0.0}{20.4833pt}{-14.0642pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{
{}{}{}}{}{ {}{}{}} {{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.39998pt}\pgfsys@invoke{
}{}{}{}{}{{}}{}{}{{}}\pgfsys@moveto{-12.73332pt}{-22.84612pt}\pgfsys@lineto{10.46669pt}{-22.84612pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{10.66667pt}{-22.84612pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}$ (2.22)

is a pullback square in $\mathscr{C}$.

In this situation, it follows that, for every $n\geq 1$, the spine
decomposition $[n]=[1]\sqcup_{[0]}\cdots\sqcup_{[0]}[1]$ induces a canonical
equivalence

$({\check{C}}p)_{n}\simeq\underbrace{P\times_{X}\cdots\times_{X}P}_{n+1\text{
factors}}\,.$ (2.23)

###### Definition 2.24

Let $\mathscr{C}$ be an $\infty$-category, and let $p\colon P\to X$ be a
morphism in $\mathscr{C}$. Then, $p$ is an _effective epimorphism_ if
${\check{C}}p\in{\mathscr{F}\mathrm{un}}({\mathbbe{\Delta}}_{+}^{\mathrm{op}},\mathscr{C})\cong{\mathscr{F}\mathrm{un}}(({\mathbbe{\Delta}}^{\mathrm{op}})^{\triangleright},\mathscr{C})$
is a colimiting cocone in $\mathscr{C}$. That is, the morphism $p\colon P\to
X$ is an effective epimorphism precisely if the induced morphism
$|{\check{C}}p|\to X$ is an equivalence.

Let $\widehat{X}\colon{\mathbbe{\Delta}}_{+}^{\mathrm{op}}\to\mathscr{C}$ be
an augmented simplicial object in an $\infty$-category $\mathscr{C}$. We
denote the morphism $\widehat{X}_{0}\to\widehat{X}_{-1}$ by $p$. Suppose that
its Čech nerve ${\check{C}}p$ exists. Observe that
$\\{p\\}=\imath^{*}\widehat{X}$ as objects in
${\mathscr{F}\mathrm{un}}(\Delta^{1},\mathscr{C})\cong{\mathscr{F}\mathrm{un}}({\mathbbe{\Delta}}_{+,\leq
0}^{\mathrm{op}},\mathscr{C})$. By the adjointness property of the right Kan
extension, there is a canonical equivalence

$\underline{{\mathscr{F}\mathrm{un}}}({\mathbbe{\Delta}}_{+,\leq
0}^{\mathrm{op}},\mathscr{C})(\imath^{*}\widehat{X},\\{p\\})\simeq\underline{{\mathscr{F}\mathrm{un}}}({\mathbbe{\Delta}}_{+}^{\mathrm{op}},\mathscr{C})(\widehat{X},{\check{C}}p)\,.$
(2.25)

The identity $\imath^{*}\widehat{X}=\\{p\\}$ thus induces a canonical morphism

$\eta\colon\widehat{X}\longrightarrow{\check{C}}p\,.$ (2.26)

We define $\infty$-topoi in terms of the Giraud-Lurie-Rezk axioms [20, Def.
6.1.0.4, Thm. 6.1.0.6]:

###### Definition 2.27

An _$\infty$ -topos_ is an $\infty$-category ${\mathbf{H}}$ satisfying the
following axioms:

  1. (1)

${\mathbf{H}}$ is presentable. In particular, ${\mathbf{H}}$ has all limits
and colimits. We denote its initial object by $\emptyset\in{\mathbf{H}}$ and
its final object by $*\in{\mathbf{H}}$.

  2. (2)

Colimits in ${\mathbf{H}}$ are universal: for any diagram $D\colon
K\to{\mathbf{H}}$, any cocone $\overline{D}\colon
K^{\triangleright}\to{\mathbf{H}}$ under $D$ with apex $Y\in{\mathbf{H}}$, and
any morphism $f\colon X\to Y$ in ${\mathbf{H}}$, the induced morphism

$\underset{K}{{\mathrm{colim}}}^{\mathbf{H}}(D\underset{\mathsf{c}Y}{\times}\mathsf{c}X)\longrightarrow\big{(}\underset{K}{{\mathrm{colim}}}^{\mathbf{H}}D\big{)}\underset{Y}{\times}X$
(2.28)

is an equivalence (on the left-hand side, $\mathsf{c}X,\mathsf{c}Y\colon
K\to{\mathbf{H}}$ are the constant diagrams with values $X$ and $Y$,
respectively, and the pullback is formed in
${\mathscr{F}\mathrm{un}}(K,{\mathbf{H}})$).

  3. (3)

Coproducts in ${\mathbf{H}}$ are disjoint: for every pair of objects
$X,Y\in{\mathbf{H}}$, the pushout diagram

${\emptyset}$${X}$${Y}$${X\sqcup Y}$ (2.29)

is also a pullback diagram.

  4. (4)

Groupoids in ${\mathbf{H}}$ are effective: given any groupoid object
$\widehat{X}\in{\mathscr{G}\mathrm{pd}}({\mathbf{H}})$, let
$p\colon\widehat{X}_{0}\to|\widehat{X}|$ denote the canonical morphism which
is part of the colimiting cocone. Then, the comparison morphism
$\eta\colon\widehat{X}\to{\check{C}}p$ constructed in (2.26) is an equivalence
of simplicial objects in ${\mathbf{H}}$. In particular, $p$ is an effective
epimorphism.

###### Example 2.30

We list some examples of $\infty$-topoi; we will mostly be using the first two
cases.

  1. (1)

The $\infty$-category of spaces ${\mathbf{S}}$ is an $\infty$-topos.

  2. (2)

Any $\infty$-category $\mathscr{P}(\mathscr{C})$ of presheaves of spaces on a
(small) $\infty$-category $\mathscr{C}$ is an $\infty$-topos.

  3. (3)

Any accessible, left-exact, reflective localisation of an $\infty$-category
$\mathscr{P}(\mathscr{C})$ of presheaves on a small $\infty$-category
$\mathscr{C}$ is an $\infty$-topos; in fact, every $\infty$-topos is
equivalent to an $\infty$-topos of this form [20, Thm. 6.1.0.6, Prop.
6.1.5.3]. $\triangleleft$

An important notion of morphism between $\infty$-topoi is that of a geometric
morphism, which is more adapted to the additional structure on $\infty$-topoi
than a mere functor of $\infty$-categories:

###### Definition 2.31

Let ${\mathbf{X}},{\mathbf{Y}}$ be $\infty$-topoi. A _geometric morphism_ of
$\infty$-topoi from ${\mathbf{X}}$ to ${\mathbf{Y}}$ is a functor
${\mathrm{F}}_{*}\colon{\mathbf{X}}\to{\mathbf{Y}}$ admitting a left exact
left adjoint ${\mathrm{F}}^{*}\colon{\mathbf{Y}}\to{\mathbf{X}}$.

One can show that the $\infty$-category ${\mathbf{S}}$ of spaces is final in
the $\infty$-category of $\infty$-topoi and geometric morphisms [20, Prop.
6.3.4.1]. That is, for every $\infty$-topos ${\mathbf{H}}$ there exists an
essentially unique geometric morphism ${\mathbf{H}}\to{\mathbf{S}}$. We will
denote the corresponding adjunction by
${\widetilde{\mathsf{c}}}:{\mathbf{S}}\rightleftarrows{\mathbf{H}}:\Gamma$ and
refer to $\Gamma$ as the _global-section functor_.

###### Example 2.32

Consider a Grothendieck $\infty$-site, i.e. a small $\infty$-category
$\mathscr{C}$ with a Grothendieck (pre)topology. Suppose $\mathscr{C}$
additionally has a final object. If ${\mathbf{H}}$ is the $\infty$-category of
sheaves of spaces on $\mathscr{C}$, then the global section functor $\Gamma$
of ${\mathbf{H}}$ agrees with the evaluation of sheaves at the final object of
$\mathscr{C}$. In particular, this applies to ${\mathbf{H}}_{\infty}$, the
$\infty$-topos of presheaves of spaces on ${\mathscr{C}\mathrm{art}}$ from
Section 2.1. $\triangleleft$

###### Definition 2.33

An $\infty$-topos ${\mathbf{H}}$ is called _cohesive_ if the adjunction
${\widetilde{\mathsf{c}}}:{\mathbf{H}}\rightleftarrows{\mathbf{S}}:\Gamma$ can
be extended to a triple adjunction $\Pi\dashv\mathsf{c}\dashv\Gamma\dashv
codisc$, in which the left adjoint $\Pi$ preserves finite products and in
which the right adjoint $codisc$ is fully faithful.

Cohesive $\infty$-topoi have been studied extensively in [24] and related
works.

###### Theorem 2.34

The $\infty$-topos ${\mathbf{H}}_{\infty}$ is cohesive, i.e. there exists a
triple adjunction $\Pi\dashv{\widetilde{\mathsf{c}}}\dashv\Gamma\dashv codisc$
as in Definition 2.33, and there is a canonical equivalence

$\Pi\simeq\mathrm{S}_{e}\,.$ (2.35)

###### Remark 2.36

The fact that ${\mathbf{H}}_{\infty}$ is cohesive is not new, see [24]. The
second statement has been indicated in [2] and has been worked out in detail
in a model categorical presentation in [9]. Here, we give an
$\infty$-categorical proof of this fact for completeness. $\triangleleft$

###### Proof.

The $\infty$-topos
${\mathbf{H}}_{\infty}=\mathscr{P}({\mathscr{C}\mathrm{art}})$ admits a right-
adjoint to its global-section functor $\Gamma$ by abstract arguments:
evaluation of a presheaf at any object preserves colimits, and since both
${\mathbf{H}}_{\infty}$ and ${\mathbf{S}}$ are presentable, $\Gamma$ must
admit a further right adjoint. It is well-known that this can actually be
extended into a triple adjunction which establishes that
${\mathbf{H}}_{\infty}$ is cohesive [24].

For the second part of the statement, we show that $\mathrm{S}_{e}$ is left-
adjoint to the functor ${\widetilde{\mathsf{c}}}$. Recall from Section 2.1
that, in this situation, ${\widetilde{\mathsf{c}}}$ simply sends a space
$K\in{\mathbf{S}}$ to the constant presheaf on ${\mathscr{C}\mathrm{art}}$
with value $K$. Further, recall from the proof of Proposition 2.9 (and [9,
Prop. 2.13]) that the $I$-local objects in ${\mathbf{H}}_{\infty}$ are
precisely the essentially constant presheaves, i.e. those
$X\in{\mathbf{H}}_{\infty}$ for which the canonical morphism $X(*)\to X(c)$ is
an equivalence for every $c\in{\mathscr{C}\mathrm{art}}$. Equivalently, $X$ is
$I$-local if and only if the canonical morphism
${\widetilde{\mathsf{c}}}\Gamma X\to X$ is an equivalence in
${\mathbf{H}}_{\infty}$. Further, by Theorem 2.6 the right adjoint
${\mathrm{R}}_{e}$ to $\mathrm{S}_{e}$ factors through the localisation
$L_{I}{\mathbf{H}}_{\infty}\subset{\mathbf{H}}_{\infty}$; this is precisely
the full $\infty$-subcategory of ${\mathbf{H}}_{\infty}$ on the $I$-local
objects.

Consider the two adjunctions
$\mathrm{S}_{e}:{\mathbf{H}}_{\infty}\rightleftarrows{\mathbf{S}}:{\mathrm{R}}_{e}$
and
${\widetilde{\mathsf{c}}}:{\mathbf{S}}\rightleftarrows{\mathbf{H}}_{\infty}:\Gamma$.
They induce an adjunction

${\mathrm{S}_{e}\circ{\widetilde{\mathsf{c}}}:{\mathbf{S}}}$${{\mathbf{S}}:\Gamma\circ{\mathrm{R}}_{e}\,.}$$\scriptstyle{\perp}$
(2.37)

By the definition (2.5) of $\mathrm{S}_{e}$, for any space $K\in{\mathbf{S}}$
we have a canonical natural equivalence

$\displaystyle\mathrm{S}_{e}\circ{\widetilde{\mathsf{c}}}(K)$
$\displaystyle=\underset{{\mathbbe{\Delta}}^{\mathrm{op}}}{{\mathrm{colim}}}^{\mathbf{S}}\big{(}{\widetilde{\mathsf{c}}}(K)(\Delta_{e})\big{)}\simeq
K\,,$ (2.38)

because left-hand side is the colimit of a constant diagram over an indexing
category whose nerve is contractible in the Kan-Quillen model structure on
${\mathscr{S}\mathrm{et}_{\Delta}}$ (see Lemma A.18, Example A.20). In other
words, there is a canonical natural equivalence
$\mathrm{S}_{e}\circ{\widetilde{\mathsf{c}}}\simeq 1_{\mathbf{S}}$.
Consequently, there is also a canonical equivalence on the right adjoints,
$\Gamma\circ{\mathrm{R}}_{e}\simeq 1_{\mathbf{S}}$. We obtain natural
equivalences

${\widetilde{\mathsf{c}}}\simeq{\widetilde{\mathsf{c}}}\circ\Gamma\circ{\mathrm{R}}_{e}\simeq{\mathrm{R}}_{e}\,.$
(2.39)

In the second equivalence we have used that ${\mathrm{R}}_{e}$ takes values in
$L_{I}{\mathbf{H}}_{\infty}\subset{\mathbf{H}}_{\infty}$ and that on objects
in $L_{I}{\mathbf{H}}_{\infty}$ the morphism
${\widetilde{\mathsf{c}}}\circ\Gamma\to 1_{{\mathbf{H}}_{\infty}}$ is an
equivalence. From the equivalence
${\mathrm{R}}_{e}\simeq{\widetilde{\mathsf{c}}}$ and the adjunction
$\mathrm{S}_{e}\dashv{\mathrm{R}}_{e}$ we infer that $\mathrm{S}_{e}$ is a
further left adjoint to ${\widetilde{\mathsf{c}}}$. Hence, it is equivalent to
the functor $\Pi$. ∎

Theorem 2.34 shows that the smooth singular complex functor
$\mathrm{S}_{e}\colon{\mathbf{H}}_{\infty}\to{\mathbf{S}}$ has a deep
homotopical meaning for assigning homotopy types to objects in
${\mathbf{H}}_{\infty}$ and for studying these homotopy types. It also
provides an additional, refined, perspective on the good homotopical
properties of the functor $\mathrm{S}_{e}$ that were found and studied in [9].
Finally, note that it also follows that there is a natural equivalence

$\mathrm{S}_{e}(F)=\underset{{\mathbbe{\Delta}}^{\mathrm{op}}}{{\mathrm{colim}}}^{\mathbf{S}}\big{(}F(\Delta_{e})\big{)}\simeq\underset{{\mathscr{C}\mathrm{art}}^{\mathrm{op}}}{{\mathrm{colim}}}^{\mathbf{S}}(F)\,.$
(2.40)

That is, $\mathrm{S}_{e}$ computes the colimit of
${\mathscr{C}\mathrm{art}}^{\mathrm{op}}$-shaped diagrams of spaces.

