natural transformation $1_{\mathscr{C}\mathrm{art}}\to
1_{\mathscr{C}\mathrm{art}}$. Note that the 2-category ${\mathbf{H}}_{\leq 1}$
is canonically equivalent to the 2-category of pseudo-functors
${\mathscr{C}\mathrm{art}}^{\mathrm{op}}\to{\mathscr{G}\mathrm{pd}}$ from
${\mathscr{C}\mathrm{art}}^{\mathrm{op}}$ to the 2-category of groupoids via
the Grothendieck construction. We make the following definitions; for more
background, see [3, 25].

###### Definition 4.8

_[ 3]_ The _2-category of smooth 2-groups_ is the 2-category of group objects
in the 2-category ${\mathbf{H}}_{\leq 1}$.

###### Example 4.9

Let $H$ be a Lie group. We associate to it the following category, denoted by
$\textstyle{\int}\underline{H}$: its objects are pairs $(c,h)$ of a cartesian
space $c\in{\mathscr{C}\mathrm{art}}$ and a smooth map $h\colon c\to H$. A
morphism $(c,h)\to(c^{\prime},h^{\prime})$ is a smooth map $f\colon c\to
c^{\prime}$ such that $h^{\prime}\circ f=h$, and the category
$\textstyle{\int}\underline{H}$ comes with a canonical projection functor
$\textstyle{\int}\underline{H}\to{\mathscr{C}\mathrm{art}}$. The product on
$H$-valued maps turns $\textstyle{\int}\underline{H}$ into a smooth 2-group in
the sense of Definition 4.8. Note that $\textstyle{\int}\underline{H}$ is
simply the Grothendieck construction of the presheaf of sets $\underline{H}$
on ${\mathscr{C}\mathrm{art}}$. $\triangleleft$

###### Example 4.10

Let $M$ be a manifold, and define a category ${\mathrm{HLB}}^{M}$ as follows:
its objects are pairs $(c,L)$ of a cartesian space
$c\in{\mathscr{C}\mathrm{art}}$ and a hermitean line bundle
$L\in{\mathrm{HLB}}(c\times M)$. A morphism $(c,L)\to(c^{\prime},L^{\prime})$
is a pair $(f,\psi)$ of a smooth map $f\colon c\to c^{\prime}$ and an
isomorphism $\psi\colon L\to(f\times 1_{M})^{*}L^{\prime}$ of hermitean line
bundles over $c$. This category comes with a projection functor
${\mathrm{HLB}}^{M}\to{\mathscr{C}\mathrm{art}}$. The tensor product of
hermitean line bundles turns ${\mathrm{HLB}}^{M}$ into a smooth 2-group.
$\triangleleft$

Let $M$ be a manifold, and let $\mathcal{G}\in{\mathscr{G}\mathrm{rb}}(M)$ be
a gerbe on $M$. Further, let $H$ be a connected Lie group acting smoothly on
$M$ from the left; we denote the action by $\Phi\colon H\times M\to M$. Given
these data, we define a category ${\mathrm{Sym}}(\mathcal{G})$ as follows: an
object in ${\mathrm{Sym}}(\mathcal{G})$ is a triple $(c,h,\mathcal{A})$, where
$c\in{\mathscr{C}\mathrm{art}}$ is a cartesian space and where $h\colon c\to
H$ is a smooth map. These give rise to a smooth map $\Phi_{h}\colon c\times
M\to c\times M$, defined as the composition

$\Phi_{h}\colon c\times M\xrightarrow{\Delta\times 1_{M}}c\times c\times
M\xrightarrow{1_{c}\times h\times 1_{M}}c\times H\times
M\xrightarrow{1_{c}\times\Phi}c\times M\,,$ (4.11)

where $\Delta\colon c\to c\times c$ is the diagonal map. Then, $\mathcal{A}$
is a 1-isomorphism

$\mathcal{A}\colon\mathrm{pr}_{M}^{*}\mathcal{G}\longrightarrow\Phi_{h}^{*}\mathcal{G}$
(4.12)

of gerbes on the manifold $c\times M$. A morphism
$(c,h,\mathcal{A})\to(c^{\prime},h^{\prime},\mathcal{A}^{\prime})$ is a pair
$(f,\psi)$, where $f$ is a smooth map $f\colon c\to c^{\prime}$ such that
$h^{\prime}\circ f=h$, and where $\psi$ is a 2-isomorphism
$\psi\colon\mathcal{A}\longrightarrow(f\times 1_{M})^{*}\mathcal{A}^{\prime}$.
(Here we have implicitly used that there is a canonical 1-isomorphism
$(f\times
1_{M})^{*}\Phi_{h^{\prime}}^{*}\mathcal{G}\cong\Phi_{h}^{*}\mathcal{G}$.)
Observe that there is a projection functor
$p\colon{\mathrm{Sym}}(\mathcal{G})\to\textstyle{\int}\underline{H}$, acting
as $(c,h,\mathcal{A})\mapsto(c,h)$ and $(f,\psi)\mapsto f$.

###### Remark 4.13

In this set-up, the following statements hold true:

  1. (1)

The connectedness of $H$ ensures that the functor $p$ is surjective on
objects. It is an essentially surjective Grothendieck fibration in groupoids
[3, Thm. 5.27].

  2. (2)

The equivalence
${\mathscr{G}\mathrm{rb}}(N)(\mathcal{G}^{\prime},\mathcal{G}^{\prime})\simeq({\mathrm{HLB}}(N),\otimes)$
for any gerbe $\mathcal{G}^{\prime}$ on any manifold $N$ implies that the
diagram

${{\mathrm{HLB}}^{M}}$${{\mathrm{Sym}}(\mathcal{G})}$${*}$${\textstyle{\int}\underline{H}}$$\scriptstyle{e_{H}}$
(4.14)

is a pullback in ${\mathbf{H}}_{\leq 1}$, where $e_{H}$ is the functor that
sends $c\in{\mathscr{C}\mathrm{art}}$ to the constant map $c\to H$ with value
the unit element of $H$. Since $p$ is a Grothendieck fibration in groupoids,
this pullback is even a homotopy pullback [3, App. A.1]. $\triangleleft$

###### Theorem 4.15

_[ 3, Thms. 5.23, 5.27]_ Let $\Phi\colon H\times M\to M$ be a smooth action of
a connected Lie group $H$ on a manifold $M$. Let
$\mathcal{G}\in{\mathscr{G}\mathrm{rb}}(M)$ be a bundle gerbe on $M$.

  1. (1)

${\mathrm{Sym}}(\mathcal{G})$ is a smooth 2-group.

  2. (2)

The functor $p$ fits into a sequence

${\mathrm{HLB}}^{M}\overset{i}{\longrightarrow}{\mathrm{Sym}}(\mathcal{G})\overset{p}{\longrightarrow}\textstyle{\int}\underline{H}$
(4.16)

of smooth 2-groups. Further, $p$ is a Grothendieck fibration in groupoids and
surjective on objects.

The nerve functor
$N\colon{\mathscr{C}\mathrm{at}}\to{\mathscr{C}\mathrm{at}}_{\infty}$ induces
a functor $N\colon{\mathbf{H}}_{\leq 1}\to{\mathbf{H}}_{\infty}$ (where we
have used the canonical equivalence between ${\mathbf{H}}_{\leq 1}$ and the
2-category of pseudo-functors
${\mathscr{C}\mathrm{art}}^{\mathrm{op}}\to{\mathscr{G}\mathrm{pd}}$ from
${\mathscr{C}\mathrm{art}}^{\mathrm{op}}$ to the 2-category of groupoids).
This functor, in particular, preserves final objects and products, so that it
maps smooth 2-groups to group objects in ${\mathbf{H}}_{\infty}$. Our smooth
string group model will be obtained by applying this functor to the sequence
(4.16).

###  4.3 A smooth string group model

We can now state the main theorem of this section. It provides a new smooth
model for smooth string group extensions which fits Definition 4.3, but which
lies outside the scope of the stricter definition via the pullback (4.4). Note
that applying the nerve functor $N$ to
$\textstyle{\int}\underline{H}\in{\mathbf{H}}_{\leq 1}$ yields the familiar
presheaf of spaces $\underline{H}\in{\mathbf{H}}_{\infty}$, defined via
$\underline{H}(c)={\mathscr{M}\mathrm{fd}}(c,H)$ for cartesian spaces
$c\in{\mathscr{C}\mathrm{art}}$.

