unramified at $p$, let $\Gamma$ be its Galois group and let
$F=\mathsf{F}\otimes\mathbb{Q}_{p}$. Let
${(\mathsf{G},\mathsf{X})}\to{(\mathsf{H}_{1},\mathsf{Y}_{1})}\to{(\mathsf{H},\mathsf{Y})}$
and
$\iota_{H_{1}}:{(\mathsf{H}_{1},\mathsf{Y}_{1})}\to(\mathsf{G}_{W},\mathsf{H}_{W})$
be as in Section 3.6, where we recall that
$W=V\otimes_{\mathbb{Q}}\mathsf{F}$. Let $\mathcal{H}_{1}$ denote the
Bruhat–Tits stabilizer group scheme associated to the image of $x$ under the
morphism of buildings $B(G,\mathbb{Q}_{p})\to B(H_{1},\mathbb{Q}_{p})$, and
let
$\mathcal{H}=\operatorname{Res}_{\mathcal{O}_{F}/\mathbb{Z}_{p}}\mathcal{G}_{\mathcal{O}_{F}}$
be the Bruhat–Tits stabilizer scheme associated to the image of $x$ in
$B(H,\mathbb{Q}_{p})$.

Recall from [KisinZhou, Section 2.4.8] that there is a natural map
$\beta:\mathcal{H}_{1}\to\mathcal{H}$.

######  Lemma 4.3.5.

If Hypothesis 4.3.2 holds, then the natural map
$\mathcal{H}_{1}\to\mathcal{H}$ is a closed immersion.

###### Proof.

By [KisinZhou, Proposition 2.4.9], it suffices to show that the centraliser
$T_{1}$ of a maximal $\breve{\mathbb{Q}}_{p}$ split torus in $H_{1}$ is an
$R$-smooth torus (in the sense of [KisinZhou, Section 5.1.2]). This is
automatic when $G$ is tamely ramified, see [KisinZhou, Proposition 2.4.6].
Otherwise, we observe that

(4.3.2) $\displaystyle
H_{1}^{\mathrm{Sp}}=\operatorname{Res}_{\mathsf{F}/\mathbb{Q}}\left(\operatorname{Res}_{\mathsf{K}_{i}/\mathbb{Q}}\mathsf{H}_{i}^{\mathrm{Sp}}\right)_{\mathsf{F}}$

and this implies as in the proof of [KisinZhou, Proposition 5.2.6] that
$T_{1}$ is $R$-smooth. ∎

#### 4.3.6.

We recall the following commutative diagram of closed embeddings of Shimura
data from Section 3.6

(4.3.3)
${{(\mathsf{G},\mathsf{X})}}$${(\mathsf{H}_{1},\mathsf{Y}_{1})}$${(\mathsf{G}_{V},\mathsf{H}_{V})}$${(\mathsf{G}_{V,\mathsf{F}},\mathsf{H}_{V,\mathsf{F}})}$${(\mathsf{G}_{W},\mathsf{H}_{W}).}$$\scriptstyle{\iota}$$\scriptstyle{\iota_{H_{1}}}$

Let $\mathcal{G}_{W}$ be the parahoric group scheme of $G_{W}$ that is the
stabilizer of the lattice
$\Lambda_{W}=\Lambda\otimes_{\mathbb{Z}_{p}}\mathcal{O}_{F}$ and let
$\mathcal{GL}_{W}$ be the reductive integral model of $\operatorname{GL}_{W}$
corresponding to $\Lambda_{W}$. We will also consider the parahoric model
$\mathcal{G}_{V,F}$ of $\mathsf{G}_{V,\mathsf{F}}$ corresponding to
$\Lambda\otimes\mathcal{O}_{F}$. We will write
$M_{p,F}=\mathcal{G}_{V,F}(\mathbb{Z}_{p})$ and
$M_{p,W}=\mathcal{G}_{W}(\mathbb{Z}_{p})$.

######  Lemma 4.3.7.

The morphism $\iota_{H_{1}}$ is a good Hodge embedding with respect to the
lattice $\Lambda_{W}=\Lambda\otimes_{\mathbb{Z}_{p}}\mathcal{O}_{F}\subset
W\otimes\mathbb{Q}_{p}$.