##  3 Principal $\infty$-bundles and group extensions in $\infty$-topoi

In this section, starting from the theory introduced in [21], we develop
characterisations of principal $\infty$-bundles and of extensions of group
objects in $\infty$-topoi. These characterisations are interesting already in
their own right, but in Section 4 they will also allow us to transfer the
definition of string group extensions from ${\mathbf{S}}$ to
${\mathbf{H}}_{\infty}$ and to construct explicit smooth models for the string
group.

###  3.1 Groups and group extensions

Here, we recall the definitions of group objects and their extensions in
$\infty$-topoi [21]. We investigate how to compute limits of group and
groupoid objects in $\infty$-topoi, and how group objects and their
classifying objects behave under functors between $\infty$-topoi that preserve
finite products and geometric realisations.

Let ${\mathbf{H}}$ be an $\infty$-topos, and let
${\mathscr{G}\mathrm{pd}}({\mathbf{H}})$ be the $\infty$-category of groupoid
objects in ${\mathbf{H}}$. Further, let
${\mathrm{EEpi}}({\mathbf{H}})\subset{\mathscr{F}\mathrm{un}}(\Delta^{1},{\mathbf{H}})$
denote the full $\infty$-subcategory on the effective epimorphisms in
${\mathbf{H}}$. Recall that by Definition 2.27(4) there is a canonical
equivalence

${\mathscr{G}\mathrm{pd}}({\mathbf{H}})\simeq{\mathrm{EEpi}}({\mathbf{H}})\,,$
(3.1)

given by forming colimits and Čech nerves, respectively.

###### Lemma 3.2

In an $\infty$-topos ${\mathbf{H}}$, effective epimorphisms are stable under
pullback and under pushout.

###### Proof.

The fact that effective epimorphisms are stable under pullback is [20, Prop.
6.2.3.15]. The stability under pushouts follows from the facts that the
effective epimorphisms in ${\mathbf{H}}$ are precisely the
$0$-connective111Note that there is a shift in counting between [20] and the
nLab: A morphism $f$ in ${\mathbf{H}}$ is $n$-connect _ive_ in the conventions
of [20] if and only if it is $(n{-}1)$-connect _ed_ in the conventions used on
the nLab. morphisms [20, Def. 6.5.1.10], and that $n$-connected morphisms in
an $\infty$-topos are stable under pushout for any $n\leq-1$ [20, Prop.
6.5.1.17]. (The $n$-connected morphisms are even stable under colimits, since
they form the left class of an orthogonal factorisation system on
${\mathbf{H}}$.) ∎

###### Definition 3.3

Let $\mathscr{C}$ be an $\infty$-category. Let
${\mathscr{G}\mathrm{rp}}(\mathscr{C})\subset{\mathscr{G}\mathrm{pd}}(\mathscr{C})$
denote the full $\infty$-subcategory on those groupoid objects where $X_{0}$
is a final object of $\mathscr{C}$. We call
${\mathscr{G}\mathrm{rp}}(\mathscr{C})$ the $\infty$-category of _group
objects_ in $\mathscr{C}$.

###### Proposition 3.4

For any $\infty$-topos ${\mathbf{H}}$, there are reflective localisations

${{\mathscr{F}\mathrm{un}}({\mathbbe{\Delta}}^{\mathrm{op}},{\mathbf{H}})}$${{\mathscr{G}\mathrm{pd}}({\mathbf{H}})}$${{\mathscr{G}\mathrm{rp}}({\mathbf{H}})\,.}$$\scriptstyle{\perp}$$\scriptstyle{\perp}$
(3.5)

###### Proof.

First, the right adjoints in the above sequence of adjunctions are fully
faithful by definition. The first morphism has a left adjoint by [20, Prop.
6.1.2.9]. For the second left adjoint, we use the equivalence (3.1)222This
proof goes back to a mathoverflow answer by Jacob Lurie, see
https://mathoverflow.net/questions/140639/is-the-category-of-group-objects-in-
an-infty-1-topos-reflective-as-a-subcat/140742#140742.: this equivalence
induces a commutative square

${{\mathscr{G}\mathrm{pd}}({\mathbf{H}})}$${{\mathscr{G}\mathrm{rp}}({\mathbf{H}})}$${{\mathrm{EEpi}}({\mathbf{H}})}$${{\mathrm{EEpi}}_{*}({\mathbf{H}})}$$\scriptstyle{\simeq}$$\scriptstyle{\simeq}$
(3.6)

where ${\mathrm{EEpi}}_{*}({\mathbf{H}})\subset{\mathrm{EEpi}}({\mathbf{H}})$
is the full $\infty$-subcategory on those effective epimorphisms $f\colon
X_{0}\to X_{-1}$ where $X_{0}$ is a final object. A left adjoint to the bottom
morphism is given by the functor that sends an effective epimorphism $f\colon
X_{0}\to X_{-1}$ to the morphism $g\colon*\to X_{-1}\sqcup_{X_{0}}*$ induced
by the pushout. Since $f$ is an effective epimorphism, Lemma 3.2 implies that
so is $g$. ∎

For a group object $\widehat{G}\in{\mathscr{G}\mathrm{rp}}({\mathbf{H}})$ in
an $\infty$-topos ${\mathbf{H}}$, we set

$G\coloneqq\widehat{G}_{1}\in{\mathbf{H}}\quad\text{and}\quad{\mathrm{B}}G\coloneqq\underset{{\mathbbe{\Delta}}^{\mathrm{op}}}{{\mathrm{colim}}}^{\mathbf{H}}\,\widehat{G}=|{\widehat{G}}|\in{\mathbf{H}}\,.$
(3.7)

Note that in an $\infty$-topos ${\mathbf{H}}$, the map
$X_{0}\to{\mathrm{colim}}^{\mathbf{H}}_{{\mathbbe{\Delta}}^{\mathrm{op}}}\widehat{X}$
is an effective epimorphism for any groupoid object
$\widehat{X}\in{\mathscr{G}\mathrm{pd}}({\mathbf{H}})$. Hence, given a group
object $\widehat{G}$ in ${\mathbf{H}}$, the morphism $*\simeq
G_{0}\to{\mathrm{B}}G$ is an effective epimorphism. Moreover, the functor
${\mathrm{B}}$ is part of an equivalence [20, Lemma 7.2.2.11] (see also [21,
Thm. 2.19])

${{\mathbf{H}}^{*/}_{\geq
1}}$${{\mathscr{G}\mathrm{rp}}({\mathbf{H}})\,,}$$\scriptstyle{\perp}$$\scriptstyle{\Omega}$$\scriptstyle{{\mathrm{B}}}$
(3.8)

where ${\mathbf{H}}^{*/}_{\geq 1}$ is the $\infty$-category of pointed,
connected objects in ${\mathbf{H}}$.

Unravelling the definition, we obtain that a group object in an
$\infty$-category $\mathscr{C}$ with a final object $*\in\mathscr{C}$ is a
equivalently simplicial object ${\widehat{G}}$ in $\mathscr{C}$ such that
${\widehat{G}}_{0}\simeq*$ and, for any $[n]\in{\mathbbe{\Delta}}$ and any
partition $[n]=S\cup S^{\prime}$ as finite sets with $S\cap
S^{\prime}\cong\\{*\\}$ consisting of a single element, the diagram

${\widehat{G}_{n}}$${\widehat{G}(S)}$${\widehat{G}(S^{\prime})}$${\widehat{G}_{0}\simeq*}$
(3.9)

is a pullback diagram in $\mathscr{C}$. That is, there is a canonical
equivalence
$\widehat{G}_{n}\overset{\simeq}{\longrightarrow}\widehat{G}(S)\times\widehat{G}(S^{\prime})$.
In particular, iterating this for the spine partition
$[n]=[1]\sqcup_{[0]}\cdots\sqcup_{[0]}[1]$, we obtain a canonical equivalence

$\widehat{G}_{n}\overset{\simeq}{\longrightarrow}G^{n}\,.$ (3.10)

###### Proposition 3.11

Let ${\mathrm{L}}\colon{\mathbf{H}}\to{\mathbf{H}}^{\prime}$ be a functor of
$\infty$-topoi.

  1. (1)

If ${\mathrm{L}}$ preserves finite products, then it preserves group objects.

  2. (2)

If ${\mathrm{L}}$ additionally preserves geometric realisations, then, for any
group object $\widehat{G}$ in ${\mathbf{H}}$, there is a canonical equivalence

${\mathrm{B}}({\mathrm{L}}G)\simeq{\mathrm{L}}({\mathrm{B}}G)\,.$ (3.12)

###### Proof.

The first part of the Proposition is known [20]; we include its proof only for
completeness. Any functor $F\colon\mathscr{C}\to\mathscr{D}$ between
$\infty$-categories preserves simplicial objects, i.e. it induces a functor
${\mathscr{F}\mathrm{un}}({\mathbbe{\Delta}}^{\mathrm{op}},\mathscr{C})\longrightarrow{\mathscr{F}\mathrm{un}}({\mathbbe{\Delta}}^{\mathrm{op}},\mathscr{D})$.
Suppose that
${\widehat{A}}\in{\mathscr{F}\mathrm{un}}({\mathbbe{\Delta}}^{\mathrm{op}},{\mathbf{H}})$
is a group object in ${\mathbf{H}}$. Since ${\mathrm{L}}$ preserves finite
products, it preserves final objects, so that
$({\mathrm{L}}\widehat{A})_{0}\simeq*$ is final in ${\mathbf{H}}^{\prime}$.
For $n\neq 0$ and any partition $[n]=S\cup S^{\prime}$ with $S\cap
S^{\prime}\cong\\{*\\}$, we obtain a commutative diagram

${({\mathrm{L}}\widehat{A})_{n}={\mathrm{L}}(\widehat{A}_{n})}$${{\mathrm{L}}\big{(}\widehat{A}(S)\times\widehat{A}(S^{\prime})\big{)}}$${{\mathrm{L}}\widehat{A}(S)\times{\mathrm{L}}\widehat{A}(S^{\prime})}$$\scriptstyle{\simeq}$$\scriptstyle{\simeq}$
(3.13)

The top morphism is an equivalence since $\widehat{A}$ is a group object in
${\mathbf{H}}$ and the vertical morphism is an equivalence since
${\mathrm{L}}$ preserves products. This proves claim (1). Using that
${\mathrm{B}}A={\mathrm{colim}}^{\mathbf{H}}_{{\mathbbe{\Delta}}^{\mathrm{op}}}\widehat{A}=|\widehat{A}|$,
the second part is now immediate. ∎

###### Remark 3.14

We will prove a number of statements about functors as in Proposition 3.11(2),
i.e. functors between $\infty$-topoi which preserve geometric realisations and
finite products. An important class of such functors arises is given by the
additional left-adjoints of cohesive $\infty$-topoi—see Definition 2.33. In
particular, the functor
$\mathrm{S}_{e}\colon{\mathbf{H}}_{\infty}\to{\mathbf{S}}$ from Section 2.1 is
of this type by Theorem 2.34. $\triangleleft$

###### Lemma 3.15

Let ${\mathbf{H}}$ be an $\infty$-topos.

  1. (1)

A morphism $\widehat{X}\to\widehat{Y}$ in
${\mathscr{G}\mathrm{pd}}({\mathbf{H}})$ is an equivalence if and only if
$X_{i}\to Y_{i}$ is an equivalence in ${\mathbf{H}}$ for $i=0,1$.

  2. (2)

A morphism $\widehat{G}\to\widehat{H}$ in
${\mathscr{G}\mathrm{rp}}({\mathbf{H}})$ is an equivalence if and only if
$G\to H$ is an equivalence in ${\mathbf{H}}$.

###### Proof.

Proposition 3.4 implies that an equivalence of groupoid objects
$\widehat{X}\overset{\simeq}{\longrightarrow}\widehat{Y}$ in ${\mathbf{H}}$ is
the same as an objectwise equivalence of the underlying simplicial objects in
${\mathbf{H}}$: $\widehat{X}$ and $\widehat{Y}$ are local objects in
${\mathscr{F}\mathrm{un}}({\mathbbe{\Delta}}^{\mathrm{op}},{\mathbf{H}})$ with
respect to the localisation
${\mathscr{G}\mathrm{pd}}({\mathbf{H}})\subset{\mathscr{F}\mathrm{un}}({\mathbbe{\Delta}}^{\mathrm{op}},{\mathbf{H}})$,
so that the local equivalences between them are precisely the original, i.e.
the levelwise, equivalences. In particular, this implies the ‘only if’ part of
claim (1).

Conversely, if we are given a morphism
$\widehat{X}\overset{\simeq}{\longrightarrow}\widehat{Y}$ of groupoid objects
in ${\mathbf{H}}$ such that $\widehat{X}_{i}\to\widehat{Y}_{i}$ is an
equivalence for $i=0,1$, then it follows that
$\widehat{X}\overset{\simeq}{\longrightarrow}\widehat{Y}$ is a levelwise
equivalence of simplicial objects; this is because there are canonical
equivalences
$\widehat{X}_{n}\simeq\widehat{X}_{1}\times_{\widehat{X}_{0}}\cdots\times_{\widehat{X}_{0}}\widehat{X}_{1}$,
natural in $\widehat{X}\in{\mathscr{G}\mathrm{pd}}({\mathbf{H}})$. It then
follows that the morphism $\widehat{X}\to\widehat{Y}$ is also an equivalence
in ${\mathscr{G}\mathrm{pd}}({\mathbf{H}})$.