###### Theorem 4.17

Let $H$ be a compact, simple, simply connected Lie group. 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 (see (4.7)).
Then, the sequence

${N\,\widehat{{\mathrm{HLB}}^{H}}}$${N\widehat{{\mathrm{Sym}}(\mathcal{G})}}$${\underline{{\widehat{H}}}}$$\scriptstyle{\widehat{Ni}}$$\scriptstyle{\widehat{Np}}$
(4.18)

is a smooth string group extension of $H$.

The proof of Theorem 4.17 will occupy the remainder of this section. By
Definition 4.3 we have to show that the sequence (4.18) is an extension of
group objects in ${\mathbf{H}}_{\infty}$ and that its image under the functor
$\mathrm{S}_{e}\colon{\mathbf{H}}_{\infty}\to{\mathbf{S}}$ is a string group
extension in ${\mathbf{S}}$ in the sense of Definition 4.1.

###### Proposition 4.19

The sequence (4.18) is an extension of group objects in the $\infty$-topos
${\mathbf{H}}_{\infty}$.

###### Proof.

The nerve functor
$N\colon{\mathscr{C}\mathrm{at}}\to{\mathscr{C}\mathrm{at}}_{\infty}$ is a
right adjoint and hence maps products in ${\mathbf{H}}_{\leq 1}$ to products
in ${\mathbf{H}}_{\infty}$, and it maps the final object in
${\mathbf{H}}_{\leq 1}$ to the final object in ${\mathbf{H}}_{\infty}$.
Consequently, it preserves group objects and group actions.

We will now use the characterisation of group extensions from Theorem 3.99(4)
to show that the sequence (4.18) of group objects in ${\mathbf{H}}_{\infty}$
is an extension of group objects. That is, we have to show that
$N{\mathrm{Sym}}(\mathcal{G})$ with the $N{\mathrm{HLB}}^{H}$-action induced
by the morphism $\widehat{Ni}$ (cf. Proposition 3.47) is an
$N\,{\mathrm{HLB}}^{M}$-principal $\infty$-bundle over $\underline{H}$.
According to the characterisation of principal $\infty$-bundles in Proposition
3.58, it suffices to prove that the morphism $Np$ is an effective epimorphism
and that the action of $N\,{\mathrm{HLB}}^{M}$ on
$N{\mathrm{Sym}}(\mathcal{G})$ is principal.

We start by showing that the morphism $Np$ is an effective epimorphism: by [3,
Sec. 5.1] the restriction $p_{|c}$ of $p$ to any fibre is essentially
surjective, hence $Np_{|c}$ is surjective on connected components. Since
${\mathbf{H}}_{\infty}$ is a presheaf $\infty$-topos (in which limits and
colimits are computed objectwise), a morphism in ${\mathbf{H}}_{\infty}$ is an
effective epimorphism if and only if it is objectwise an effective epimorphism
in ${\mathbf{S}}$. The effective epimorphisms in ${\mathbf{S}}$, however, are
exactly those morphisms which are surjective on connected components [20, Cor.
7.2.1.15]. Therefore, $Np$ is an effective epimorphism in
${\mathbf{H}}_{\infty}$.

The action of $N\,{\mathrm{HLB}}^{H}$ on $N{\mathrm{Sym}}(\mathcal{G})$ is
principal with respect to $Np$ as was shown in [3, Thm. 5.27]. (There, the
principality condition was shown on the level of the sequence (4.16) of smooth
2-groups—this suffices for the $\infty$-categorical context used here because
of Lemma 3.56 and because the nerve functor is a right adjoint.) Therefore,
the sequence (4.18) is a group extension in ${\mathbf{H}}_{\infty}$. ∎

It thus remains to show that the image of the sequence (4.18) under
$\mathrm{S}_{e}$ is a string group extension in ${\mathbf{S}}$. To that end,
we first show the following lemma:

###### Lemma 4.20

Let $X$ be a connected manifold with $\mathrm{H}^{2}(X;\mathbb{Z})\cong 0$.

  1. (1)

The object $N\,{\mathrm{HLB}}^{X}\in{\mathbf{H}}_{\infty}$ is equivalent to
the object ${\mathrm{B}}(\underline{\mathsf{U}(1)}^{\underline{X}})$.

  2. (2)

Suppose that also $\pi_{1}(N)=0$. Then, the object
$N\,{\mathrm{HLB}}^{X}\in{\mathbf{H}}_{\infty}$ is $I$-locally equivalent to
${\mathrm{B}}\underline{\mathsf{U}(1)}\in{\mathbf{H}}_{\infty}$.

Both equivalences are even established by morphisms of group objects in
${\mathbf{H}}_{\infty}$.

Since $\mathrm{S}_{e}$ maps $I$-local equivalences in ${\mathbf{H}}_{\infty}$
to equivalences of spaces, applying Lemma 4.20 to $X=H$ establishes axiom (1)
of Definition 4.1 for the image of the sequence (4.18) under the functor
$\mathrm{S}_{e}$.

###### Proof.

We proceed in parallel to the proof of [3, Thm. 8.7]: since any
$c\in{\mathscr{C}\mathrm{art}}$ is contractible and since
$\mathrm{H}^{2}(X;\mathbb{Z})\cong 0$, it follows that any hermitean line
bundle on $c\times N$ is trivialisable. Consequently, ${\mathrm{HLB}}^{X}(c)$
is equivalent to the groupoid with one object and morphisms given by the group
$\underline{\mathsf{U}(1)}(c)$ of smooth maps from $c$ to $\mathsf{U}(1)$.
This induces an equivalence
$N\,{\mathrm{HLB}}^{X}\simeq{\mathrm{B}}(\underline{\mathsf{U}(1)}^{\underline{X}})$
in ${\mathbf{H}}_{\infty}$, which extends to a morphism of group objects in
${\mathbf{H}}_{\infty}$. This proves (1).

Next, since $\pi_{1}(X)$ is trivial, there exists a smooth homotopy
equivalence
$\operatorname{ev}_{e}\colon\underline{\mathsf{U}(1)}^{\underline{X}}\to\underline{\mathsf{U}(1)}$,
given by restricting a smooth map $c\times X\to\mathsf{U}(1)$ to
$c\times\\{x\\}$, where $x\in X$ is any point. A homotopy inverse to
$\operatorname{ev}_{x}$ is given by pulling a smooth map $c\to\mathsf{U}(1)$
back along the projection $c\times X\to c$ [3, Lemma 8.9]. In particular,
$\operatorname{ev}_{x}$ is an $I$-local equivalence [9, Cor. 3.16].

Observe that $\operatorname{ev}_{x}$ induces a morphism of group objects

$\widehat{\operatorname{ev}}_{x}\colon\big{(}\underline{\mathsf{U}(1)}^{\underline{X}}\big{)}\,\widehat{}\longrightarrow\widehat{\underline{\mathsf{U}(1)}}\,.$
(4.21)

Since $\operatorname{ev}_{x}$ is an $I$-local equivalence in
${\mathbf{H}}_{\infty}$ and $I$-local equivalences are closed under finite
products (Proposition 2.9), the morphism $\widehat{\operatorname{ev}}_{x}$ is
a levelwise $I$-local equivalence of simplicial objects in
${\mathbf{H}}_{\infty}$.