###### Proof.

We need to check that the morphism $\iota_{H_{1}}$ satisfies the following
three properties

  1. (i)

$\iota_{H_{1}}(H_{1})$ contains the scalars.

  2. (ii)

$\iota_{H_{1}}$ extends to a closed immersion
$\iota_{H_{1}}:\mathcal{H}_{1}\to\mathcal{GL}_{W}$.

  3. (iii)

The natural map of local models
$\mathbb{M}_{\mathcal{H}_{1},\mu}\to\mathbb{M}_{\mathcal{GL}_{W},\mu}$ induced
by $\mathcal{H}_{1}\to\mathcal{GL}_{W}$ is a closed immersion.777This
condition is equivalent to condition (3) of [KisinZhou, Definition 3.3.15]
because there is a unique natural map extending the natural map on the generic
fiber. Moreover, the local models used by [KisinZhou] agree with ours because
theirs also satisfy the Scholze–Weinstein conjecture (which means that they
have the correct associated v-sheaf), see [KisinZhou, Proposition 3.3.10].

We know that $\iota_{H_{1}}(H_{1})$ contains the scalars because $\iota(G)$
contains the scalars of $G_{V}$ and the image of the natural map $G_{V,F}\to
G_{W}$ contains the scalars. Property (ii) is a consequence of Lemma 4.3.5
together with the fact that $\mathcal{H}\to\mathcal{GL}_{W}$ is a closed
immersion because it is the restriction of scalars along
$\operatorname{Spec}\mathcal{O}_{F}\to\operatorname{Spec}\mathbb{Z}_{p}$ of
(the base change to $\mathcal{O}_{F}$ of) the closed immersion
$\mathcal{G}\to\mathcal{GL}_{V}$. Property (iii) follows because under our
assumptions we have functorial isomorphism (see Lemma 2.4.3)

(4.3.4) $\displaystyle\mathbb{M}_{\mathcal{GL}_{W},\mu}\otimes\mathcal{O}_{F}$
$\displaystyle\simeq\prod_{i=1}^{d}\mathbb{M}_{\mathcal{GL}_{V},\mu}\otimes\mathcal{O}_{F}$
(4.3.5)
$\displaystyle\mathbb{M}_{\mathcal{H}_{1},\mu}\mathcal{O}_{F}\simeq\mathbb{M}_{\mathcal{H},\mu}\otimes\mathcal{O}_{F}$
$\displaystyle\simeq\prod_{i=1}^{d}\mathbb{M}_{\mathcal{G},\mu}\otimes\mathcal{O}_{F}$

and property (iii) holds for $\iota:\mathcal{G}\to\mathcal{G}_{V}$ by
assumption. ∎

#### 4.3.8.

We let $K_{p}=\mathcal{H}(\mathbb{Z}_{p})$ which gives
$K_{p,1}=\mathcal{H}_{1}(\mathbb{Z}_{p})$ and
$K_{p}^{\Gamma}=\mathcal{G}(\mathbb{Z}_{p})$. We recall the construction of
$\mathscr{S}_{K_{p,1}}{(\mathsf{H}_{1},\mathsf{Y}_{1})}$ and
$\mathscr{S}_{K_{p}^{\Gamma}}{(\mathsf{G},\mathsf{X})}$ from [KisinZhou,
Section 5.1.4], and note that they come with natural maps