The same line of argument shows the second claim. ∎

###### Lemma 3.16

Let ${\mathbf{H}}$ be an $\infty$-topos, and let
$K\in{\mathscr{S}\mathrm{et}_{\Delta}}$ be a simplicial set.

  1. (1)

A diagram $\widehat{X}\colon
K^{\triangleleft}\to{\mathscr{G}\mathrm{pd}}({\mathbf{H}})$ of groupoid
objects in ${\mathbf{H}}$ is a limit diagram if and only if the composition
$\iota\widehat{X}\colon
K^{\triangleleft}\to{\mathscr{G}\mathrm{pd}}({\mathbf{H}})\hookrightarrow{\mathscr{F}\mathrm{un}}({\mathbbe{\Delta}}^{\mathrm{op}},{\mathbf{H}})$
is a limit diagram.

  2. (2)

A diagram $\widehat{X}\colon
K^{\triangleleft}\to{\mathscr{G}\mathrm{pd}}({\mathbf{H}})$ of groupoid
objects in ${\mathbf{H}}$ is a limit diagram if and only if the induced
diagrams $\widehat{X}_{i}\colon K^{\triangleleft}\to{\mathbf{H}}$ are limit
diagrams for $i=0,1$.

  3. (3)

A diagram $\widehat{G}\colon
K^{\triangleleft}\to{\mathscr{G}\mathrm{rp}}({\mathbf{H}})$ of group objects
in ${\mathbf{H}}$ is a limit diagram if and only if the the composition
$\jmath\widehat{G}\colon
K^{\triangleleft}\to{\mathscr{G}\mathrm{rp}}({\mathbf{H}})\hookrightarrow{\mathscr{G}\mathrm{pd}}({\mathbf{H}})$
is a limit diagram.

  4. (4)

A diagram $\widehat{G}\colon
K^{\triangleleft}\to{\mathscr{G}\mathrm{rp}}({\mathbf{H}})$ of group objects
in ${\mathbf{H}}$ is a limit diagram if and only if the induced diagram
$\widehat{G}_{1}=G\colon K^{\triangleleft}\to{\mathbf{H}}$ is a limit diagram.

###### Proof.

The ‘only if’ direction of claims (1) and (2) is readily seen as follows: we
first note that since the inclusion
${\mathscr{G}\mathrm{pd}}({\mathbf{H}})\subset{\mathscr{F}\mathrm{un}}({\mathbbe{\Delta}}^{\mathrm{op}},{\mathbf{H}})$
is a right adjoint, we have that if $\widehat{X}\colon
K^{\triangleleft}\to{\mathscr{G}\mathrm{pd}}({\mathbf{H}})$ is a limit
diagram, then so is $\widehat{X}\colon
K^{\triangleleft}\to{\mathscr{F}\mathrm{un}}({\mathbbe{\Delta}}^{\mathrm{op}},{\mathbf{H}})$.
Further, since limits in presheaf (or diagram) categories are computed
pointwise, this is equivalent to the functor $\widehat{X}_{i}\colon
K^{\triangleleft}\to{\mathbf{H}}$ being a limit diagram in ${\mathbf{H}}$ for
every $[i]\in{\mathbbe{\Delta}}$.

For the converse direction in claim (1), we first show that limits of diagrams
in ${\mathscr{G}\mathrm{pd}}({\mathbf{H}})$ can be computed in
${\mathscr{F}\mathrm{un}}({\mathbbe{\Delta}}^{\mathrm{op}},{\mathbf{H}})$.
More precisely, a functor $\widehat{X}\colon
K^{\triangleleft}\to{\mathscr{G}\mathrm{pd}}({\mathbf{H}})$ is a limit diagram
whenever its composition with the inclusion
$\iota\colon{\mathscr{G}\mathrm{pd}}({\mathbf{H}})\hookrightarrow{\mathscr{F}\mathrm{un}}({\mathbbe{\Delta}}^{\mathrm{op}},{\mathbf{H}})$
is so, i.e. the inclusion reflects limits. Equivalently, the
$\infty$-subcategory
${\mathscr{G}\mathrm{pd}}({\mathbf{H}})\hookrightarrow{\mathscr{F}\mathrm{un}}({\mathbbe{\Delta}}^{\mathrm{op}},{\mathbf{H}})$
is closed under limits in
${\mathscr{F}\mathrm{un}}({\mathbbe{\Delta}}^{\mathrm{op}},{\mathbf{H}})$.
This is seen as follows: consider a functor $\widehat{X}\colon
K^{\triangleleft}\to{\mathscr{G}\mathrm{pd}}({\mathbf{H}})$ and a
decomposition $[n]=S\cup S^{\prime}$ with $S\cap S^{\prime}=\\{*\\}$. This
induces an equivalence

$\widehat{X}_{n}\overset{\simeq}{\longrightarrow}\widehat{X}(S)\underset{\widehat{X}_{0}}{\times}\widehat{X}(S^{\prime})$
(3.17)

in ${\mathscr{F}\mathrm{un}}(K^{\triangleleft},{\mathbf{H}})$. Setting
$\widehat{Y}\coloneqq{\mathrm{lim}}_{K}^{{\mathscr{F}\mathrm{un}}({\mathbbe{\Delta}}^{\mathrm{op}},{\mathbf{H}})}(\iota\widehat{X})$
and using that limits commute with limits, we have

$\displaystyle\widehat{Y}_{n}$
$\displaystyle\simeq\big{(}\underset{K}{{\mathrm{lim}}}^{{\mathscr{F}\mathrm{un}}({\mathbbe{\Delta}}^{\mathrm{op}},{\mathbf{H}})}(\iota\widehat{X})\big{)}_{n}$
(3.18)
$\displaystyle\simeq\underset{K}{{\mathrm{lim}}}^{{\mathbf{H}}}(\iota\widehat{X}_{n})$
(3.19)
$\displaystyle\simeq\underset{K}{{\mathrm{lim}}}^{{\mathbf{H}}}\big{(}\iota\widehat{X}(S)\underset{\iota\widehat{X}_{0}}{\times}\iota\widehat{X}(S^{\prime})\big{)}$
(3.20)
$\displaystyle\simeq\big{(}\underset{K}{{\mathrm{lim}}}^{{\mathbf{H}}}\iota\widehat{X}(S)\big{)}\underset{(\underset{K}{{\mathrm{lim}}}^{{\mathbf{H}}}\iota\widehat{X}_{0})}{\times}\big{(}\underset{K}{{\mathrm{lim}}}^{{\mathbf{H}}}\iota\widehat{X}(S^{\prime})\big{)}$
(3.21)
$\displaystyle\simeq\widehat{Y}(S)\times_{\widehat{Y}_{0}}\widehat{Y}(S^{\prime})\,,$
(3.22)

which shows that
$\widehat{Y}\in{\mathscr{F}\mathrm{un}}({\mathbbe{\Delta}}^{\mathrm{op}},{\mathbf{H}})$
is local with respect to the localisation
${\mathscr{G}\mathrm{pd}}({\mathbf{H}})\hookrightarrow{\mathscr{F}\mathrm{un}}({\mathbbe{\Delta}}^{\mathrm{op}},{\mathbf{H}})$,
i.e. that $\widehat{Y}\in{\mathscr{G}\mathrm{pd}}({\mathbf{H}})$. Since the
inclusion
$\iota\colon{\mathscr{G}\mathrm{pd}}({\mathbf{H}})\hookrightarrow{\mathscr{F}\mathrm{un}}({\mathbbe{\Delta}}^{\mathrm{op}},{\mathbf{H}})$
is fully faithful, $\widehat{Y}$ is also a limit of the diagram
$\widehat{X}\colon K\to{\mathscr{G}\mathrm{pd}}({\mathbf{H}})$. Consequently,
if $\iota\widehat{X}\colon
K^{\triangleleft}\to{\mathscr{F}\mathrm{un}}({\mathbbe{\Delta}}^{\mathrm{op}},{\mathbf{H}})$
is a limit diagram, then $\widehat{X}\colon
K^{\triangleleft}\to{\mathscr{G}\mathrm{pd}}({\mathbf{H}})$ is a limit
diagram.

For the converse direction in claim (2), suppose that $\widehat{X}\colon
K^{\triangleleft}\to{\mathscr{G}\mathrm{pd}}({\mathbf{H}})$ is a diagram such
that the functors $\widehat{X}_{i}\colon K^{\triangleleft}\to{\mathbf{H}}$ are
limit diagrams for $i=0,1$. By part (1) it suffices to show that the
composition $\iota\widehat{X}\colon
K^{\triangleleft}\to{\mathscr{F}\mathrm{un}}({\mathbbe{\Delta}}^{\mathrm{op}},{\mathbf{H}})$
is a limit diagram; that is, it suffices to show that $\widehat{X}_{i}\colon
K^{\triangleleft}\to{\mathbf{H}}$ is a limit diagram for every
$[i]\in{\mathbbe{\Delta}}$.

Since $\widehat{X}\colon
K^{\triangleleft}\to{\mathscr{F}\mathrm{un}}({\mathbbe{\Delta}}^{\mathrm{op}},{\mathbf{H}})$
is valued in groupoid objects, and since limits in ${\mathbf{H}}$ commute [20,
Lemma 5.5.2.3], it follows from (3.18) that for every
$[n]\in{\mathbbe{\Delta}}$ the diagram $\widehat{X}_{n}\colon
K^{\triangleleft}\to{\mathbf{H}}$ is equivalent to a limit diagram
$\widehat{X}_{1}\times_{\widehat{X}_{0}}\cdots\times_{\widehat{X}_{0}}\widehat{X}_{1}\colon
K^{\triangleleft}\to{\mathbf{H}}$, and is hence a limit diagram itself.

The proof of claim (3) proceeds along the exact same line as the proof of part
(2): the key insight is the fact that if $\widehat{G}\colon
K^{\triangleleft}\to{\mathscr{G}\mathrm{rp}}({\mathbf{H}})$ is a diagram such
that the composition $\jmath\widehat{G}\colon
K^{\triangleleft}\to{\mathscr{G}\mathrm{pd}}({\mathbf{H}})$ is a limit
diagram, then
${\mathrm{lim}}_{K}^{{\mathscr{G}\mathrm{pd}}({\mathbf{H}})}(\jmath\widehat{G})$
is still local with respect to the localisation
${\mathscr{G}\mathrm{rp}}({\mathbf{H}})\subset{\mathscr{G}\mathrm{pd}}({\mathbf{H}})$.

Claim (4) is then the combination of claims (2) and (3). ∎

Having established several properties of the $\infty$-category of group
objects in ${\mathbf{H}}$, we now define extensions of group objects:

###### Definition 3.23

_[ 21, Def. 4.26]_ Let $\widehat{A}$ and $\widehat{H}$ be group objects in an
$\infty$-topos ${\mathbf{H}}$. An _extension of group objects of $\widehat{H}$
by $\widehat{A}$_ is a sequence
${\widehat{A}}\to{\widehat{G}}\to{\widehat{H}}$ in the $\infty$-category
${\mathscr{G}\mathrm{rp}}({\mathbf{H}})$ such that the sequence
${\mathrm{B}}A\to{\mathrm{B}}G\to{\mathrm{B}}H$ is a fibre sequence in
${\mathbf{H}}^{*/}_{\geq 1}$.

###### Remark 3.24

This definition of a group extension has advantages from a theoretical
perspective. Nevertheless, it appears that there should be a simpler
definition that more directly generalises extensions of groups in
${\mathscr{S}\mathrm{et}}$ to the $\infty$-categorical setting. For group
objects in ${\mathscr{S}\mathrm{et}}$, a group extension is a sequence $A\to
G\to H$ of group homomorphisms such that $A$ is the fibre of the morphism
$G\to H$ at the identity element of $H$. We will prove in Theorem 3.99 that
one can indeed define group extensions in $\infty$-topoi along these
principles. $\triangleleft$

###  3.2 Group actions in $\infty$-categories

We now investigate actions of group objects in $\infty$-topoi. For a
simplicial set $K\in{\mathscr{S}\mathrm{et}_{\Delta}}$, we let
$\operatorname{obj}(K)$ be the set $K_{0}$ of vertices of $K$, seen as a
discrete simplicial set. Let $J\coloneqq\Delta^{1}[f^{-1}]$ be the
localisation of $\Delta^{1}$ at its non-trivial edge (see e.g. [11, Sec.
3.3]).

###### Lemma 3.25

Let $\mathscr{C}$ be an $\infty$-category, and let $K$ be a simplicial set.

  1. (1)

The inclusion $\iota\colon\operatorname{obj}(K)\hookrightarrow K$ induces a
morphism

$\iota^{*}\colon\mathscr{C}^{K}={\mathscr{F}\mathrm{un}}(K,\mathscr{C})\longrightarrow{\mathscr{F}\mathrm{un}}\big{(}\operatorname{obj}(K),\mathscr{C}\big{)}=\mathscr{C}^{\operatorname{obj}(K)}$
(3.26)

of simplicial sets, which is a fibration between fibrant objects in the Joyal
model structure.

  2. (2)

Consider either of the inclusions $\Delta^{\\{i\\}}\hookrightarrow J$, where
$i=0,1$. The induced morphism

${\mathscr{F}\mathrm{un}}(J,\mathscr{C}^{K})\longrightarrow\mathscr{C}^{K}\underset{\mathscr{C}^{\operatorname{obj}(K)}}{\times}{\mathscr{F}\mathrm{un}}\big{(}J,\mathscr{C}^{\operatorname{obj}(K)}\big{)}$
(3.27)

is a trivial Kan fibration.