Further, the class $W_{I}$ of $I$-local equivalences in
${\mathbf{H}}_{\infty}$ is strongly saturated [20, Lemma 5.5.4.11]. In
particular, the full subcategory of
${\mathscr{F}\mathrm{un}}(\Delta^{1},{\mathbf{H}}_{\infty})$ on the $I$-local
equivalences is stable under colimits. Therefore, taking the colimit in
${\mathbf{H}}_{\infty}$ of simplicial objects (i.e. taking geometric
realisations), we obtain an $I$-local equivalence

${\mathrm{B}}\operatorname{ev}_{x}\colon{\mathrm{B}}\big{(}\underline{\mathsf{U}(1)}^{\underline{X}}\big{)}\longrightarrow{\mathrm{B}}\underline{\mathsf{U}(1)}$
(4.22)

in ${\mathbf{H}}_{\infty}$. Composing with the morphism constructed in part
(1), we now obtain the desired $I$-local equivalence
$N\,{\mathrm{HLB}}^{X}\longrightarrow{\mathrm{B}}\underline{\mathsf{U}(1)}$ in
${\mathbf{H}}_{\infty}$. ∎

We are thus left to show that the sequence of group objects in ${\mathbf{S}}$
obtained by applying the functor $\mathrm{S}_{e}$ to the sequence (4.18) of
group objects in ${\mathbf{H}}_{\infty}$ satisfies axiom (2) of Definition
4.1. That is, we have to show that the principal $\infty$-bundle of spaces

$\big{(}\mathrm{S}_{e}N{\mathrm{Sym}}(\mathcal{G})\big{)}{//}\big{(}\mathrm{S}_{e}N\,{\mathrm{HLB}}^{H}\big{)}\longrightarrow\mathrm{S}_{e}\underline{H}$
(4.23)

represents a generator of $\mathrm{H}^{3}(H;\mathbb{Z})\cong\mathbb{Z}$. This
is best checked using Čech cohomology.

Recall that a differentiably good open covering of
$c\in{\mathscr{C}\mathrm{art}}$ is an open covering $\\{c_{a}\hookrightarrow
c\\}_{a\in\Lambda}$ such that every finite non-empty intersection of the
images of the patches $c_{a}$ is again a cartesian space. The differentiably
good open coverings endow ${\mathscr{C}\mathrm{art}}$ with a Grothendieck
pretopology $\tau$ [14, 24].

###### Lemma 4.24

Let $X\in{\mathscr{M}\mathrm{fd}}$ be a connected manifold, and let
$k\in\mathbb{N}_{0}$. Then,

  1. (1)

In ${\mathbf{H}}_{\infty}$, there is an $I$-local equivalence

${\mathrm{B}}^{k}(\underline{\mathsf{U}(1)}^{\underline{X}})\simeq{\mathrm{B}}^{k}\underline{\mathsf{U}(1)}\,.$
(4.25)

  2. (2)

Suppose $X$ is simply connected. Then, the presheaf
${\mathrm{B}}^{k}(\underline{\mathsf{U}(1)}^{\underline{X}})$ satisfies
descent with respect to the Grothendieck pretopology $\tau$.

###### Proof.

Ad (1): This is an iteration of the argument in the proof of Lemma 4.20(2).

Ad (2): We prove this claim by induction. For $k=0$, we have to check that the
functor ${\mathscr{C}\mathrm{art}}^{\mathrm{op}}\to{\mathbf{S}}$,
$c\mapsto{\mathscr{M}\mathrm{fd}}(c\times X,\mathsf{U}(1))$ satisfies descent
with respect to good open coverings of $c$. However, this follows directly
from the fact that, for any manifold $Y$, the assignment functor

${\mathscr{O}\mathrm{p}}(Y)^{\mathrm{op}}\to{\mathscr{S}\mathrm{et}}\,,\qquad
U\mapsto{\mathscr{M}\mathrm{fd}}\big{(}U,\mathsf{U}(1)\big{)}$ (4.26)

defines a sheaf on $Y$, where ${\mathscr{O}\mathrm{p}}(Y)$ is the category of
open subsets of $Y$ and their inclusions.

Suppose that ${\mathrm{B}}^{l}(\underline{\mathsf{U}(1)}^{\underline{X}})$ is
a sheaf on ${\mathscr{C}\mathrm{art}}$ for all $l=0,\ldots,k$. Let
${\mathscr{U}}=\\{c_{a}\hookrightarrow c\\}_{a\in\Lambda}$ be a differentiably
good open covering of $c$. We have to show that the canonical morphism

$q^{*}\colon{\mathrm{B}}^{k+1}\big{(}\underline{\mathsf{U}(1)}^{\underline{X}}\big{)}(c)\longrightarrow\underset{n\in{\mathbbe{\Delta}}}{{\mathrm{lim}}}^{\mathbf{S}}\,\underline{{\mathbf{H}}}_{\infty}\big{(}{\check{C}}{\mathscr{U}}_{n},{\mathrm{B}}^{k+1}\underline{\mathsf{U}(1)}^{\underline{X}}\big{)}$
(4.27)

is an equivalence of spaces. Here,
${\check{C}}{\mathscr{U}}\in{\mathscr{F}\mathrm{un}}({\mathbbe{\Delta}}^{\mathrm{op}},{\mathbf{H}}_{\infty})$
is the Čech nerve of the covering ${\mathscr{U}}$.

We first show that $q^{*}$ is essentially surjective; that is, it induces a
bijection on isomorphism classes of objects. Since limits and colimits in
${\mathbf{H}}_{\infty}={\mathscr{F}\mathrm{un}}({\mathscr{C}\mathrm{art}}^{\mathrm{op}},{\mathbf{S}})$
are computed pointwise, we have isomorphisms

$\pi_{0}\,{\mathrm{B}}^{k+1}\big{(}\underline{\mathsf{U}(1)}^{\underline{X}}\big{)}(c)\cong\pi_{0}\,{\mathrm{B}}^{k+1}\big{(}\underline{\mathsf{U}(1)}^{\underline{X}}(c)\big{)}=*\,.$
(4.28)

On the other hand, we have that

$\pi_{0}\,\underset{n\in{\mathbbe{\Delta}}}{{\mathrm{lim}}}^{\mathbf{S}}\,\underline{{\mathbf{H}}}_{\infty}\big{(}{\check{C}}{\mathscr{U}}_{n},{\mathrm{B}}^{k+1}\underline{\mathsf{U}(1)}^{\underline{X}}\big{)}\cong\check{\mathrm{H}}^{k+1}\big{(}{\mathscr{U}};\underline{\mathsf{U}(1)}^{\underline{X}}\big{)}\cong\mathrm{H}^{k+2}(c;\mathbb{Z})\cong*\,,$
(4.29)

where on the right-hand side we have the usual Čech cohomology group with
respect to the covering ${\mathscr{U}}$ of the sheaf of abelian groups on $c$
given by

${\mathscr{O}\mathrm{p}}(c)^{\mathrm{op}}\to{\mathscr{A}\mathrm{b}}\,,\qquad
U\mapsto{\mathscr{M}\mathrm{fd}}\big{(}U\times X,\mathsf{U}(1)\big{)}\,.$
(4.30)

By a slight abuse of notation, we also denote this sheaf by
$\underline{\mathsf{U}(1)}^{\underline{X}}$.

This Čech cohomology is indeed isomorphic to the full sheaf cohomology of
$\underline{\mathsf{U}(1)}^{\underline{X}}$: first, we observe that since $X$
is simply connected, there is a short exact sequence

$\mathbb{Z}\longrightarrow\underline{\mathbb{R}}^{\underline{X}}\longrightarrow\underline{\mathsf{U}(1)}^{\underline{X}}\,.$
(4.31)

We further observe that the sheaf $\underline{\mathbb{R}}^{\underline{X}}$ is
fine (it admits partitions of unity, for instance those induced from the
canonical map
$\underline{\mathbb{R}}\to\underline{\mathbb{R}}^{\underline{X}}$) when seen
as a sheaf on the open subsets of any manifold. Therefore, for any manifold
$Y$, we have a canonical isomorphism

$\mathrm{H}^{l}\big{(}Y;\underline{\mathsf{U}(1)}^{\underline{X}}\big{)}\cong\mathrm{H}^{l+1}(Y;\mathbb{Z})$
(4.32)

for every $l\geq 1$. From this, we see that

$\mathrm{H}^{l}\big{(}d;\underline{\mathsf{U}(1)}^{\underline{X}}\big{)}\cong*$
(4.33)

for every cartesian space $d\in{\mathscr{C}\mathrm{art}}$ and $l\geq 1$. Since
the covering ${\mathscr{U}}$ of $c$ is differentiably good, it follows from
[4, Thm. 1.3.6] that there is a canonical isomorphism

$\check{\mathrm{H}}^{k}\big{(}{\mathscr{U}};\underline{\mathsf{U}(1)}^{\underline{X}}\big{)}\cong\mathrm{H}^{k}\big{(}c;\underline{\mathsf{U}(1)}^{\underline{X}}\big{)}\,.$
(4.34)

This proves the isomorphisms in (4.29), and hence that the morphism $q^{*}$
from (4.27) is bijective on connected components.