(4.3.6)
$\displaystyle\iota:\mathscr{S}_{K_{p}^{\Gamma}}{(\mathsf{G},\mathsf{X})}$
$\displaystyle\to\mathscr{S}_{M_{p}}(\mathsf{G}_{V},\mathsf{H}_{V})$ (4.3.7)
$\displaystyle\iota_{H_{1}}:\mathscr{S}_{K_{p,1}}{(\mathsf{H}_{1},\mathsf{Y}_{1})}$
$\displaystyle\to\mathscr{S}_{M_{p,W}}(\mathsf{G}_{W},\mathsf{H}_{W}),$

where $M_{p}=\mathcal{G}_{V}(\mathbb{Z}_{p})$ and
$M_{p,W}=\mathcal{G}_{W}(\mathbb{Z}_{p})$. The targets of these morphisms have
a moduli interpretation in terms of (weakly) polarized abelian schemes
$(A,\lambda)$ up to prime-to-$p$ isogeny with prime-to-$p$ level structures.
By [KisinZhou, Proposition 5.2.2], the $\mathcal{O}_{E}$ scheme
$\mathscr{S}_{K_{p}^{\Gamma}}{(\mathsf{G},\mathsf{X})}$ is a flat
$\mathsf{G}(\mathbb{A}_{f}^{p})$-equivariant extension of
$\mathbf{Sh}_{K_{p}^{\Gamma}}{(\mathsf{G},\mathsf{X})}_{E}$. Moreover
$\mathscr{S}_{K_{p}^{\Gamma}}{(\mathsf{G},\mathsf{X})}$ fits in a local model
diagram

$\textstyle{\widetilde{\mathscr{S}}_{K_{p}^{\Gamma}}{(\mathsf{G},\mathsf{X})}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\scriptstyle{q}$$\textstyle{\mathscr{S}_{K_{p}^{\Gamma}}{(\mathsf{G},\mathsf{X})}}$$\textstyle{\mathbb{M}_{\mathcal{G},\mu},}$

where $q$ is a $\mathcal{G}$-torsor and $\pi$ is pro-smooth of relative
dimension $\dim G$. We have a similar local model diagram for
$\mathscr{S}_{K_{p,1}}{(\mathsf{H}_{1},\mathsf{Y}_{1})}$ given by

$\textstyle{\widetilde{\mathscr{S}}_{K_{p,1}}{(\mathsf{H}_{1},\mathsf{Y}_{1})}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi_{1}}$$\scriptstyle{q_{1}}$$\textstyle{\mathscr{S}_{K_{p,1}}{(\mathsf{H}_{1},\mathsf{Y}_{1})}}$$\textstyle{\mathbb{M}_{\mathcal{H},\mu},}$

where $q_{1}$ is an $\mathcal{H}_{1}$-torsor and $\pi_{1}$ is pro-smooth of
relative dimension $\dim H_{1}$.

######  Remark 4.3.9.

The map $\pi$ has an entirely classical description over the generic fiber,
see [KisinPappas, Proposition 4.2.26]. The torsor of trivializations of the
first de-Rham cohomology of the universal abelian variety over
$\mathscr{S}_{M_{p}}(\mathsf{G}_{V},\mathsf{H}_{V})$ has a natural reduction
to a $\mathcal{G}$-bundle $\mathcal{V}_{dR}$. The morphism $\pi$ is given by
assigning to a trivialization $\phi$ of the first de-Rham cohomology, the pre-
image $\phi^{-1}(\mathcal{F}^{1})$ of the nontrivial step of the Hodge
filtration.

#### 4.3.10.

We require the following results on the local model diagram.

######  Proposition 4.3.11.

  1. (1)

There exists a $\Gamma$-action on
$\widetilde{\mathscr{S}}_{K_{p,1}}{(\mathsf{H}_{1},\mathsf{Y}_{1})}$ extending
that which fits in a commutative diagram

(4.3.8)
${\widetilde{\mathscr{S}}_{K_{p,1}}{(\mathsf{H}_{1},\mathsf{Y}_{1})}}$${\widetilde{\mathscr{S}}_{M_{p,F}}(\mathsf{G}_{V,\mathsf{F}},\mathsf{H}_{V,\mathsf{F}})}$${\mathscr{S}_{K_{p,1}}{(\mathsf{H}_{1},\mathsf{Y}_{1})}}$${\mathscr{S}_{M_{p,F}}(\mathsf{G}_{V,\mathsf{F}},\mathsf{H}_{V,\mathsf{F}})}$${\mathbb{M}_{\mathcal{H},\mu}}$${\mathbb{M}_{\mathcal{G}_{V,F},\mu},}$$\scriptstyle{q_{1}}$$\scriptstyle{\widetilde{\iota}_{H_{1}}}$$\scriptstyle{\pi_{F,V}}$$\scriptstyle{q_{F,V}}$$\scriptstyle{\iota_{H_{1}}}$$\scriptstyle{\pi_{1}}$$\scriptstyle{\iota_{H_{1},p}}$