  3. (3)

Let $g\colon K\to\mathscr{C}$ and
$g^{\prime}\colon\operatorname{obj}(K)\to\mathscr{C}$ be functors. For any
equivalence $\eta\colon\iota^{*}g\overset{\simeq}{\longrightarrow}g^{\prime}$,
consider the space of pairs $(\hat{g}^{\prime},\hat{\eta})$, where
$\hat{g}^{\prime}$ is a lift of $g^{\prime}$ to a functor
$\hat{g}^{\prime}\colon K\to\mathscr{C}$, and where $\hat{\eta}$ is an
equivalence $g\overset{\simeq}{\longrightarrow}\hat{g}^{\prime}$ such that
$\iota^{*}\hat{\eta}=\eta$. This space is a contractible Kan complex.

###### Proof.

Part (1) follows since $\operatorname{obj}(K)\hookrightarrow K$ is a
cofibration in the Joyal model category
${\mathscr{S}\mathrm{et}_{\Delta}}_{J}$, $\mathscr{C}$ is a fibrant object in
${\mathscr{S}\mathrm{et}_{\Delta}}_{J}$, and
${\mathscr{S}\mathrm{et}_{\Delta}}_{J}$ is a (closed) symmetric monoidal model
category.

For part (2), we apply [11, Cor. 3.6.4] to the categorical anodyne extension
$\Delta^{\\{i\\}}\hookrightarrow J=\Delta^{1}[f^{-1}]$ and the Joyal fibration
(i.e. isofibration) from part (1).

Part (3) is obtained by taking the fibre of the morphism from part (2), which
is a contractible Kan complex since it is the fibre of a trivial Kan
fibration. This fibre is equivalently described as the space of lifts in the
commutative diagram

${\Delta^{\\{i\\}}}$${\mathscr{C}^{K}}$${J}$${\mathscr{C}^{\operatorname{obj}(K)}}$$\scriptstyle{g}$$\scriptstyle{\iota^{*}}$$\scriptstyle{\eta}$
(3.28)

which is precisely the space of pairs $(\hat{g}^{\prime},\hat{\eta})$ of lifts
$\hat{g}^{\prime}\colon K\to\mathscr{C}$ of $g^{\prime}$ and equivalences
$\hat{\eta}\colon g\overset{\simeq}{\longrightarrow}\hat{g}^{\prime}$ such
that $\iota^{*}\hat{\eta}=\eta$. ∎

###### Example 3.29

Let $\widehat{G}$ be a group object in an $\infty$-category $\mathscr{C}$ with
a final object. This is, in particular, a simplicial object
$\widehat{G}\colon{\mathbbe{\Delta}}^{\mathrm{op}}\to\mathscr{C}$ (we suppress
the canonical inclusion functors
${\mathscr{G}\mathrm{rp}}(\mathscr{C})\hookrightarrow{\mathscr{G}\mathrm{pd}}(\mathscr{C})\hookrightarrow{\mathscr{F}\mathrm{un}}({\mathbbe{\Delta}}^{\mathrm{op}},\mathscr{C})$).
Consider the functor

$[0]\star(-)\colon{\mathbbe{\Delta}}\longrightarrow{\mathbbe{\Delta}}\,,\qquad[n]\longmapsto[0]\star[n]\cong[n+1]\,,$
(3.30)

where $\star$ denotes the join of categories (and where we view partially
ordered sets as categories). The induced pullback functor

$\operatorname{Dec}^{0}\coloneqq\big{(}[0]\star(-)\big{)}^{*}\colon{\mathscr{F}\mathrm{un}}({\mathbbe{\Delta}}^{\mathrm{op}},\mathscr{C})\longrightarrow{\mathscr{F}\mathrm{un}}({\mathbbe{\Delta}}^{\mathrm{op}},\mathscr{C})$
(3.31)

is also called the _decalage_ functor; see [28] for more background. For any
$n\geq 1$, the partition $[n]=\\{0,1\\}\sqcup_{\\{1\\}}\\{1,\ldots,n\\}$
induces an equivalence

$\gamma_{n}\colon(\operatorname{Dec}^{0}\widehat{G})_{n}=\widehat{G}_{n+1}\simeq
G\times\widehat{G}_{n}\,.$ (3.32)

We can phrase this as an equivalence of functors
$\operatorname{Dec}^{0}\widehat{G}\simeq
G\times\widehat{G}\colon\operatorname{obj}({\mathbbe{\Delta}}^{\mathrm{op}})\to\mathscr{C}$.
From Lemma 3.25 we obtain that there exists an essentially unique way to lift
these data to a functor ${\mathbbe{\Delta}}^{\mathrm{op}}\to\mathscr{C}$,
which we denote by $G{//}G$, and an equivalence
$\gamma\colon\operatorname{Dec}^{0}\widehat{G}\overset{\simeq}{\longrightarrow}G{//}G$
in ${\mathscr{F}\mathrm{un}}({\mathbbe{\Delta}}^{\mathrm{op}},\mathscr{C})$,
whose components are exactly the equivalences $\gamma_{n}$ from (3.32). One
can now check that $G{//}G$ is the simplicial object in $\mathscr{C}$ that
describes the right action of ${\widehat{G}}$ on itself via the group
multiplication in ${\widehat{G}}$. $\triangleleft$

###### Definition 3.33

Let $\mathscr{C}$ be an $\infty$-category with pullbacks and a final object,
let ${\widehat{G}}$ be a group object in $\mathscr{C}$, and let
$P\in\mathscr{C}$ be an object in $\mathscr{C}$. An _action of ${\widehat{G}}$
on $P$_ is a simplicial object
$P{//}G\in{\mathscr{F}\mathrm{un}}({\mathbbe{\Delta}}^{\mathrm{op}},\mathscr{C})$
such that

  1. (1)

for each $n\in\mathbb{N}_{0}$, we have $(P{//}G)_{n}=P\times G^{n-1}$,

  2. (2)

the morphism $d_{1}\colon P\times G\to P$ agrees with the canonical projection
onto $P$, the morphism $s_{0}\colon P\to P\times G$ agrees with the morphism
$1_{P}\times(*\to G)$, and

  3. (3)

the collapse morphism $P\to*$ induces a morphism $P{//}G\to{\widehat{G}}$ in
${\mathscr{F}\mathrm{un}}({\mathbbe{\Delta}}^{\mathrm{op}},\mathscr{C})$.

Given a group action $P{//}G$, we set $a\coloneqq d_{0}\colon P\times G\to P$.
It follows by the pasting law for pullbacks that there are canonical
equivalences of morphisms between $d_{0}\colon P\times G^{n}\to P\times
G^{n-1}$ and $a\times 1_{G^{n-1}}\colon P\times G^{n}\to P\times G^{n-1}$, and
similarly between $d_{n}\colon P\times G^{n}\to P\times G^{n-1}$ and the
projection onto the first $n$ factors.

###### Remark 3.34

Definition 3.33 is taken from [21, Def. 3.1] almost verbatim, but it differs
from that source in that we do not _require_ group actions to be groupoid
objects. Instead, we show in Theorem 3.37 that this is a _consequence_ of the
axioms in Definition 3.33. A second (minor) difference is that we also fix the
level-zero degeneracy map $s_{0}\colon P\to P\times G$. $\triangleleft$

###### Example 3.35

For any group object $\widehat{G}\in{\mathscr{G}\mathrm{rp}}(\mathscr{C})$
there is a canonical trivial action $*{//}G$ on the final object
$*\in\mathscr{C}$, coming from the canonical equivalence
$*\times\widehat{G}\simeq\widehat{G}$ of simplicial objects. That is, there is
a canonical equivalence ${\widehat{G}}\simeq*{//}G$ in
${\mathscr{F}\mathrm{un}}({\mathbbe{\Delta}}^{\mathrm{op}},\mathscr{C})$.
$\triangleleft$

###### Example 3.36

We can now give a precise meaning to the last sentence of Example 3.29: the
object
$G{//}G\in{\mathscr{F}\mathrm{un}}({\mathbbe{\Delta}}^{\mathrm{op}},\mathscr{C})$
is an action of $G$ on itself via right multiplication. $\triangleleft$

Given an action of a group object ${\widehat{G}}$ on an object $P$ in
$\mathscr{C}$, we would like to think of the simplicial object $P{//}G$ as the
_action groupoid_ associated with this action. This is indeed justified:

###### Theorem 3.37

Let $\mathscr{C}$ be an $\infty$-category with finite limits, let
${\widehat{G}}\in{\mathscr{G}\mathrm{rp}}(\mathscr{C})$ be a group object in
$\mathscr{C}$, and let
$P{//}G\in{\mathscr{F}\mathrm{un}}({\mathbbe{\Delta}}^{\mathrm{op}},\mathscr{C})$
be an action of ${\widehat{G}}$ on an object $P\in\mathscr{C}$. Then, $P{//}G$
is a groupoid object in $\mathscr{C}$.

###### Remark 3.38

Theorem 3.37 is important for us since will need to show that functors
${\mathrm{L}}\colon{\mathbf{H}}\to{\mathbf{H}}^{\prime}$ between
$\infty$-topoi that preserve finite products and geometric realisations map
group actions to group actions (see Theorem 3.60). In [21], group actions are
_defined_ to be groupoid objects, but functors
${\mathrm{L}}\colon{\mathbf{H}}\to{\mathbf{H}}^{\prime}$ as above do not
preserve groupoid objects in general. However, Theorem 3.37 shows that, as in
the classical case of (set-theoretic) group actions, actions of group objects
in $\infty$-topoi are _automatically_ a groupoid objects. $\triangleleft$

We prove Theorem 3.37 in Appendix A. For the remainder of this section, let
${\mathbf{H}}$ be an $\infty$-topos.

###### Definition 3.39

Let ${\widehat{G}}\in{\mathscr{G}\mathrm{rp}}({\mathbf{H}})$ be a group
object. A _$G$ -action over an object $X\in{\mathbf{H}}$_ is an augmented
simplicial object
$\widehat{X}\in{\mathscr{F}\mathrm{un}}({\mathbbe{\Delta}}_{+}^{\mathrm{op}},{\mathbf{H}})$
whose underlying simplicial object is a $G$-action $P{//}G$ on some object
$P\in{\mathbf{H}}$, and whose augmenting object is $X$, i.e.
$\widehat{X}_{-1}=X$. Writing $p\colon P\to X$ for the morphism
$\widehat{X}_{|{\mathbbe{\Delta}}_{+,\leq 0}}$, we also denote a $G$-action
over $X$ by

$P{//}G\overset{p}{\longrightarrow}X\quad\in{\mathscr{F}\mathrm{un}}({\mathbbe{\Delta}}_{+}^{\mathrm{op}},{\mathbf{H}})\,.$
(3.40)

A _morphism of $G$-actions over $X\in{\mathbf{H}}$_,

$(P{//}G\to X)\overset{f}{\longrightarrow}(Q{//}G\to X)\,,$ (3.41)

is a morphism $f$ in
${\mathscr{F}\mathrm{un}}({\mathbbe{\Delta}}_{+}^{\mathrm{op}},{\mathbf{H}})$
as above such that

  1. (1)

$f_{-1}=1_{X}$ is the identity on $X$, and

  2. (2)

the collapse morphisms $P\to*$ and $Q\to*$ induce a (weakly) commutative
diagram

${P{//}G}$${Q{//}G}$${*{//}G}$$\scriptstyle{f_{|{\mathbbe{\Delta}}^{\mathrm{op}}}}$
(3.42)

of simplicial objects in ${\mathbf{H}}$.

The _$\infty$ -category of $G$-actions over $X\in{\mathbf{H}}$_ is the full
$\infty$-subcategory of
${\mathscr{F}\mathrm{un}}({\mathbbe{\Delta}}^{\mathrm{op}},{\mathbf{H}})_{/(X\times(*{//}G))}$
on those objects whose underlying simplicial object is a $G$-action.

Observe that an ordinary $G$-action is equivalent to a $G$-action over the
final object $*\in{\mathbf{H}}$.

###### Example 3.43

For a group object $\widehat{G}\in{\mathscr{G}\mathrm{rp}}({\mathbf{H}})$ and
an action $P{//}G$ of $\widehat{G}$ on an object $P\in{\mathbf{H}}$, let
$q\colon({\mathbbe{\Delta}}^{\mathrm{op}})^{\triangleright}\to{\mathbf{H}}$ be
a colimiting cocone of the simplicial diagram $P{//}G$ in ${\mathbf{H}}$.
Observing that
$({\mathbbe{\Delta}}^{\mathrm{op}})^{\triangleright}\cong({\mathbbe{\Delta}}^{\triangleleft})^{\mathrm{op}}\cong({\mathbbe{\Delta}}_{+})^{\mathrm{op}}$,
this defines an augmented simplicial object in ${\mathbf{H}}$, which we denote
as

$q\colon
P{//}G\longrightarrow\underset{{\mathbbe{\Delta}}^{\mathrm{op}}}{{\mathrm{colim}}}^{\mathbf{H}}(P{//}G)=|P{//}G|\,.$
(3.44)

Therefore, the data $P{//}G\to|P{//}G|$ form a $G$-action over $|P{//}G|$. In
particular, the canonical morphism $*{//}G\to{\mathrm{B}}G$ is of this form.
$\triangleleft$

Another example of a morphism of this type is the collapse morphism
$G{//}G\to*$, as we show now:

###### Proposition 3.45

If $\widehat{G}$ is a group object in ${\mathbf{H}}$, then the canonical
morphism

$|G{//}G|\overset{\simeq}{\longrightarrow}*$ (3.46)

is an equivalence.

###### Proof.

Since ${\mathbf{H}}$ is presentable, there exists a combinatorial simplicial
model category ${\mathscr{M}}$ and an equivalence of $\infty$-categories
${\mathbf{H}}\simeq N({\mathscr{M}}^{\circ})$ [20, Prop. A.3.7.6]. Under this
equivalence, colimits in ${\mathbf{H}}$ over diagrams indexed by ordinary
categories correspond to homotopy colimits in ${\mathscr{M}}$ [20, Cor.
4.2.4.8]. It now suffices to observe that any simplicial object in
${\mathscr{M}}$ obtained as the decalage of another simplicial object has an
augmentation and extra degeneracies [23, 28]. ∎

Any morphism ${\widehat{A}}\to{\widehat{G}}$ of group objects induces an
action of ${\widehat{A}}$ on $G$ by the following construction:

###### Proposition 3.47

Let $\widehat{f}\colon{\widehat{A}}\to{\widehat{G}}$ be a morphism in
${\mathscr{G}\mathrm{rp}}({\mathbf{H}})$. Define a simplicial object $G{//}A$
as the pullback

${G{//}A}$${G{//}G}$${*{//}A}$${*{//}G}$ (3.48)

in ${\mathscr{F}\mathrm{un}}({\mathbbe{\Delta}}^{\mathrm{op}},{\mathbf{H}})$.
Then, $G{//}A$ is an action of ${\widehat{A}}$ on $G$.