It remains to check that $q^{*}$ is an isomorphism on all higher homotopy
groups. We will achieve this by comparing the automorphisms of the unique
object in the source and target space of $q^{*}$. On the source side, this
automorphism space is given as the pullback of spaces

${\Omega{\mathrm{B}}^{k+1}\big{(}\underline{\mathsf{U}(1)}^{\underline{X}}\big{)}(c)}$${*}$${*}$${{\mathrm{B}}^{k+1}\big{(}\underline{\mathsf{U}(1)}^{\underline{X}}\big{)}(c)}$
(4.35)

and there is a canonical equivalence

$\Omega{\mathrm{B}}^{k+1}\big{(}\underline{\mathsf{U}(1)}^{\underline{X}}\big{)}(c)\simeq{\mathrm{B}}^{k}\big{(}\underline{\mathsf{U}(1)}^{\underline{X}}(c)\big{)}\,.$
(4.36)

On the target side of $q^{*}$, the automorphism space of the (essentially)
unique object is the pullback

${\Omega\,\underset{n\in{\mathbbe{\Delta}}}{{\mathrm{lim}}}^{\mathbf{S}}\,\underline{{\mathbf{H}}}_{\infty}\big{(}{\check{C}}{\mathscr{U}}_{n},{\mathrm{B}}^{k+1}\underline{\mathsf{U}(1)}^{\underline{X}}\big{)}}$${*}$${*}$${\underset{n\in{\mathbbe{\Delta}}}{{\mathrm{lim}}}^{\mathbf{S}}\,\underline{{\mathbf{H}}}_{\infty}\big{(}{\check{C}}{\mathscr{U}}_{n},{\mathrm{B}}^{k+1}\underline{\mathsf{U}(1)}^{\underline{X}}\big{)}}$
(4.37)

Since limits in ${\mathbf{H}}_{\infty}$ are computed objectwise, there are
canonical equivalences

$\displaystyle\Omega\,\underset{n\in{\mathbbe{\Delta}}}{{\mathrm{lim}}}^{\mathbf{S}}\,\underline{{\mathbf{H}}}_{\infty}\big{(}{\check{C}}{\mathscr{U}}_{n},{\mathrm{B}}^{k+1}\underline{\mathsf{U}(1)}^{\underline{X}}\big{)}$
$\displaystyle\simeq\underset{n\in{\mathbbe{\Delta}}}{{\mathrm{lim}}}^{\mathbf{S}}\,\underline{{\mathbf{H}}}_{\infty}\big{(}{\check{C}}{\mathscr{U}}_{n},\Omega\,{\mathrm{B}}^{k+1}\underline{\mathsf{U}(1)}^{\underline{X}}\big{)}$
(4.38)
$\displaystyle\simeq\underset{n\in{\mathbbe{\Delta}}}{{\mathrm{lim}}}^{\mathbf{S}}\,\underline{{\mathbf{H}}}_{\infty}\big{(}{\check{C}}{\mathscr{U}}_{n},{\mathrm{B}}^{k}\underline{\mathsf{U}(1)}^{\underline{X}}\big{)}\,.$
(4.39)

However, by the induction hypothesis, the presheaf
${\mathrm{B}}^{k}\underline{\mathsf{U}(1)}^{\underline{X}}$ is a sheaf, so
that $q^{*}$ induces an equivalence between the automorphism spaces. This
proves that $q^{*}$ is indeed an equivalence. ∎

Another application of [8, Thm. 1.2] now implies that the presheaf of spaces

$\underline{{\mathbf{H}}}_{\infty}\big{(}-,{\mathrm{B}}^{n}\big{(}\underline{\mathsf{U}(1)}^{\underline{H}}\big{)}\big{)}\colon{\mathscr{M}\mathrm{fd}}^{\mathrm{op}}\longrightarrow{\mathbf{S}}$
(4.40)

satisfies descent with respect to open coverings (and even surjective
submersions). Consequently, given any open covering $\\{c_{a}\hookrightarrow
H\\}_{a\in\Lambda}$, whose Čech nerve we denote by
${\mathscr{V}}\to\underline{H}$, the canonical morphism

$\underline{{\mathbf{H}}}_{\infty}(\underline{H},{\mathrm{B}}^{n}\big{(}\underline{\mathsf{U}(1)}^{\underline{H}}\big{)}\big{)}\cong\underset{{\mathbbe{\Delta}}}{{\mathrm{lim}}}^{\mathbf{S}}\,\underline{{\mathbf{H}}}_{\infty}\big{(}{\mathscr{V}},{\mathrm{B}}^{n}\big{(}\underline{\mathsf{U}(1)}^{\underline{H}}\big{)}\big{)}\,.$
(4.41)

is an equivalence of spaces because
${\mathrm{B}}^{n}(\underline{\mathsf{U}(1)}^{\underline{H}})$ is local with
respect to the class of morphisms consisting of the Čech nerves of coverings.
Therefore, there is an isomorphism

$\pi_{0}\underline{{\mathbf{H}}}_{\infty}(\underline{H},{\mathrm{B}}^{n}\big{(}\underline{\mathsf{U}(1)}^{\underline{H}}\big{)}\big{)}\cong\check{\mathrm{H}}^{n}\big{(}H;\underline{\mathsf{U}(1)}^{\underline{H}}\big{)}\,,$
(4.42)

which can be represented explicitly by composing any morphism
$\underline{H}\to{\mathrm{B}}^{n}(\underline{\mathsf{U}(1)}^{\underline{H}})$
with any Čech nerve ${\mathscr{V}}\to\underline{H}$ of an open covering of
$H$. (Alternatively, this can be seen directly in the presentation of
${\mathbf{H}}_{\infty}$ by the projective model structure on simplicial
presheaves on ${\mathscr{C}\mathrm{art}}$.) Let
$\operatorname{ev}_{e}\colon\underline{\mathsf{U}(1)}^{\underline{H}}\longrightarrow\underline{\mathsf{U}(1)}$
be the morphism induced by pullback along the base-point inclusion
$*\hookrightarrow\underline{H}$. We obtain a commutative diagram

${\pi_{0}\underline{{\mathbf{H}}}_{\infty}(\underline{H},{\mathrm{B}}^{n}\big{(}\underline{\mathsf{U}(1)}^{\underline{H}}\big{)}\big{)}}$${\pi_{0}\underline{{\mathbf{H}}}_{\infty}(\underline{H},{\mathrm{B}}^{n}\underline{\mathsf{U}(1)}\big{)}}$${\check{\mathrm{H}}^{n}\big{(}H;\underline{\mathsf{U}(1)}^{\underline{H}}\big{)}}$${\check{\mathrm{H}}^{n}\big{(}H;\underline{\mathsf{U}(1)}\big{)}}$$\scriptstyle{\cong}$$\scriptstyle{({\mathrm{B}}^{n}\operatorname{ev}_{e})_{*}}$$\scriptstyle{\cong}$$\scriptstyle{(\operatorname{ev}_{e})_{*}}$
(4.43)

It was shown in [3, Prop. 8.11] that the bottom horizontal morphism is an
isomorphism for all $n\in\mathbb{N}$ (with $n>0$). Consider the morphisms

$\displaystyle\pi_{0}\underline{{\mathbf{H}}}_{\infty}\big{(}\underline{H},{\mathrm{B}}^{n}\big{(}\underline{\mathsf{U}(1)}^{\underline{H}}\big{)}\big{)}\longrightarrow$
$\displaystyle\pi_{0}\underline{{\mathbf{S}}}\big{(}\mathrm{S}_{e}\underline{H},\mathrm{S}_{e}{\mathrm{B}}^{n}\big{(}\underline{\mathsf{U}(1)}^{\underline{H}}\big{)}\big{)}$
(4.44)
$\displaystyle\cong\pi_{0}\underline{{\mathbf{S}}}\big{(}H,{\mathrm{B}}^{n}\big{(}\mathrm{S}_{e}\underline{\mathsf{U}(1)}^{\underline{H}}\big{)}\big{)}$
(4.45)
$\displaystyle\cong\pi_{0}\underline{{\mathbf{S}}}\big{(}H,{\mathrm{B}}^{n}\mathrm{S}_{e}\underline{\mathsf{U}(1)}\big{)}$
(4.46)
$\displaystyle\cong\pi_{0}\underline{{\mathbf{S}}}\big{(}H,{\mathrm{B}}^{n}\mathsf{U}(1)\big{)}\,.$
(4.47)