###### Proof.

We check the axioms in Definition 3.33: axiom (1) follows from the pasting law
for pullbacks and the diagram

${(G{//}A)_{n}}$${(G{//}G)_{n}}$${G}$${(*{//}A)_{n}}$${(*{//}G)_{n}}$${*}$
(3.49)

in which the right-hand square is a pullback for any $n\in\mathbb{N}_{0}$ by
construction of $G{//}G$.

Axiom (2) is readily seen from applying the maps $d_{1}$ and $s_{0}$ to the
diagram (3.48), for $n=0,1$. Axiom (3) follows since the morphism
$G{//}A\longrightarrow*{//}A$ induced by the above diagram agrees with the
morphism obtained by collapsing the first factor $G$. ∎

###  3.3 Principal $\infty$-bundles

In this subsection, we characterise principal $\infty$-bundles and group
extensions in $\infty$-topoi. Throughout this section, let ${\mathbf{H}}$ be
an $\infty$-topos and let
${\widehat{G}}\in{\mathscr{G}\mathrm{rp}}({\mathbf{H}})$ be a group object in
${\mathbf{H}}$.

###### Definition 3.50

_[ 21, Def. 3.4]_ A _$G$ -principal $\infty$-bundle on $X\in{\mathbf{H}}$_ is
a $G$-action $P{//}G\to X$ over $X$ such that the augmented simplicial object
$P{//}G\to X$ is a colimiting cocone for the simplicial diagram
$P{//}G\in{\mathscr{F}\mathrm{un}}({\mathbbe{\Delta}}^{\mathrm{op}},{\mathbf{H}})$.
In other words, the augmenting map $p\colon P\to X$ induces an equivalence
${\mathrm{colim}}^{\mathbf{H}}_{{\mathbbe{\Delta}}^{\mathrm{op}}}(P{//}G)\overset{\simeq}{\longrightarrow}X$
in ${\mathbf{H}}$.

A _morphism of $G$-principal $\infty$-bundles on $X$_, denoted $(P{//}G\to
X)\longrightarrow(Q{//}G\to X)$, is a morphism of the underlying $G$-actions
over $X$. The $\infty$-category ${\mathscr{B}\mathrm{un}}_{G}(X)$ of
$G$-principal $\infty$-bundles over $X$ is the full $\infty$-subcategory of
${\mathscr{F}\mathrm{un}}({\mathbbe{\Delta}}^{\mathrm{op}},{\mathbf{H}})_{/(X\times(*{//}G))}$
(cf. Definition 3.39) on the $G$-principal $\infty$-bundles on $X$.

###### Example 3.51

Let ${\widehat{G}}$ be a group object in ${\mathbf{H}}$. For any $G$-action
$P{//}G$ in ${\mathbf{H}}$, the morphism $P{//}G\to|P{//}G|$ is a principal
$G$-bundle in ${\mathbf{H}}$ over $|P{//}G|$. As concrete examples of this
type, we have already seen that $G{//}G$ turns $G$ into a principal $G$-bundle
over $*\in{\mathbf{H}}$ (Proposition (3.45)), and that $*{//}G$ turns $*$ into
a principal $G$-bundle over ${\mathrm{B}}G$ (by the definition of
${\mathrm{B}}G$). $\triangleleft$

We now provide an alternative characterisation of principal $\infty$-bundles
in an $\infty$-topoi. Let
${\widehat{G}}\in{\mathscr{G}\mathrm{rp}}({\mathbf{H}})$ be a group object in
${\mathbf{H}}$, and let $p\colon P{//}G\to X$ be a $G$-action over an object
$X\in{\mathbf{H}}$. Let $\imath\colon{\mathbbe{\Delta}}_{+,\leq
0}^{\mathrm{op}}\hookrightarrow{\mathbbe{\Delta}}_{+}^{\mathrm{op}}$ be the
inclusion. The identity provides a canonical equivalence

$\eta\colon\\{p\\}=\imath^{*}(P{//}G\to
X)\overset{\simeq}{\longrightarrow}\imath^{*}({\check{C}}p)=\\{p\\}$ (3.52)

in ${\mathscr{F}\mathrm{un}}({\mathbbe{\Delta}}_{+,\leq
0}^{\mathrm{op}},{\mathbf{H}})\simeq{\mathscr{F}\mathrm{un}}(\Delta^{1},{\mathbf{H}})$.
Since right Kan extension is a right adjoint, there is an equivalence

$\underline{{\mathscr{F}\mathrm{un}}}({\mathbbe{\Delta}}_{+,\leq
0}^{\mathrm{op}},{\mathbf{H}})\big{(}\imath^{*}(P{//}G\to
X),\\{p\\}\big{)}\simeq\underline{{\mathscr{F}\mathrm{un}}}({\mathbbe{\Delta}}_{+}^{\mathrm{op}},{\mathbf{H}})\big{(}(P{//}G\to
X),{\check{C}}p\big{)}$ (3.53)

of mapping spaces (compare also (2.26)). We denote the image of $\eta$ under
this equivalence by

$\alpha\colon(P{//}G\to X)\longrightarrow{\check{C}}p\,.$ (3.54)

Observe that, by construction, the restriction of $\alpha$ along $\imath$ is
$\eta$. We will not distinguish notationally between $\alpha$ as defined here
and its restriction along the inclusion
${\mathbbe{\Delta}}^{\mathrm{op}}\subset{\mathbbe{\Delta}}_{+}^{\mathrm{op}}$
(since $\alpha_{-1}=1_{X}$).

###### Definition 3.55

A $G$-action $P{//}G\longrightarrow X$ over $X\in{\mathbf{H}}$ is called
_principal_ if the canonical morphism $\alpha\colon
P{//}G\longrightarrow{\check{C}}p$ is an equivalence in
${\mathscr{F}\mathrm{un}}({\mathbbe{\Delta}}^{\mathrm{op}},{\mathbf{H}})$.

This is an $\infty$-categorical version of the principality condition for a
group action. It is, in fact, equivalent to the usual principality
condition—that the morphism $P\times G\to P\times_{X}P$ is an equivalence—in
the following sense (in particular, this implies the converse to [21, Prop.
3.7]):

###### Lemma 3.56

Let $P{//}G\to X$ be a $G$-action over $X\in{\mathbf{H}}$. The following are
equivalent:

  1. (1)

The $G$-action is principal.

  2. (2)

The diagram

${P\times
G}$${P}$${P}$${X}$$\scriptstyle{d_{1}=\mathrm{pr}_{P}}$$\scriptstyle{a=d_{0}}$$\scriptstyle{p}$$\scriptstyle{p}$
(3.57)

is a pullback diagram in ${\mathbf{H}}$.

###### Proof.

(1) implies (2) since the action $P{//}G\overset{p}{\longrightarrow}X$ is
principal precisely if it is equivalent, as an augmented simplicial object in
${\mathbf{H}}$, to the Čech nerve ${\check{C}}p=\mathrm{Ran}_{\iota}\\{p\\}$.
Thus, the implication follows from Proposition 2.21.

Conversely, (2) also implies (1): we know from Theorem 3.37 that $P{//}G$ is a
groupoid object. If we additionally have that (3.57) is a pullback diagram,
then we can again apply Proposition 2.21 to obtain the claim. ∎

We can use Lemma 3.56 to give a characterisation of principal $\infty$-bundles
which can be understood as encoding directly the classical criteria for
principal bundles: a locally trivial map $p\colon P\to X$ and a principal
$G$-action over $X$.

###### Proposition 3.58

Let $P{//}G\xrightarrow{p}X$ be a $G$-action over an object
$X\in{\mathbf{H}}$. The following are equivalent:

  1. (1)

$P{//}G\overset{p}{\longrightarrow}X$ is a principal $\infty$-bundle (in the
sense of Definition 3.50).

  2. (2)

The morphism $p$ is an effective epimorphism and the action $P{//}G$ is
principal.

###### Proof.

First, observe that since $P{//}G$ is a groupoid object in ${\mathbf{H}}$, and
since by assumption the canonical morphism $|P{//}G|\to X$ is an equivalence,
it follows from Definition 2.27(4) that the canonical morphism $\alpha\colon
P{//}G\to{\check{C}}p$ is an equivalence in
${\mathscr{F}\mathrm{un}}({\mathbbe{\Delta}}^{\mathrm{op}},{\mathbf{H}})$. In
particular, $p$ is an effective epimorphism. Further, it has been shown in
[21, Prop. 3.7] that if $P{//}G\to X$ is a principal bundle, then the action
$P{//}G$ satisfies condition (2) of Lemma 3.56, and so the action is
principal.

To see the other direction, consider the commutative diagram

${{|P{//}G|}}$${{|{\check{C}}p|}}$${X}$$\scriptstyle{|\alpha|}$ (3.59)

In this case, both the top and the right-hand morphisms in diagram are
equivalences. It thus follows that also the left-hand morphism is an
equivalence, which amounts to the fact that $P{//}G\to X$ is a principal
$G$-bundle in the sense of Definition 3.50. ∎

###### Theorem 3.60

Let ${\mathrm{L}}\colon{\mathbf{H}}\to{\mathbf{H}}^{\prime}$ be a functor
between $\infty$-topoi which preserves geometric realisations and finite
products. Suppose ${\widehat{G}}$ is any group object in ${\mathbf{H}}$.

  1. (1)

${\mathrm{L}}$ maps $G$-actions $P{//}G\xrightarrow{p}X$ over
$X\in{\mathbf{H}}$ to ${\mathrm{L}}G$-actions
${\mathrm{L}}P{//}{\mathrm{L}}G\xrightarrow{{\mathrm{L}}p}{\mathrm{L}}X$ over
${\mathrm{L}}X\in{\mathbf{H}}^{\prime}$.

  2. (2)

If the action $P{//}G\to X$ is a principal $G$-bundle, then the action
${\mathrm{L}}P{//}{\mathrm{L}}G\xrightarrow{{\mathrm{L}}p}{\mathrm{L}}X$ is a
principal ${\mathrm{L}}G$-bundle.

###### Proof.

Since ${\mathrm{L}}$ preserves finite products, the first claim follows
readily from Definition 3.33.

For the second claim, recall that $P{//}G\to X$ is a principal $G$-bundle
precisely if the map $|P{//}G|\to X$ is an equivalence. Applying the functor
${\mathrm{L}}$ to this morphism, we obtain an equivalence
${\mathrm{L}}|P{//}G|\overset{\simeq}{\longrightarrow}{\mathrm{L}}X$. Since
${\mathrm{L}}$ preserves geometric realisations, and using claim (1), we
obtain canonical equivalences

$|{\mathrm{L}}P{//}{\mathrm{L}}G|\overset{\simeq}{\longrightarrow}{\mathrm{L}}|P{//}G|\overset{\simeq}{\longrightarrow}{\mathrm{L}}X\,,$
(3.61)

which establishes the action
${\mathrm{L}}P{//}{\mathrm{L}}G\xrightarrow{{\mathrm{L}}p}{\mathrm{L}}X$ as a
principal ${\mathrm{L}}G$-bundle over $X$. ∎

###### Remark 3.62

The proof of Theorem 3.60 would fail if it were not automatic that group
actions are groupoid objects (Theorem 3.37), since ${\mathrm{L}}$ does not
preserve groupoid objects in general. $\triangleleft$

###### Proposition 3.63

Let ${\widehat{G}}$ be a group object in ${\mathbf{H}}$, and let $P{//}G\to Y$
be a $G$-principal $\infty$-bundle in ${\mathbf{H}}$. For any morphism
$f\colon X\to Y$ in ${\mathbf{H}}$, there is a canonical $G$-action over $X$
on the pullback $Q\coloneqq X\times_{Y}P$ that makes $Q{//}G\to X$ into a
$G$-principal $\infty$-bundle on $X$.

###### Proof.

Let
$\mathsf{c}\colon{\mathbf{H}}\longrightarrow{\mathscr{F}\mathrm{un}}({\mathbbe{\Delta}}^{\mathrm{op}},{\mathbf{H}})$
be the constant-diagram functor. Consider the pullback diagram

${\mathsf{c}X\times_{\mathsf{c}Y}(P{//}G)}$${P{//}G}$${\mathsf{c}X}$${\mathsf{c}Y}$$\scriptstyle{\hat{f}}$$\scriptstyle{f^{*}p}$$\scriptstyle{p}$$\scriptstyle{\mathsf{c}f}$
(3.64)

in ${\mathscr{F}\mathrm{un}}({\mathbbe{\Delta}}^{\mathrm{op}},{\mathbf{H}})$
(or, equivalently, in ${\mathscr{G}\mathrm{pd}}({\mathbf{H}})$). For any
$[n]\in{\mathbbe{\Delta}}$ there exists a canonical equivalence

$\big{(}\mathsf{c}X\times_{\mathsf{c}Y}(P{//}G)\big{)}_{n}\simeq
X\times_{Y}(P\times G^{n-1})\simeq(X\times_{Y}P)\times G^{n-1}\,.$ (3.65)

We use Lemma 3.25 to obtain from these equivalences a canonical pair (up to
contractible choices) of an object
$(X\times_{Y}P){//}G\in{\mathscr{F}\mathrm{un}}({\mathbbe{\Delta}}^{\mathrm{op}},{\mathbf{H}})$,
with $((X\times_{Y}P){//}G)_{n}=(X\times_{Y}P)\times G^{n}$ for all
$n\in\mathbb{N}_{0}$, together with an equivalence

$(X\times_{Y}P){//}G\overset{\simeq}{\longrightarrow}\mathsf{c}X\times_{\mathsf{c}Y}(P{//}G)$
(3.66)

of simplicial objects in ${\mathbf{H}}$. By a slight abuse of notation, we
also denote the composition

$(X\times_{Y}P){//}G\overset{\simeq}{\longrightarrow}\mathsf{c}X\times_{\mathsf{c}Y}(P{//}G)\longrightarrow\mathsf{c}X$
(3.67)

by $f^{*}p$. It follows by construction that
$(X\times_{Y}P){//}G\overset{f^{*}p}{\longrightarrow}X$ is a $G$-action over
$X$. We are hence left to show that it is a principal $\infty$-bundle.

To that end, we will show that the morphism

$|(X\times_{Y}P){//}G|\longrightarrow X$ (3.68)

is an equivalence (compare Definition 3.50). Diagram (3.64) is a diagram of
the form
$\Delta^{1}\times\Delta^{1}\longrightarrow{\mathscr{F}\mathrm{un}}({\mathbbe{\Delta}}^{\mathrm{op}},{\mathbf{H}})$.
Composing with the functor
${\mathrm{colim}}_{{\mathbbe{\Delta}}^{\mathrm{op}}}^{\mathbf{H}}=|{-}|\colon{\mathscr{F}\mathrm{un}}({\mathbbe{\Delta}}^{\mathrm{op}},{\mathbf{H}})\longrightarrow{\mathbf{H}}$
we obtain a diagram

${{|(X\times_{Y}P){//}G|}}$${{|P{//}G|}}$${X}$${Y}$$\scriptstyle{|\hat{f}|}$$\scriptstyle{f^{*}p}$$\scriptstyle{p}$$\scriptstyle{\simeq}$$\scriptstyle{f}$
(3.69)

in ${\mathbf{H}}$. The right-hand morphism is an equivalence since $P{//}G\to
Y$ is assumed to be a principal $\infty$-bundle. Using the equivalence (3.66),
diagram (3.69) is equivalent to the diagram

${{|\mathsf{c}X\times_{\mathsf{c}Y}(P{//}G)|}}$${{|P{//}G|}}$${X}$${Y}$$\scriptstyle{|\hat{f}|}$$\scriptstyle{f^{*}p}$$\scriptstyle{p}$$\scriptstyle{\simeq}$$\scriptstyle{f}$
(3.70)

By the universality of colimits in ${\mathbf{H}}$, we have a canonical
equivalence

$|\mathsf{c}X\times_{\mathsf{c}Y}(P{//}G)|\simeq X\times_{Y}|P{//}G|\,.$
(3.71)

This establishes that the morphism $f^{*}p$ in diagram (3.70) is the pullback
of an equivalence in ${\mathbf{H}}$, and hence that $f^{*}p$ is an equivalence
itself. ∎

One can now even show that every $G$-principal $\infty$-bundle arises as a
pullback of the bundle $(*{//}G)\to{\mathrm{B}}G$. This insight is not new,
but has been observed in [21, Prop. 3.13, Thm. 3.17] already. However, in
Section 4.3 it will be important to have a good understanding of the
classifying map of a principal $\infty$-bundle, and so we include a brief
treatment of these maps. We start with two short technical lemmas, before
constructing for each $G$-principal $\infty$-bundle in ${\mathbf{H}}$ its
classifying map.

###### Lemma 3.72

Let ${\widehat{G}}$ be a group object in ${\mathbf{H}}$, and let
$f\colon*\to{\mathrm{B}}G$ be the base point of ${\mathrm{B}}G$. The pullback
of the canonical bundle $(*{//}G)\to{\mathrm{B}}G$ along $f$ agrees with the
bundle $G{//}G\to*$.

###### Proof.

Consider the commutative square

${G{//}G}$${*{//}G}$${*}$${\mathsf{c}{\mathrm{B}}G}$ (3.73)

in ${\mathscr{F}\mathrm{un}}({\mathbbe{\Delta}}^{\mathrm{op}},{\mathbf{H}})$.
By the canonical equivalence $G\simeq\Omega{\mathrm{B}}G$, this diagram is
level-wise a pullback, i.e. it is a pullback diagram in
${\mathscr{F}\mathrm{un}}({\mathbbe{\Delta}}^{\mathrm{op}},{\mathbf{H}})$.
That proves the claim by Proposition 3.63. ∎

###### Definition 3.74

A $G$-principal $\infty$-bundle $P{//}G\to X$ is _trivial_ if it is equivalent
in ${\mathscr{B}\mathrm{un}}_{G}(X)$ to the _trivial $G$-principal
$\infty$-bundle_ $X\times(G{//}G)\to X$, i.e. if there is an equivalence of
simplicial objects in ${\mathbf{H}}_{/X}$ between $P{//}G$ and
$X\times(G{//}G)$ that commutes with the canonical morphisms to $*{//}G$.

###### Lemma 3.75

_[ 21, Prop. 3.12]_ For every $G$-principal $\infty$-bundle $P{//}G\to X$ in
${\mathbf{H}}$, there exists an effective epimorphism $U\to X$ such that the
pullback bundle $U\times_{X}(P{//}G)$ is trivial.

###### Proof.

We give an alternative proof to [21]. Given a $G$-principal $\infty$-bundle
$P{//}G\to X$ in ${\mathbf{H}}$, consider the effective epimorphism $P\to X$
and the pullback bundle $P\times_{X}(P{//}G)$. We have a commutative diagram

${P\times(G{//}G)}$${P\times_{X}(P{//}G)}$${P{//}G}$${\mathsf{c}P}$${\mathsf{c}X}$$\scriptstyle{\mathrm{pr}}$$\scriptstyle{a\times
1}$$\scriptstyle{\psi}$ (3.76)

in ${\mathscr{F}\mathrm{un}}({\mathbbe{\Delta}}^{\mathrm{op}},{\mathbf{H}})$,
where $a\times 1$ acts on $P$ with the first copy of $G$ and as the identity
on the remaining copies of $G$. The induced morphism $\psi$ is a morphism of
$G$-principal $\infty$-bundles (since the triangles in the diagram commute and
since $a\times 1$ is a morphism of $G$-actions). It is thus equivalent to a
morphism

$\psi^{\prime}\colon{\check{C}}(P\times G\to
P)\longrightarrow{\check{C}}\big{(}(P\times_{X}(P{//}G))\longrightarrow
P\big{)}$ (3.77)

of Čech nerves over $P$. The level-zero component of $\psi^{\prime}$ is
precisely the equivalence $P\times G\to P\times_{X}P$ which establishes that
$P{//}G\to X$ is principal (cf. Proposition 3.56). Since $\psi^{\prime}$ is
the image of $\psi$ under the right Kan extension $\mathrm{Ran}_{\iota}$
(compare Definition 2.19), it follows that $\psi^{\prime}$, and hence $\psi$,
is an equivalence. ∎

###### Proposition 3.78

For every $G$-principal $\infty$-bundle $P{//}G\to X$ in ${\mathbf{H}}$, the
diagram

${P{//}G}$${*{//}G}$${\mathsf{c}X}$${\mathsf{c}{\mathrm{B}}G}$$\scriptstyle{p}$$\scriptstyle{|p|}$
(3.79)

is a pullback diagram in
${\mathscr{F}\mathrm{un}}({\mathbbe{\Delta}}^{\mathrm{op}},{\mathbf{H}})$:
there is an equivalence $(P{//}G\to X)\simeq\times_{{\mathrm{B}}G}(*{//}G)$ of
$G$-principal $\infty$-bundles over $X$. In particular, every $G$-principal
$\infty$-bundle is a pullback of the bundle $*{//}G\to{\mathrm{B}}G$.

This is a refinement of [21, Prop. 3.13] to a statement on the level of
simplicial objects, rather than only on their zeroth level.

###### Proof.

Consider the diagram

${G{//}G}$${*{//}G}$${P\times(G{//}G)}$${P{//}G}$${*}$${\mathsf{c}{\mathrm{B}}G}$${\mathsf{c}P}$${\mathsf{c}X}$
(3.80)

in ${\mathscr{F}\mathrm{un}}({\mathbbe{\Delta}}^{\mathrm{op}},{\mathbf{H}})$.
Here, the front and back squares are pullbacks (by Lemmas 3.75 and 3.72), and
the diagram is obtained as a morphism of pullback diagrams. We need to show
that the right-hand face is a pullback square in
${\mathscr{F}\mathrm{un}}({\mathbbe{\Delta}}^{\mathrm{op}},{\mathbf{H}})$.

First, we show that the top square of (3.80) is a pullback. By Lemma 3.16 it
suffices to check this level-wise: at simplicial level $n=0$, it is trivial.
For $n\in\mathbb{N}$, the square consists of the the image under the functor
$(-)\times G^{n-1}$ of the diagram

${P\times G}$${P}$${G}$${*}$$\scriptstyle{a}$$\scriptstyle{\mathrm{pr}_{1}}$
(3.81)

This a pullback diagram: there is a commutative diagram

${P\times G}$${P\times
G}$${P}$${G}$${*}$$\scriptstyle{\mathrm{pr}_{1}}$$\scriptstyle{\mathrm{pr}_{0}}$$\scriptstyle{g}$$\scriptstyle{a}$$\scriptstyle{\mathrm{pr}_{1}}$
(3.82)

in which the dashed morphism is given by $g=(a\times
1_{G})\circ(1_{P}\times{\mathrm{inv}}\times
1_{G})\circ(1_{P}\circ\Delta_{G})$, where $\Delta_{G}\colon G\to G^{2}$ is the
diagonal morphism, and where ${\mathrm{inv}}\colon G\to G$ is the choice of an
inverse in $G$: since the group object
${\widehat{G}}\in{\mathscr{G}\mathrm{rp}}({\mathbf{H}})$ is in particular a
groupoid object, we have a diagram

${{\widehat{G}}_{1}\simeq{\widehat{G}}_{1}\times*}$${{\widehat{G}}_{1}\times{\widehat{G}}_{1}\simeq{\widehat{G}}(\Lambda^{2}_{0})}$${{\widehat{G}}_{2}}$${{\widehat{G}}_{1}\,,}$$\scriptstyle{\simeq}$$\scriptstyle{d_{0}}$
(3.83)

where we have used the characterisation of groupoid objects as certain
category objects from Proposition A.13. Choosing an inverse for the right-
facing morphism defines the morphism ${\mathrm{inv}}$.

Since $g$ is an equivalence (because ${\widehat{G}}$ is a group object),
diagram (3.81) is a pullback in ${\mathbf{H}}$, and since the span category
$\\{0,1\\}\leftarrow\\{0\\}\rightarrow\\{0,2\\}$ has contractible nerve, the
pullback (3.81) is preserved by $(-)\times G^{n-1}$ (see Lemma A.21). We thus
obtain that the top square in diagram (3.80) is a pullback.

Next, we prove that the bottom square of (3.80) is a pullback. We define
$Y\coloneqq*\times_{{\mathrm{B}}G}X\in{\mathbf{H}}$, and we consider the
diagram (omitting constant-diagram functors)

${Y\times_{X}(P{//}G)}$${Y}$${*}$${P{//}G}$${X}$${{\mathrm{B}}G}$ (3.84)

Both squares in this diagram are pullbacks in
${\mathscr{F}\mathrm{un}}({\mathbbe{\Delta}}^{\mathrm{op}},{\mathbf{H}})$, so
that the pasting law yields a canonical equivalence of simplicial objects

$Y\underset{X}{\times}(P{//}G)\simeq*\underset{{\mathrm{B}}G}{\times}(P{//}G)\,.$
(3.85)

Observe that $Y\times_{X}(P{//}G)\to Y$ is a $G$-principal $\infty$-bundle by
Proposition 3.63, so that

$Y\simeq\big{|}Y\underset{X}{\times}(P{//}G)\big{|}\simeq\big{|}*\underset{{\mathrm{B}}G}{\times}(P{//}G)\big{|}\,.$
(3.86)

Now we use that the morphism $P{//}G\to{\mathrm{B}}G$ factors through $*{//}G$
(by Definitions 3.33 and 3.50) and that $*\times_{{\mathrm{B}}G}(*{//}G)\simeq
G{//}G$ (by Lemma 3.72). Applying the pasting law to the diagram

${(P{//}G)\underset{(*{//}G)}{\times}(G{//}G)}$${G{//}G}$${*}$${P{//}G}$${*{//}G}$${{\mathrm{B}}G}$
(3.87)

in ${\mathscr{F}\mathrm{un}}({\mathbbe{\Delta}}^{\mathrm{op}},{\mathbf{H}})$,
in which both squares are pullbacks, we obtain a canonical equivalence

$*\underset{{\mathrm{B}}G}{\times}(P{//}G)\simeq(P{//}G)\underset{(*{//}G)}{\times}(G{//}G)\,.$
(3.88)

The right-hand side is precisely the pullback described by the top square in
diagram (3.80), and is hence canonically equivalent to $P\times(G{//}G)$ in
${\mathscr{F}\mathrm{un}}({\mathbbe{\Delta}}^{\mathrm{op}},{\mathbf{H}})$.
Thus, it follows that

$Y\simeq\big{|}*\underset{{\mathrm{B}}G}{\times}(P{//}G)\big{|}\simeq\big{|}P\times(G{//}G)\big{|}\simeq
P\,.$ (3.89)

The last equivalence can be seen either by combining Proposition 3.45 with the
fact that $|{-}|$ preserves finite products (because
${\mathbbe{\Delta}}^{\mathrm{op}}$ is sifted [20]), or simply by recalling
that $P\times(G{//}G)\to P$ is a $G$-principal $\infty$-bundle on $P$. This
shows that the bottom square in (3.80) is a pullback.