The first morphism is applying the functor $\mathrm{S}_{e}$. For the second
morphism we have used [9, Thm. 5.1]: for every manifold
$X\in{\mathscr{M}\mathrm{fd}}$, there is a canonical equivalence
$\mathrm{S}_{e}\underline{X}\simeq X$ in ${\mathbf{S}}$. Further, here we have
used that $\mathrm{S}_{e}$ commutes with ${\mathrm{B}}$ (Proposition 3.11).
For the third morphism, we have used that the inclusion
$\underline{\mathsf{U}(1)}\hookrightarrow\underline{\mathsf{U}(1)}^{\underline{H}}$
is an $I$-local equivalence in ${\mathbf{H}}_{\infty}$: since $H$ is connected
and simply connected, this morphism is a smooth homotopy equivalence by [3,
Lemma 8.9], and by [9, Cor. 3.16] any smooth homotopy equivalence is an
$I$-local equivalence. The last morphism again uses [9, Thm. 5.1]. Since
$\mathrm{S}_{e}$ preserves finite products, the equivalence
$\mathrm{S}_{e}\underline{\mathsf{U}(1)}\simeq\mathsf{U}(1)$ in ${\mathbf{S}}$
is compatible with the group structure333This can also be seen directly: the
comparison map $\mathrm{S}_{e}M\to M$ sends a smooth map $\Delta_{e}^{k}\to M$
to the restriction $|\Delta^{k}|\to M$, which is a $k$-simplex in
$\operatorname{Sing}(M)$—see [9, Secs. 4, 5] for details. on $\mathsf{U}(1)$.

We can describe the map (4.44) more explicitly as follows: we have already
seen above that any element in
$\pi_{0}\underline{{\mathbf{H}}}_{\infty}(\underline{H},{\mathrm{B}}^{n}(\underline{\mathsf{U}(1)}^{\underline{H}}))$
can be described as a smooth $\mathsf{U}(1)^{H}$-valued Čech cocycle with
respect to a (differentiably good) open cover ${\mathscr{V}}$ of $H$. Under
the map (4.44), these data are sent first to the same Čech cocycle, but seen
as a map of spaces, and then this resulting Čech cocycle is composed with the
evaluation $\mathsf{U}(1)^{H}\to\mathsf{U}(1)$ at the unit element in $H$.
Therefore, using the canonical isomorphism
$\pi_{0}\underline{{\mathbf{S}}}(H,{\mathrm{B}}^{n}\mathsf{U}(1))\cong\check{\mathrm{H}}^{n}(H;\mathsf{U}(1))$
and combining this with the maps (4.43) and (4.44) we obtain a commutative
diagram of abelian groups

${\pi_{0}\underline{{\mathbf{H}}}_{\infty}(\underline{H},{\mathrm{B}}^{n}\big{(}\underline{\mathsf{U}(1)}^{\underline{H}}\big{)}\big{)}}$${\pi_{0}\underline{{\mathbf{S}}}\big{(}H,{\mathrm{B}}^{n}\big{(}\mathrm{S}_{e}\underline{\mathsf{U}(1)}^{\underline{H}}\big{)}\big{)}}$${\check{\mathrm{H}}^{n}\big{(}H;\underline{\mathsf{U}(1)}^{\underline{H}}\big{)}}$${\check{\mathrm{H}}^{n}\big{(}H;\mathsf{U}(1)\big{)}}$$\scriptstyle{(\operatorname{ev}_{e})_{*}}$$\scriptstyle{(\operatorname{ev}_{e})_{*}}$
(4.48)

In this diagram, the left-hand vertical morphism is invertible as argued
before (4.42). The bottom morphism is an isomorphism by [3, Prop. 8.11] and
the fact that Čech cohomology and abelian sheaf cohomology are isomorphic on
manifolds. The right-hand vertical morphism is invertible as a consequence of
the isomorphisms in (4.43) and the fact that the map
$\operatorname{ev}_{e}\colon\underline{\mathsf{U}(1)}^{\underline{H}}\to\underline{\mathsf{U}(1)}$
is an $I$-local equivalence.

Combining diagram (4.48) with Proposition 3.97 and Lemma 4.20, we obtain that
the class in
$\mathrm{H}^{3}(H;\mathbb{Z})\cong\check{\mathrm{H}}^{2}(H,\mathsf{U}(1))$
defined by the $N{\mathrm{HLB}}^{H}$-principal $\infty$-bundle

$\big{(}{\mathrm{Sym}}(\mathcal{G})\big{)}{//}N{\mathrm{HLB}}^{H}\longrightarrow\underline{H}$
(4.49)

in ${\mathbf{H}}_{\infty}$ agrees with the class defined by the principal
$\infty$-bundle

$\big{(}\mathrm{S}_{e}N{\mathrm{Sym}}(\mathcal{G})\big{)}{//}\big{(}\mathrm{S}_{e}N\,{\mathrm{HLB}}^{H}\big{)}\longrightarrow\mathrm{S}_{e}\underline{H}\simeq
H$ (4.50)

in ${\mathbf{S}}$. Here we have used that there is an equivalence
$\widehat{N\,{\mathrm{HLB}}^{H}}\simeq{\mathrm{B}}\underline{\mathsf{U}(1)}^{\underline{H}}$
in ${\mathscr{G}\mathrm{rp}}({\mathbf{H}}_{\infty})$, so that

$\pi_{0}\underline{{\mathbf{H}}}_{\infty}\big{(}\underline{H},{\mathrm{B}}\,N\,{\mathrm{HLB}}^{H}\big{)}\simeq\pi_{0}\underline{{\mathbf{H}}}_{\infty}\big{(}\underline{H},{\mathrm{B}}^{2}\underline{\mathsf{U}(1)}^{\underline{H}}\big{)}\,.$
(4.51)

(Again, one can alternatively see the coincidence of the cohomology classes
more explicitly on the level of Čech cocycles in the presentation of
${\mathbf{H}}_{\infty}$ by the simplicial model category
$\mathscr{H}_{\infty}^{p}$: a smooth bundle represented by a smooth
$\mathsf{U}(1)^{H}$-valued cocycle on $H$ gets sent to the topological bundle
represented by the same Čech cocycle interpreted as a collection of continuous
maps.) It thus remains to compute the cohomology class associated to these
bundles. In [3, Sec. 8] it has been shown that the class in
$\mathrm{H}^{3}(H;\mathbb{Z})$ of the bundle (4.49) agrees with the class in
$\mathrm{H}^{3}(H;\mathbb{Z})$ that classifies the gerbe $\mathcal{G}$ under
the isomorphism (4.7). Since we started our construction from a so-called
_basic_ gerbe, i.e. one whose Dixmier-Douady class is a generator of
$\mathrm{H}^{3}(H;\mathbb{Z})$, this concludes the proof of Theorem 4.17.

###### Remark 4.52

We conclude with the following remarks:

  1. (1)

In [3], we suggested the smooth 2-group extension (4.18) as a model for the
string group extension of $H$. However, the necessary formalism to make this
precise was not available then—its development was the main goal of the
present paper.

  2. (2)

Moreover, in [3, Sec. 5.5] we also presented a second smooth 2-group extension

${{\mathrm{HLB}}^{H}}$${\mathfrak{Des}_{\mathsf{L}}}$${\textstyle{\int}\underline{H}}$$\scriptstyle{i}$$\scriptstyle{p}$
(4.53)

of $H$; its construction uses a connection on $\mathcal{G}$ as auxiliary data
and relies heavily on a notion of parallel transport on a gerbe $\mathcal{G}$
with connection, as developed in [3]. The extension (4.53) is then obtained as
an explicit homotopy-coherent version of an associated bundle construction. By
[3, Thm. 5.33], there is an equivalence (in ${\mathbf{H}}_{\leq 1}$) between
the smooth 2-group extension in (4.53) and (4.16), so that we automatically
obtain an equivalence between the $\infty$-bundles in ${\mathbf{H}}_{\infty}$
they induce under the nerve functor. Therefore, given the input of a basic
gerbe $\mathcal{G}$ on $H$, by Theorem 4.17 the extension (4.53) also gives
rise to a second (but equivalent) smooth string group extension

${N\widehat{{\mathrm{HLB}}^{H}}}$${N\widehat{\mathfrak{Des}_{\mathsf{L}}}}$${\underline{{\widehat{H}}}}$
(4.54)

of $H$, for any compact, connected and simply connected Lie group $H$.
$\triangleleft$

##  Appendix A Actions and category objects

In this appendix, we prove Theorem 3.37; that is, we show that group actions
$P{//}G$ in $\infty$-topoi (as in Definition 3.33) are automatically groupoid
objects.

###### Definition A.1

Let $\mathscr{C}$ be an $\infty$-category. A _category object_ in
$\mathscr{C}$ is a simplicial object
$X\in{\mathscr{F}\mathrm{un}}({\mathbbe{\Delta}}^{\mathrm{op}},\mathscr{C})$
such that for every $n\in\mathbb{N}_{0}$ the pullback
$X_{1}\times_{X_{0}}\cdots\times_{X_{0}}X_{1}$ exists in $\mathscr{C}$ and the
morphism

$X_{n}\longrightarrow
X_{1}\underset{X_{0}}{\times}\cdots\underset{X_{0}}{\times}X_{1}\,,$ (A.2)

induced by the spine decomposition
$[n]\cong[1]\sqcup_{[0]}\cdots\sqcup_{[0]}[1]$ of finite ordered sets, is an
equivalence.

Suppose $\mathscr{C}$ has a final object. In analogy with Definition 3.3, a
_monoid object_ in $\mathscr{C}$ is a category object
$*{//}M\in{\mathscr{F}\mathrm{un}}({\mathbbe{\Delta}}^{\mathrm{op}},\mathscr{C})$
such that $(*{//}M)_{0}\simeq*$ is a final object in $\mathscr{C}$. As for
group objects, we set $M\coloneqq(*{//}M)_{1}$, and it follows that there are
canonical natural equivalences $(*{//}M)_{n}\simeq M^{n-1}$. Therefore, by
Lemma 3.25 we may assume, without loss of generality, that we have
$(*{//}M)_{n}=M^{n-1}$ for any $n\in\mathbb{N}_{0}$. We set
$M{//}M\coloneqq\operatorname{Dec}^{0}(*{//}M)\in{\mathscr{F}\mathrm{un}}({\mathbbe{\Delta}}^{\mathrm{op}},\mathscr{C})$.
Monoid objects can act on objects in their ambient $\infty$-category. A monoid
action is defined precisely like a group action (Definition 3.33), but for the
reader’s convenience, we spell out the definition:

###### Definition A.3

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

  1. (1)

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

  2. (2)

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

  3. (3)

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

We set $a\coloneqq d_{0}\colon P\times M\to P$. The pasting law for pullbacks
implies that there are canonical equivalences of morphisms between
$d_{0}\colon P\times M^{n}\to P\times M^{n-1}$ and $a\times 1_{M^{n-1}}\colon
P\times M^{n}\to P\times M^{n-1}$, and similarly between $d_{n}\colon P\times
M^{n}\to P\times M^{n-1}$ and the projection onto the first $n$ factors.

###### Proposition A.4

Let $\mathscr{C}$ be an $\infty$-category with pullbacks and a final object,
let
$*{//}M\in{\mathscr{F}\mathrm{un}}({\mathbbe{\Delta}}^{\mathrm{op}},\mathscr{C})$
be a monoid object in $\mathscr{C}$, and let
$P{//}M\in{\mathscr{F}\mathrm{un}}({\mathbbe{\Delta}}^{\mathrm{op}},\mathscr{C})$
be an action of $*{//}M$ on an object $P\in\mathscr{C}$. Then, $P{//}M$ is a
category object in $\mathscr{C}$.

###### Proof.

Consider the diagram

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}{}{}{}{}{{}}{}{}{{}}\pgfsys@moveto{35.48688pt}{-20.5159pt}\pgfsys@lineto{8.47284pt}{-47.52618pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{-0.70715}{-0.70706}{0.70706}{-0.70715}{8.33144pt}{-47.66759pt}\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}{24.19122pt}{-39.52908pt}\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{$\scriptstyle{\mathrm{pr}_{0}}$} }}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} \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}}$ (A.5)

We use the following notational convention: let $I$ be a set, and consider a
product $\prod_{i\in I}C_{i}$ of objects in $\mathscr{C}$. For a subset
$J\subset I$, we let $\mathrm{pr}_{J}\colon\prod_{i\in I}C_{i}\to\prod_{j\in
J}C_{j}$ denote the canonical projection. If $J=\\{i_{0},\ldots,i_{n}\\}$ is
finite, we also write $\mathrm{pr}_{i_{0}\ldots i_{n}}$ instead of
$\mathrm{pr}_{\\{i_{0},\ldots,i_{n}\\}}$.

We can augment the above diagram to a diagram

${P\times M\times M}$${P\times M}$${M}$${P\times
M}$${P}$${*}$$\scriptstyle{\mathrm{pr}_{01}}$$\scriptstyle{a\times
1_{M}}$$\scriptstyle{\mathrm{pr}_{0}}$$\scriptstyle{\mathrm{pr}_{1}}$$\scriptstyle{a}$
(A.6)

Here, the right and the outer rectangle are pullback diagrams, and hence the
left square is a pullback diagram as well by the pasting law. It follows that
the canonical morphism

$(P{//}M)_{2}\longrightarrow(P{//}M)_{1}\underset{(P{//}M)_{0}}{\times}(P{//}M)_{1}$
(A.7)

is an equivalence in $\mathscr{C}$.

We now proceed by induction: suppose that the canonical morphism

$(P{//}M)_{k}\longrightarrow(P{//}M)_{1}\underset{(P{//}M)_{0}}{\times}\cdots\underset{(P{//}M)_{0}}{\times}(P{//}M)_{1}$
(A.8)

is an equivalence, for each $2\leq k\leq n$. By this assumption, it now
suffices to show that the morphism

$(P{//}M)_{n+1}\longrightarrow(P{//}M)_{n}\underset{(P{//}M)_{0}}{\times}(P{//}M)_{1}$
(A.9)

induced by the partition $[n+1]=[n]\sqcup_{[0]}[1]$ is an equivalence. We
again have an augmented diagram

${P\times M^{n+1}}$${P\times M}$${M}$${P\times
M^{n}}$${P}$${*}$$\scriptstyle{\mathrm{pr}_{01}}$$\scriptstyle{a^{(n)}\times
1_{M}}$$\scriptstyle{\mathrm{pr}_{0}}$$\scriptstyle{\mathrm{pr}_{1}}$$\scriptstyle{a^{(n)}}$
(A.10)

where the morphism $a^{(n)}$ is, up to canonical equivalence, the morphism

$a\circ(a\times 1_{M})\circ\cdots\circ(a\times 1_{M^{n}})\colon P\times
M^{n}\to P\,.$ (A.11)

Again, the right-hand square in this diagram is a pullback square, and the top
left object is constructed as the pullback of $P\times M^{n}\to*\leftarrow M$.
It follows by the pasting law that the left-hand square is a pullback as well,
which then implies that the top left morphism is (canonically equivalent to)
$a^{(n)}\times 1_{M}$. Since $(P{//}M)_{n+1}=P\times M^{n+1}$, and the
morphisms $P\times M^{n+1}\to P\times M^{n}$ and $P\times M^{n+1}\to P\times
M$ in (A.9) are canonically equivalent to the morphisms induced from the
partition $[n+1]=[n]\sqcup_{[0]}[1]$ this completes the proof. ∎

We recall a criterion from (the proof of) [19, Prop. 1.1.8] for when a
category object is a groupoid object. Given a simplicial object
$X\in{\mathscr{F}\mathrm{un}}({\mathbbe{\Delta}}^{\mathrm{op}},\mathscr{C})$
in an $\infty$-category $\mathscr{C}$ and a simplicial set
$K\in{\mathscr{S}\mathrm{et}_{\Delta}}$, we define an object
$X(K)\in\mathscr{C}$ as the limit (if it exists) of the diagram

$N({\mathbbe{\Delta}}_{/K})^{\mathrm{op}}\longrightarrow
N{\mathbbe{\Delta}}^{\mathrm{op}}\overset{X}{\longrightarrow}\mathscr{C}\,.$
(A.12)

###### Proposition A.13

_[ 19]_ Let $\mathscr{C}$ be an $\infty$-category with finite limits. A
category object $X$ in $\mathscr{C}$ is a groupoid object in $\mathscr{C}$ if
and only if the inclusion $\Lambda^{2}_{0}\hookrightarrow\Delta^{2}$ induces
an equivalence $X_{2}\overset{\simeq}{\longrightarrow}X(\Lambda^{2}_{0})$.