Finally, we prove that the right-hand square in (3.80) is a pullback as well.
Consider the commutative diagram of solid arrows

${P{//}G}$${X\underset{{\mathrm{B}}G}{\times}(*{//}G)}$${*{//}G}$${X}$${{\mathrm{B}}G}$$\scriptstyle{\varphi}$
(3.90)

which induces an essentially unique morphism $\varphi$ of simplicial objects
in ${\mathbf{H}}$. By the commutativity of the right-hand triangle in this
diagram $\varphi$ is even a morphism of $G$-actions, and by the commutativity
of the left-hand triangle it is a morphism of $G$-actions over $X$. Since its
source and target are $G$-principal $\infty$-bundles, $\varphi$ is equivalent
to a morphism of Čech nerves

$\varphi^{\prime}\colon{\check{C}}(P\to
X)\longrightarrow{\check{C}}\big{(}(X\underset{{\mathrm{B}}G}{\times}*)\to
X\big{)}\,.$ (3.91)

That is, $\varphi^{\prime}$ is the image under $\mathrm{Ran}_{\iota}$ (compare
Definition 2.19) of the square

${P}$${X\underset{{\mathrm{B}}G}{\times}*}$${X}$${X}$$\scriptstyle{\varphi^{\prime}_{0}}$
(3.92)

This $\varphi^{\prime}_{0}$ is an equivalence precisely because the bottom
square of (3.80) is a pullback. Consequently, the morphism $\varphi$ is an
equivalence in
${\mathscr{F}\mathrm{un}}({\mathbbe{\Delta}}^{\mathrm{op}},{\mathbf{H}})$, and
thus the right-hand face in (3.80) is a pullback. ∎

###### Corollary 3.93

Let $P{//}G\to X$ be a $G$-principal $\infty$-bundle in ${\mathbf{H}}$. For
any morphism $x\colon*\to X$, we have a pullback diagram

${G{//}G}$${P{//}G}$${*}$${X}$$\scriptstyle{x}$ (3.94)

in ${\mathscr{F}\mathrm{un}}({\mathbbe{\Delta}}^{\mathrm{op}},{\mathbf{H}})$.
In particular, any fibre of $P\to X$ is canonically equivalent to $G$ in
${\mathbf{H}}$.

###### Remark 3.95

In fact, for any group object ${\widehat{G}}$ in ${\mathbf{H}}$ and any object
$X\in{\mathbf{H}}$, there is an equivalence

${\mathscr{B}\mathrm{un}}_{G}(X)\simeq\underline{{\mathbf{H}}}(X,{\mathrm{B}}G)$
(3.96)

between the $\infty$-category of $G$-principal $\infty$-bundles on $X$ and the
mapping space $\underline{{\mathbf{H}}}(X,{\mathrm{B}}G)$ [21, Thm. 3.17].
This implies that every morphism of principal $G$-bundles on $X$ is an
equivalence. Proposition 3.78 feeds into the proof of this equivalence by
showing that the functor
$\underline{{\mathbf{H}}}(X,{\mathrm{B}}G)\to{\mathscr{B}\mathrm{un}}_{G}(X)$,
sending a morphism $X\to{\mathrm{B}}G$ to the principal $\infty$-bundle
$X\times_{{\mathrm{B}}G}(*{//}G)$, is fully faithful. $\triangleleft$

In particular, under the equivalence of Remark 3.95, the morphism $|p|\colon
X\to{\mathrm{B}}G$ in diagram (3.79) is a classifying morphism for the bundle
$P{//}G\to X$.

###### Proposition 3.97

Let ${\mathrm{L}}\colon{\mathbf{H}}\to{\mathbf{H}}^{\prime}$ be a functor of
$\infty$-topoi that preserves finite products and geometric realisations. If
$P{//}G\to X$ is a $G$-principal $\infty$-bundle in ${\mathbf{H}}$, classified
(up to canonical equivalence) by a morphism $|p|\colon X\to{\mathrm{B}}G$,
then the ${\mathrm{L}}G$-principal $\infty$-bundle
${\mathrm{L}}P{//}{\mathrm{L}}G\longrightarrow{\mathrm{L}}X$ (compare Theorem
3.60) in ${\mathbf{H}}^{\prime}$ is classified by the morphism
$|{\mathrm{L}}p|\simeq{\mathrm{L}}|p|$.

###### Proof.

Consider the commutative diagram

${{\mathrm{L}}(P{//}G)}$${{\mathrm{L}}(*{//}G)}$${{\mathrm{L}}P{//}{\mathrm{L}}G}$${*{//}{\mathrm{L}}G}$${{\mathrm{L}}|P{//}G|}$${{\mathrm{L}}{\mathrm{B}}G}$${{|{\mathrm{L}}P{//}{\mathrm{L}}G|}}$${{\mathrm{B}}{\mathrm{L}}G}$$\scriptstyle{{\mathrm{L}}p}$$\scriptstyle{q}$$\scriptstyle{\simeq}$$\scriptstyle{\simeq}$$\scriptstyle{{\mathrm{L}}|p|}$$\scriptstyle{|q|}$$\scriptstyle{\simeq}$$\scriptstyle{\simeq}$
(3.98)

The morphism $q$ is the canonical morphism induced from the collapse morphism
${\mathrm{L}}P\to*$. The front face of this diagram is a pullback in
${\mathbf{H}}^{\prime}$, witnessing $|q|$ as the classifying morphism
${\mathrm{L}}X\to{\mathrm{B}}{\mathrm{L}}G$ of the bundle
${\mathrm{L}}P{//}{\mathrm{L}}G\longrightarrow{\mathrm{L}}X$. Since all
diagonal morphisms are equivalences, the back face of the diagram is a
pullback as well, showing that ${\mathrm{L}}|p|$ is a classifying morphism of
the ${\mathrm{L}}G$-principal $\infty$-bundle
${\mathrm{L}}(P{//}G)\longrightarrow{\mathrm{L}}X$, which is equivalent to the
bundle ${\mathrm{L}}P{//}{\mathrm{L}}G\longrightarrow{\mathrm{L}}X$. Finally,
since the diagonal morphisms arise from the natural equivalences
${\mathrm{L}}\circ|{-}|\simeq|{-}|\circ{\mathrm{L}}$, it follows that
$|q|\simeq|{\mathrm{L}}p|$. ∎

We now state several alternative characterisations of group extensions in
$\infty$-topoi. These clarify the relation between the original notion of an
extension of group objects from Definition 3.23 and more direct
categorifications of several perspectives on group extensions in
${\mathscr{S}\mathrm{et}}$. The last of these alternative characterisations
will be important in Section 4.3.

###### Theorem 3.99

Let ${\mathbf{H}}$ be an $\infty$-topos, and let
${\widehat{A}}\xrightarrow{\widehat{\iota}}{\widehat{G}}\xrightarrow{\widehat{p}}{\widehat{H}}$
be a sequence of morphisms in ${\mathscr{G}\mathrm{rp}}({\mathbf{H}})$. The
following are equivalent:

  1. (1)

${\widehat{A}}\xrightarrow{\widehat{\iota}}{\widehat{G}}\xrightarrow{\widehat{p}}{\widehat{H}}$
is an extension of group objects in ${\mathbf{H}}$ (see Definition 3.23).

  2. (2)

The sequence
${\widehat{A}}\xrightarrow{\widehat{\iota}}{\widehat{G}}\xrightarrow{\widehat{p}}{\widehat{H}}$
is a fibre sequence in ${\mathscr{G}\mathrm{rp}}({\mathbf{H}})$.

  3. (3)

The sequence $A\xrightarrow{\iota}G\xrightarrow{p}H$ is a fibre sequence in
${\mathbf{H}}$.

  4. (4)

The map $p\colon G\to H$ together with the action $G{//}A$ of $A$ on $G$
induced by $\iota$ define a principal $A$-bundle over $H$.

###### Proof.

$(1)\Rightarrow(3)$: This implication has been proven in [21] already. We
import the proof for completeness: consider the diagram

${A}$${G}$${*}$${*}$${H}$${{\mathrm{B}}A}$${*}$${*}$${{\mathrm{B}}G}$${{\mathrm{B}}H}$$\scriptstyle{\iota^{\prime}}$$\scriptstyle{p^{\prime}}$$\scriptstyle{{\mathrm{B}}\iota}$$\scriptstyle{{\mathrm{B}}p}$
(3.100)

in ${\mathbf{H}}$. Every square in diagram (3.100) is a pullback square (this
assumes (1)). It thus follows that the sequence
$A\xrightarrow{\iota^{\prime}}G\xrightarrow{p^{\prime}}H$ is a fibre sequence
in ${\mathbf{H}}$. By this construction, the morphisms $\iota^{\prime}$ and
$p^{\prime}$ coincide with the morphisms $\Omega\circ{\mathrm{B}}(\iota)$ and
$\Omega\circ{\mathrm{B}}(p)$, respectively. The natural equivalence
$\Omega\circ{\mathrm{B}}\simeq{\mathrm{id}}_{\mathbf{H}}$ then yields that
also $A\xrightarrow{\iota}G\xrightarrow{p}H$ is a fibre sequence in
${\mathbf{H}}$. Observe that every vertical morphism in diagram (3.100) is an
effective epimorphism since $*\to{\mathrm{B}}G$ is an effective epimorphism
for every group object
${\widehat{G}}\in{\mathscr{G}\mathrm{rp}}({\mathbf{H}})$ and since effective
epimorphisms are stable under pullback.

$(3)\Rightarrow(2)$: This follows from Proposition 3.16.

$(2)\Rightarrow(1)$: This implication holds because the delooping functor
${\mathrm{B}}$ is a right adjoint.

$(4)\Rightarrow(3)$: Consider the commutative diagram

${A}$${G}$${*}$${*}$${H}$${{\mathrm{B}}A}$ (3.101)

The outer rectangle is a pullback diagram by construction of ${\mathrm{B}}A$,
and the right-hand square is a pullback by Corollary 3.93. Thus, the
implication follows by applying the pasting law for pullbacks.

$(1)\Rightarrow(4)$: Recall that the morphism $*\to{\mathrm{B}}H$ is an
effective epimorphism and that effective epimorphisms are stable under
pullback. It then readily follows from the pasting diagram (3.100) that
$p\colon G\to H$ is an effective epimorphism. By Proposition 3.63, the
morphism $H\to{\mathrm{B}}A$ induces an $A$-principal $\infty$-bundle over $H$
as the pullback of $*{//}A\to{\mathrm{B}}A$. We know from the pasting
construction (3.100) that the level-zero object of the induced pullback
principal $\infty$-bundle is (canonically) equivalent to $G$. Further, we have
that the map $G\to H$ induced from the pullback construction coincides with
$p\colon G\to H$ under this equivalence (as in the proof of
“$(1)\Rightarrow(3)$”).

It thus remains to show that the action of $A$ on $G$ obtained via the upper
central pullback square in (3.100) and the pullback construction in
Proposition 3.63 coincides with the action $G{//}A$ induced from the morphism
$\widehat{\iota}\colon{\widehat{A}}\to{\widehat{G}}$ in
${\mathscr{G}\mathrm{rp}}({\mathbf{H}})$ (cf. Proposition 3.47).

To that end, we consider the following diagram in
${\mathscr{F}\mathrm{un}}({\mathbbe{\Delta}}^{\mathrm{op}},{\mathbf{H}})$:

${G{//}G}$${*{//}G}$${(H\times_{{\mathrm{B}}A}*){//}A}$${*{//}A}$${*}$${\mathsf{c}{\mathrm{B}}G}$${\mathsf{c}H}$${\mathsf{c}{\mathrm{B}}A}$$\scriptstyle{\mathrm{pr}_{H}}$$\scriptstyle{g}$$\scriptstyle{\iota}$$\scriptstyle{\mathsf{c}{\mathrm{B}}\iota}$
(3.102)

The simplicial object in the front, upper-left corner is obtained from the
pullback construction for $A$-principal $\infty$-bundles (Proposition 3.63).
Therefore, the front face of the cube in diagram (3.102) is a pullback square
in ${\mathscr{F}\mathrm{un}}({\mathbbe{\Delta}}^{\mathrm{op}},{\mathbf{H}})$.
Further, the bottom square is a pullback by assuming (1) and by (3.100), and
the back square is a pullback diagram because it is so objectwise (see also
Lemma 3.72). Forgetting for a moment about the two upper left objects, the
remaining diagram can be viewed as a morphism of cospans in ${\mathbf{H}}$.
This induces an essentially unique morphism of the left and right pullback
squares, and this is how we define the upper-left diagonal morphism, labelled
$g$, and the remaining coherence data of the cube (3.102).

Applying the pasting law to the front and bottom square, we deduce that the
diagonal square

${(H\times_{{\mathrm{B}}A}*){//}A}$${*{//}A}$${*}$${\mathsf{c}{\mathrm{B}}G}$
(3.103)

is a pullback diagram in
${\mathscr{F}\mathrm{un}}({\mathbbe{\Delta}}^{\mathrm{op}},{\mathbf{H}})$. Its
vertical morphisms are the diagonals of the left and right faces of the cube
in diagram (3.102). This, in turn, yields a commutative diagram

${(H\times_{{\mathrm{B}}A}*){//}A}$${G{//}G}$${*}$${*{//}A}$${*{//}G}$${\mathsf{c}{\mathrm{B}}G}$
(3.104)

in which the outer square and the right-hand square are pullbacks. (These
squares are precisely the top and the back squares of diagram (3.102).) The
pasting law for pullbacks thus implies that the left-hand square is a pullback
diagram in
${\mathscr{F}\mathrm{un}}({\mathbbe{\Delta}}^{\mathrm{op}},{\mathbf{H}})$ as
well. By construction of the action $G{//}A$ induced from the morphism
$\widehat{\iota}\colon{\widehat{A}}\to{\widehat{G}}$ in
${\mathscr{G}\mathrm{rp}}({\mathbf{H}})$ (Proposition 3.47) as a pullback in
${\mathscr{F}\mathrm{un}}({\mathbbe{\Delta}}^{\mathrm{op}},{\mathbf{H}})$, we
thus obtain a canonical equivalence

$(H\times_{{\mathrm{B}}A}*){//}A\simeq G{//}A$ (3.105)

in ${\mathscr{G}\mathrm{pd}}({\mathbf{H}})$. That is, the two actions of $A$
on $G$ agree, as desired. Since the action $(H\times_{{\mathrm{B}}A}*){//}A$
is principal by Proposition 3.63, it follows that the action $G{//}A$ is
principal as well. This proves the claim. ∎

###### Corollary 3.106

Suppose
${\widehat{A}}\xrightarrow{\widehat{\iota}}{\widehat{G}}\xrightarrow{\widehat{p}}{\widehat{H}}$
is an extension of group objects in ${\mathbf{H}}$. Then, there is a canonical
equivalence in ${\mathbf{H}}$,

$|G{//}A|\simeq H\,.$ (3.107)

This is the $\infty$-categorical analogue of the canonical isomorphism
$G/A\cong H$ for ordinary (set-theoretic) group extensions $A\to G\to H$.