Let $\mathscr{I}$ be the span category, depicted as
$\\{0,1\\}\leftarrow\\{0\\}\rightarrow\\{0,2\\}$. Consider the functor
$D\colon\mathscr{I}\to{\mathbbe{\Delta}}_{/\Lambda^{2}_{0}}$, which sends the
object $\\{0\\}\in\mathscr{I}$ to the canonical inclusion
$\Delta^{\\{0\\}}\hookrightarrow\Lambda^{2}_{0}$ and the object $\\{0,i\\}$ to
the canonical inclusion $\Delta^{\\{0,i\\}}\hookrightarrow\Lambda^{2}_{0}$,
for $i=0,2$.

###### Lemma A.14

Let $D\colon\mathscr{I}\to{\mathbbe{\Delta}}_{/\Lambda^{2}_{0}}$ be defined as
above, and let $\mathscr{C}$ be an $\infty$-category with finite limits. The
following statements hold true:

  1. (1)

The functor $D\colon\mathscr{I}\to{\mathbbe{\Delta}}_{/\Lambda^{2}_{0}}$ is
cofinal.

  2. (2)

For any
$X\in{\mathscr{F}\mathrm{un}}({\mathbbe{\Delta}}^{\mathrm{op}},\mathscr{C})$,
the diagram

${X(\Lambda^{2}_{0})}$${X_{1}}$${X_{1}}$${X_{0}}$$\scriptstyle{\iota_{0,1}^{*}}$$\scriptstyle{\iota_{0,2}^{*}}$$\scriptstyle{d_{1}}$$\scriptstyle{d_{1}}$
(A.15)

is a pullback diagram in $\mathscr{C}$, where $\iota_{0,i}^{*}$ denotes the
morphism $X(\Delta^{\\{0,i\\}}\hookrightarrow\Lambda^{2}_{0})$.

  3. (3)

A category object
$X\in{\mathscr{F}\mathrm{un}}({\mathbbe{\Delta}}^{\mathrm{op}},\mathscr{C})$
in $\mathscr{C}$ is a groupoid object precisely if the diagram

${X_{2}}$${X_{1}}$${X_{1}}$${X_{0}}$$\scriptstyle{d_{2}}$$\scriptstyle{d_{1}}$$\scriptstyle{d_{1}}$$\scriptstyle{d_{1}}$
(A.16)

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

###### Proof.

For claim (1), note that an object of ${\mathbbe{\Delta}}_{/\Lambda^{2}_{0}}$
is a pair $([n],\varphi)$ of an object $[n]\in{\mathbbe{\Delta}}$ and a
morphism of simplicial sets $\varphi\colon\Delta^{n}\to\Lambda^{2}_{0}$. We
show that, for each object $([n],\varphi)$ of
${\mathbbe{\Delta}}_{/\Lambda^{2}_{0}}$, the slice category
$([n],\varphi)_{/D}$ is contractible.

Since the horn $\Lambda^{2}_{0}$ fits into a pushout diagram

${\Delta^{\\{0\\}}}$${\Delta^{\\{0,1\\}}}$${\Delta^{\\{0,2\\}}}$${\Lambda^{2}_{0}}$$\scriptstyle{d^{1}}$$\scriptstyle{d^{1}}$
(A.17)

in ${\mathscr{S}\mathrm{et}_{\Delta}}$, the morphism
$\varphi\colon\Delta^{n}\to\Lambda^{2}_{0}$ is either the constant map at the
apex of the horn, i.e. $\varphi$ factors as
$\varphi\colon\Delta^{n}\to\Delta^{\\{0\\}}\hookrightarrow\Lambda^{2}_{0}$, or
it factors through a unique map $\Delta^{n}\to\Delta^{\\{0,i\\}}$, for $i=0$
or $i=2$, but not through the apex
$\Delta^{\\{0\\}}\hookrightarrow\Lambda^{2}_{0}$. (One can see this either by
writing $\Delta^{n}=N[n]$ and $\Lambda^{2}_{0}=N\mathscr{I}$ and using that
the nerve functor is fully faithful, or by using that
${\mathscr{S}\mathrm{et}_{\Delta}}(\Delta^{n},-)=(-)_{n}$ preserves colimits.)

In the first case, the slice category $([n],\varphi)_{/D}$ is the category
describing spans; in other words, it is isomorphic to $\mathscr{I}$, and we
have
$|N\mathscr{I}|\cong|\Delta^{1}\sqcup_{\Delta^{0}}\Delta^{1}|\cong|\Delta^{1}|\sqcup_{|\Delta^{0}|}|\Delta^{1}|\simeq*$.
In the other cases, the slice category $([n],\varphi)_{/D}$ is the final
category, and hence contractible as well.

Claim (2) now follows directly from the definition of $X(K)$, for
$K\in{\mathscr{S}\mathrm{et}_{\Delta}}$, together with part (1) (after taking
opposites), and claim (3) then follows by combining claim (2) with Proposition
A.13. ∎

###### Lemma A.18

Let $K$ be a simplicial set, let $\mathscr{C}$ be an $\infty$-category, and
let $C\in\mathscr{C}$ be an object. Let
$\mathsf{c}\colon\mathscr{C}\to{\mathscr{F}\mathrm{un}}(K,\mathscr{C})$ denote
the constant-diagram functor.

  1. (1)

If $K$ is contractible, i.e. $K\simeq*$ in ${\mathscr{S}\mathrm{et}_{\Delta}}$
with the Kan-Quillen model structure, and
${\mathrm{colim}}^{\mathscr{C}}_{K}(\mathsf{c}C)$ exists in $\mathscr{C}$,
then the canonical morphism
${\mathrm{colim}}^{\mathscr{C}}_{K}(\mathsf{c}C)\to C$ in $\mathscr{C}$ is an
equivalence.

  2. (2)

Dually, if $K$ is contractible and
${\mathrm{lim}}^{\mathscr{C}}_{K}(\mathsf{c}C)$ exists in $\mathscr{C}$, then
the canonical morphism $C\to{\mathrm{lim}}^{\mathscr{C}}_{K}(\mathsf{c}C)$ in
$\mathscr{C}$ is an equivalence.

###### Proof.

By the definition of $\mathsf{c}$, there is a commutative diagram

${K}$${\mathscr{C}}$${*}$$\scriptstyle{coll}$$\scriptstyle{\mathsf{c}C}$$\scriptstyle{C}$
(A.19)

in ${\mathbf{S}}$. By [20, Cor. 4.1.2.6, Thm. 4.1.3.1], the morphism $coll$ is
cofinal if and only if the simplicial set $K\times*\cong K$ is contractible,
i.e. precisely if $coll\colon K\to*$ is an equivalence in
${\mathscr{S}\mathrm{et}_{\Delta}}$ (in the Kan-Quillen model structure). The
first claim then follows from the fact that cofinal morphisms preserve
colimits [20, Prop. 4.1.1.8]. The second statement follows by duality. ∎

###### Example A.20

We need the following two specific cases in which Lemma A.18 applies:

  1. (1)

The nerve $N\mathscr{I}\in{\mathscr{S}\mathrm{et}_{\Delta}}$ is contractible,
as already seen in the proof of Lemma A.14.