###### Corollary 3.108

Let ${\mathrm{L}}\colon{\mathbf{H}}\to{\mathbf{H}}^{\prime}$ be a functor
between $\infty$-topoi that preserves geometric realisations and finite
products. Suppose
${\widehat{A}}\xrightarrow{\widehat{\iota}}{\widehat{G}}\xrightarrow{\widehat{p}}{\widehat{H}}$
is an extension of group objects in ${\mathbf{H}}$. Then, the sequence
${\mathrm{L}}{\widehat{A}}\xrightarrow{{\mathrm{L}}\widehat{\iota}}{\mathrm{L}}{\widehat{G}}\xrightarrow{{\mathrm{L}}\widehat{p}}{\mathrm{L}}{\widehat{H}}$
is an extension of group objects in ${\mathbf{H}}^{\prime}$.

###### Proof.

This statement now follows from combining Theorem 3.60 and Theorem 3.99. ∎

##  4 Homotopy-theoretic smooth string group models

In this section, we present a definition of string group extensions within the
$\infty$-category ${\mathbf{H}}_{\infty}$ of smooth spaces. This relies on the
singular complex functor
$\mathrm{S}_{e}\colon{\mathbf{H}}_{\infty}\to{\mathbf{S}}$ for smooth spaces
from Section 2 and the theory of group extensions in $\infty$-topoi from
Section 3. We begin by recalling the definition of a string group extension in
the $\infty$-category ${\mathbf{S}}$ of spaces. Then, we use our results thus
far to transfer this definition to ${\mathbf{H}}_{\infty}$ along the functor
$\mathrm{S}_{e}$, leading to a homotopy-theoretic definition of smooth string
group extensions (Definition 4.3).

After recalling some background on bundle gerbes in Section 4.2, we provide
new smooth models for the string group in Section 4.3, building on recent
constructions of smooth 2-group extensions in [3]. (Already in that paper,
evidence was given that these smooth 2-group extensions can model the string
group; here we provide a full formal framework and proof for that conjecture.)

###  4.1 The definition of smooth string groups

The definition of a string group via the Whitehead tower (see Section 1) is
originally purely homotopy-theoretic. In particular, in a string group
extension $A\to{\mathrm{String}}(H)\to H$ the extending group $A$ is not
fixed, but only its homotopy type. So far, to our knowledge there does not
exist a definition of string group extensions in a smooth context that
contains this flexibility—the extending group $A$ is usually chosen ad hoc to
be some smooth version of ${\mathrm{B}}\mathsf{U}(1)$. Here, we provide a
smooth version of the original homotopy-theoretic definition (see Definition
4.3). In particular, only the underlying homotopy type of the extending smooth
group $A$ is fixed in this definition.

Recall that any compact, simple, and simply connected Lie group $H$ is also
2-connected and satisfies $\mathrm{H}^{3}(H;\mathbb{Z})\cong\mathbb{Z}$. Any
Lie group $H$ defines a group object ${\widehat{H}}$ in the $\infty$-topos of
spaces ${\mathbf{S}}$. We start by reformulating the definition of a string
group extension of topological groups within the $\infty$-category of spaces:

###### Definition 4.1

Let $H$ be a compact, simple, and simply connected Lie group, and denote by
${\widehat{H}}\in{\mathscr{G}\mathrm{rp}}({\mathbf{S}})$ its associated group
object in ${\mathbf{S}}$. A _string group extension of $H$_ is an extension of
group objects ${\widehat{A}}\xrightarrow{\ \widehat{\iota}\
}\widehat{{\mathrm{String}}(H)}\xrightarrow{\ \widehat{p}\ }{\widehat{H}}$ in
${\mathbf{S}}$ such that

  1. (1)

$A$ is an Eilenberg-MacLane space $K(\mathbb{Z},2)$, and

  2. (2)

under the isomorphism

$\pi_{0}\underline{{\mathbf{S}}}(H,{\mathrm{B}}A)\cong\pi_{0}\underline{{\mathbf{S}}}\big{(}H,K(\mathbb{Z},3)\big{)}\cong\mathrm{H}^{3}(H;\mathbb{Z})\cong\mathbb{Z}\,,$
(4.2)

the classifying morphism $H\to{\mathrm{B}}A$ of the $A$-principal
$\infty$-bundle ${\mathrm{String}}(H){//}A\to H$ (compare Remark 3.95 and
Theorem 3.99(4)) represents a generator of $\mathbb{Z}$.

Given condition (1), condition (2) is equivalent to saying that the map $G\to
H$ of spaces induces an isomorphism $\pi_{i}(G)\to\pi_{i}(H)$ for $i\neq 3$
and that $\pi_{3}(G)\cong 0$. This is a consequence of the Hurewicz Theorem,
the Universal Coefficient Theorem, and the long exact sequence of homotopy
groups associated to a (homotopy) fibre sequence of spaces. That is, $G$ is a
3-connected approximation to $H$.

Recall the $\infty$-topos
${\mathbf{H}}_{\infty}=\mathscr{P}({\mathscr{C}\mathrm{art}})$ from Section
2.1. There we also introduced the localisation $L_{I}{\mathbf{H}}_{\infty}$ of
${\mathbf{H}}_{\infty}$ at the set $I=\\{c\times\mathbb{R}\to
c\,|\,c\in{\mathscr{C}\mathrm{art}}\\}$ and the smooth singular complex
functor $\mathrm{S}_{e}\colon{\mathbf{H}}_{\infty}\to{\mathbf{S}}$. Further,
recall the fully faithful embedding
$\underline{(-)}\colon{\mathscr{M}\mathrm{fd}}\hookrightarrow{\mathbf{H}}_{\infty}$,
with $\underline{M}(c)={\mathscr{M}\mathrm{fd}}(c,M)$; under this embedding,
any Lie group $H$ gives rise to a group object $\underline{{\widehat{H}}}$ in
${\mathbf{H}}_{\infty}$. We can now use our results from Section 3 to transfer
the definition of a string group extension to the $\infty$-topos
${\mathbf{H}}_{\infty}$:

###### Definition 4.3

Let $H$ be a compact, simple, and simply connected Lie group. A _smooth string
group extension of $H$_ is an extension ${\widehat{A}}\xrightarrow{\
\widehat{\iota}\ }\widehat{{\mathrm{String}}(\underline{H})}\xrightarrow{\
\widehat{p}\ }\widehat{\underline{H}}$ of group objects in
${\mathbf{H}}_{\infty}$ such that its image under $\mathrm{S}_{e}$ is a string
group extension in ${\mathbf{S}}$.

Note that by Theorem 3.108 the functor $\mathrm{S}_{e}$ maps group extensions
in ${\mathbf{H}}_{\infty}$ to group extensions in ${\mathbf{S}}$. Further,
even though $\mathrm{S}_{e}$ induces an equivalence between ${\mathbf{S}}$ and
the localisation $L_{I}{\mathbf{H}}_{\infty}$ rather than the full
$\infty$-category ${\mathbf{H}}_{\infty}$, we do not need to demand that $A$,
${\mathrm{String}}(H)$ and $H$ are local objects, because $\mathrm{S}_{e}$
sends all $I$-local equivalences in ${\mathbf{H}}_{\infty}$ to equivalences in
${\mathbf{S}}$ (Theorem 2.6(1)).

Definition 4.3 is a generalisation as well as a weakening of the following
approach to smooth string group extensions (see, for instance, [13]): there,
one works in the localisation $L_{\tau}{\mathbf{H}}_{\infty}$ of
${\mathbf{H}}_{\infty}$ at the differentiably good open coverings $\\{c_{a}\to
c\\}_{a\in\Lambda}$ of cartesian spaces $c\in{\mathscr{C}\mathrm{art}}$, and
one defines a string group extension of $H$ via the pullback

${{\mathrm{B}}{\mathrm{String}}(\underline{H})}$${*}$${{\mathrm{B}}\underline{H}}$${{\mathrm{B}}^{3}\underline{{\mathrm{U}}(1)}}$$\scriptstyle{\frac{1}{2}p_{1}}$
(4.4)

Here, $\frac{1}{2}p_{1}$ denotes the fractional first Pontryagin class, which
is a generator of
$\mathrm{H}^{4}({\mathrm{B}}\underline{H};\mathbb{Z})\cong\mathbb{Z}$.
However, this definition of ${\mathrm{String}}(\underline{H})$ is considerably
stricter than the original perception of ${\mathrm{String}}(H)$ as a
3-connected covering of $H$ by another group object (Definition 4.1). For
instance, the definition of a string group extension based on (4.4) enforces
that ${\mathrm{String}}(\underline{H})\to\underline{H}$ is a
${\mathrm{B}}\underline{\mathsf{U}(1)}$-principal $\infty$-bundle. (Note that
if ${\mathbf{H}}$ is an $\infty$-topos and
${\widehat{A}}\in{\mathscr{G}\mathrm{rp}}({\mathbf{H}})$ is a group object
whose multiplication lifts to an $\mathbb{E}_{2}$-algebra structure, then
${\mathrm{B}}A$ is canonically the level-zero object of a group object in
${\mathbf{H}}$ [21].) However, from the purely homotopy-theoretic point of
view, not the actual fibre of this map should be fixed, but the homotopy type
of its underlying space (which must be a $K(\mathbb{Z};2)$). Definition 4.3
emphasises this latter, homotopy-theoretic aspect of string group extensions.

More concretely, for smooth string group extensions
${\widehat{A}}\xrightarrow{\ \widehat{\iota}\
}\widehat{{\mathrm{String}}(\underline{H})}\xrightarrow{\ \widehat{p}\
}\widehat{\underline{H}}$ in the sense of Definition 4.3 it is enough if there
is an $I$-local equivalence $A\simeq\underline{\mathsf{U}(1)}$ in
${\mathbf{H}}_{\infty}$. Therefore, this setup is considerably more general
than working with the pullback (4.4); in particular, two different smooth
string group extensions of a Lie group $H$ need not be equivalent in
${\mathbf{H}}_{\infty}$, but only in $L_{I}{\mathbf{H}}_{\infty}$.

###### Remark 4.5

It will be interesting to see a Lie-algebra version of Definition 4.3. The
$\infty$-groups
$\widehat{A}\in{\mathscr{G}\mathrm{rp}}({\mathbf{H}}_{\infty})$ that we allow
to appear in string group extensions can have much larger Lie algebras than
those which appear in the stricter definition via (4.4). This is true, in
particular, for the smooth string group extension we present in Section 4.3
below. There might hence be a Lie-algebra version of $I$-local equivalences of
group objects in ${\mathbf{H}}_{\infty}$. $\triangleleft$

We now work towards establishing a new string group model that fits Definition
4.3, but does not fit into the pullback (4.4). (The reason will be that the
extension is not by ${\mathrm{B}}\underline{\mathsf{U}(1)}$, but by a much
larger, but also homotopically correct, $\infty$-group.)

###  4.2 Bundle gerbes and their symmetries

Before we can present our smooth string group extension, we need to recall
some background on bundle gerbes. We will not give full definitions or details
here; for these, we refer the reader to [30, 10, 5, 7]. Bundle gerbes provide
an explicit, geometric model for categorified line bundles. We point out that
there also exists a notion of connection on a bundle gerbe, but here we will
only be working with bundle gerbes _without_ connection. (This is the main
technical cause for the distinction between our smooth string group model and
that in [13].) Further, bundle gerbes are a very geometric, 2-categorical
model for higher line bundles, so that our smooth string group model will
arise as an explicitly defined smooth 2-group, making it particularly
tangible.

To any manifold $M$, we can assign a symmetric monoidal 2-groupoid
$({\mathscr{G}\mathrm{rb}}(M),\otimes)$ of bundle gerbes on $M$. Given a
bundle gerbe $\mathcal{G}\in{\mathscr{G}\mathrm{rb}}(M)$, the monoidal
groupoid ${\mathscr{G}\mathrm{rb}}(M)(\mathcal{G},\mathcal{G})$ of
automorphisms of $\mathcal{G}$ is canonically equivalent to the symmetric
monoidal groupoid $({\mathrm{HLB}}(M),\otimes)$ of hermitean line bundles on
$M$ with the usual tensor product (which we also denote by $\otimes$). Note
that $({\mathrm{HLB}}(M),\otimes)$ is even a _2-group_ ; that is, it is a
symmetric monoidal groupoid in which every object has an inverse with respect
to the monoidal product. Every smooth map $f\colon N\to M$ of manifolds
induces a monoidal 2-functor

$f^{*}\colon{\mathscr{G}\mathrm{rb}}(M)\longrightarrow{\mathscr{G}\mathrm{rb}}(N)\,.$
(4.6)

Isomorphism classes of gerbes are in canonical bijection with the third
integer cohomology of $M$: there is an isomorphism of abelian groups

$\pi_{0}\big{(}{\mathscr{G}\mathrm{rb}}(M),\otimes\big{)}\cong\mathrm{H}^{3}(M;\mathbb{Z})\,.$
(4.7)

The class associated to a gerbe $\mathcal{G}$ under this isomorphism is called
the _Dixmier-Douady class of $\mathcal{G}$_.

We let ${\mathbf{H}}_{\leq 1}$ denote the following 2-category: its objects
are functors $\pi\colon\mathscr{C}\to{\mathscr{C}\mathrm{art}}$ that are
Grothendieck fibrations in groupoids (that is, $\pi$ is a Grothendieck
fibration and all its fibres are groupoids). Its morphisms
$(\pi\colon\mathscr{C}\to{\mathscr{C}\mathrm{art}})\longrightarrow(\pi^{\prime}\colon\mathscr{C}^{\prime}\to{\mathscr{C}\mathrm{art}})$
are functors $F\colon\mathscr{C}\to\mathscr{C}^{\prime}$ such that
$\pi^{\prime}\circ F=\pi$, and its 2-morphisms are natural transformations
$\eta\colon F\to F^{\prime}$ such that $\pi^{\prime}\eta$ is the identity