  2. (2)

The inclusion $\\{[0]\\}\hookrightarrow{\mathbbe{\Delta}}$ is the inclusion of
a final object. Thus, the nerve
$N{\mathbbe{\Delta}}\in{\mathscr{S}\mathrm{et}_{\Delta}}$ is contractible in
${\mathscr{S}\mathrm{et}_{\Delta}}$. $\triangleleft$

###### Lemma A.21

Let $K\in{\mathscr{S}\mathrm{et}_{\Delta}}$ be contractible (in the Kan-
Quillen model structure) and let $\mathscr{C}$ be an $\infty$-category
admitting limits of shape $K$. Let $P\in\mathscr{C}$ be any object.

  1. (1)

The constant diagram functor
$\mathsf{c}\colon\mathscr{C}\to{\mathscr{F}\mathrm{un}}(K,\mathscr{C})$ is
fully faithful.

  2. (2)

If $\mathscr{C}$ admits finite products, then the functor
$P\times(-)\colon\mathscr{C}\to\mathscr{C}$ preserves limits of shape $K$.

###### Proof.

Let $C,D\in\mathscr{C}$ be any objects. To see (1), we use the adjunction
$\mathsf{c}\dashv{\mathrm{lim}}^{\mathscr{C}}_{K}$ and Lemma A.18, which yield
canonical equivalences

$\underline{\mathscr{C}^{K}}(\mathsf{c}C,\mathsf{c}D)\simeq\underline{\mathscr{C}}(C,{\mathrm{lim}}^{\mathscr{C}}_{K}\,\mathsf{c}D)\simeq\underline{\mathscr{C}}(C,D)\,.$
(A.22)

For claim (2), let $C,P\in\mathscr{C}$ be objects, and let $F\colon
K\to\mathscr{C}$ be a diagram. We now have canonical equivalences

$\displaystyle\underline{\mathscr{C}}\big{(}C,{\mathrm{lim}}^{\mathscr{C}}_{K}(\mathsf{c}P\times
F)\big{)}$
$\displaystyle\simeq\underline{\mathscr{C}^{K}}(\mathsf{c}C,\mathsf{c}P\times
F)$ (A.23)
$\displaystyle\simeq\underline{\mathscr{C}^{K}}(\mathsf{c}C,\mathsf{c}P)\times\underline{\mathscr{C}^{K}}(\mathsf{c}C,F)$
(A.24)
$\displaystyle\simeq\underline{\mathscr{C}}(C,P)\times\underline{\mathscr{C}}(C,{\mathrm{lim}}^{\mathscr{C}}_{K}F)$
(A.25)
$\displaystyle\simeq\underline{\mathscr{C}}(C,P\times{\mathrm{lim}}^{\mathscr{C}}_{K}F)\,.$
(A.26)

In the third equivalence we have used part (1), i.e. that $\mathsf{c}$ is
fully faithful here. Then, the statement follows from the Yoneda Lemma. ∎

###### Remark A.27

If we were working with categories instead of $\infty$-categories, it would
suffice to have an indexing category $\mathscr{J}$ (instead of $K$) which is
connected, rather than contractible, to prove an analogue of Lemma A.21.
$\triangleleft$

We can now prove Theorem 3.37:

###### Proof of Theorem 3.37.

Since every group object in $\mathscr{C}$ is in particular a monoid object in
$\mathscr{C}$, it follows from Proposition A.4 that $P{//}G$ is a category
object in $\mathscr{C}$. We now use Lemma A.14 to show that it is even a
groupoid object. By that lemma, it suffices to check that the diagram

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{{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}{}{}{{{}{}}}{}{{}}{}{}{}{{{}{}}}\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@endscope}}{{}}}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-4.13805pt}{-34.82852pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
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}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\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}}$ (A.28)

is a pullback diagram in $\mathscr{C}$, where we have used axioms (1) and (2)
of Definition A.3 and their consequences pointed out after Definition 3.33.

Our goal now is to split off the factor $P$ in diagram (A.28). To that end,
consider the diagram

${P}$${P}$${P\times G^{2}}$${P\times
G}$${G^{2}}$${G}$$\scriptstyle{\mathrm{pr}_{0}\simeq
d_{1}d_{2}}$$\scriptstyle{\mathrm{pr}_{12}}$$\scriptstyle{d_{1}}$$\scriptstyle{\mathrm{pr}_{0}=d_{1}}$$\scriptstyle{\mathrm{pr}_{1}}$$\scriptstyle{d_{1}^{G}}$
(A.29)

The bottom rectangle in diagram (A.29) commutes by axiom (3) of Definition
3.33. The top rectangle commutes because $P{//}G$ is a simplicial object in
$\mathscr{C}$, so we have a canonical equivalence $d_{1}d_{2}\simeq
d_{1}d_{1}$. This establishes the morphism $d_{1}$ as a morphism of binary
products in $\mathscr{C}$. As such, it is induced by the morphisms
$1_{P}\colon P\to P$ and $d_{1}^{G}\colon G^{2}\to G$. Thus, there is a
canonical equivalence

$d_{1}\simeq 1_{P}\times d_{1}^{G}\,,$ (A.30)

of morphisms $(P{//}G)_{2}\to(P{//}G)_{1}$ in ${\mathbf{H}}$. We thus have an
equivalence of diagrams

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}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{32.38191pt}{-27.61465pt}\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}{-4.13805pt}{-34.82852pt}\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{$\scriptstyle{d_{1}=\mathrm{pr}_{0}}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\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}}\quad\simeq\quad
P\times\left(\leavevmode\hbox to71.83pt{\vbox
to58.66pt{\pgfpicture\makeatletter\hbox{\hskip
36.16869pt\lower-29.38118pt\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}{-35.65652pt}{-29.28134pt}\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
9.63678pt\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}{-5.33124pt}{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{${G^{2}}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}}}&\hskip
9.63678pt\hfil&\hfil\hskip 43.80272pt\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}{-3.93124pt}{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{${G}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}&\hskip
8.23679pt\hfil\cr\vskip 28.45276pt\cr\hfil\hskip
8.23679pt\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}{-3.93124pt}{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{${G}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}&\hskip
8.23679pt\hfil&\hfil\hskip 42.37148pt\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}{-2.5pt}{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{${*}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}&\hskip
6.80554pt\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{-16.18297pt}{15.82416pt}\pgfsys@lineto{18.58301pt}{15.82416pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{18.78299pt}{15.82416pt}\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}{-1.4217pt}{19.4658pt}\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{$\scriptstyle{d_{2}^{G}}$} }}\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{-26.01974pt}{9.46445pt}\pgfsys@lineto{-26.01974pt}{-18.18835pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{0.0}{-1.0}{1.0}{0.0}{-26.01974pt}{-18.38834pt}\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}{-34.01591pt}{-7.03137pt}\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{$\scriptstyle{d_{1}^{G}}$} }}\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{27.41975pt}{9.46445pt}\pgfsys@lineto{27.41975pt}{-20.3689pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{0.0}{-1.0}{1.0}{0.0}{27.41975pt}{-20.56888pt}\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{-17.58296pt}{-26.78134pt}\pgfsys@lineto{20.01425pt}{-26.78134pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{20.21423pt}{-26.78134pt}\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}}\right)$ (A.31)

By the definition of ${\widehat{G}}$ as a groupoid object with $G_{0}\simeq*$,
we have that the diagram

${G^{2}}$${G}$${G}$${*}$$\scriptstyle{d_{2}^{G}}$$\scriptstyle{d_{1}^{G}}$
(A.32)

is a pullback diagram in $\mathscr{C}$ by Lemma A.13. It now follows from
Lemma A.21 that the functor $P\times(-)$ sends this pullback diagram to a
pullback diagram. Consequently, the square (A.28) is a pullback diagram in
${\mathbf{H}}$. ∎

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Universität Hamburg, Fachbereich Mathematik, Bereich Algebra und
Zahlentheorie,
Bundesstraße 55, 20146 Hamburg, Germany
<EMAIL_ADDRESS>