and a $\Gamma$-action on
$\widetilde{\mathscr{S}}_{M_{p,F}}(\mathsf{G}_{V,\mathsf{F}},\mathsf{H}_{V,\mathsf{F}})$
such that $\pi_{1},\pi_{F,V},q_{1},q_{F,V},\iota_{H_{1},p},\iota_{H_{1}}$ are
$\Gamma$-equivariant.

###### Proof.

To prove the Proposition it suffices to consider the case
${(\mathsf{G},\mathsf{X})}=(\mathsf{G}_{V},\mathsf{H}_{V})$, whence
${(\mathsf{H}_{1},\mathsf{Y}_{1})}=(\mathsf{G}_{V,\mathsf{F}},\mathsf{H}_{V,\mathsf{F}})$.
Indeed, we already know that the $\Gamma$-action on
$\mathbf{Sh}_{K_{1}^{\prime}}{(\mathsf{H}_{1},\mathsf{Y}_{1})}$ extends to
$\mathscr{S}_{K_{1}^{\prime}}{(\mathsf{H}_{1},\mathsf{Y}_{1})}$ by
[DanielsVHKimZhangII, Corollary 4.0.11]. Thus we are only trying to show that
the action extends to the $\mathcal{H}_{1}$-torsor, and this admits a closed
immersion into the pullback of
$\widetilde{\mathscr{S}}_{M_{p,F}}(\mathsf{G}_{V,\mathsf{F}},\mathsf{H}_{V,\mathsf{F}})\to\mathscr{S}_{M_{p,F}}(\mathsf{G}_{V,\mathsf{F}},\mathsf{H}_{V,\mathsf{F}})$
along the $\Gamma$-equivariant natural map $\iota_{H_{1}}$. Moreover, the
$\Gamma$-equivariance of the induced maps can be checked on the generic fiber
and then over $\mathbb{C}$, where it is clear from uniformization.

We may also assume that $\Lambda_{V}\subset V$ is a self-dual
$\mathbb{Z}_{p}$-lattice whence we may assume the same for $\Lambda_{W}\subset
V_{F}$. To define the action of $\Gamma$ on the torsor
$\widetilde{\mathscr{S}}_{M_{p,F}}(\mathsf{G}_{V,\mathsf{F}},\mathsf{H}_{V,\mathsf{F}})$
we may make use of the moduli problem associated to this torsor. Recall that
$\mathscr{S}_{M_{p,F}}(\mathsf{G}_{V,\mathsf{F}},\mathsf{H}_{V,\mathsf{F}})$
classifies, over $S$ a $\mathbb{Z}_{p}$-scheme, tuples
$(A,\lambda,i,\overline{\eta})$ where:

  1. (1)

$h:A\to S$ is an abelian scheme of dimension
$[\mathsf{F}:\mathbb{Q}]\cdot\text{dim}(V)/2$ considered up to prime-to-p
isogeny.

  2. (2)

$\lambda$ is a $\mathbb{Z}_{(p)}^{\times}$-multiple of a principal
polarization.

  3. (3)

$i:\mathcal{O}_{\mathsf{F},(p)}\to\operatorname{End}_{S}(A)$ is a map of rings
such that the Rosati involution associated to $\lambda$ induces the identity
on $\mathcal{O}_{F}$ and such that $i$ satisfies a determinant condition as in
[KottwitzPoints, Section 5].

  4. (4)

$\eta$ is a trivialization
$\eta:V^{p}(A)\xrightarrow{\sim}V_{F}\otimes\mathbb{A}_{f}^{p}$ compatible
with the $\mathcal{O}_{F}$-action and with the polarization up to a scalar in
$\mathbb{A}_{f}^{p,\times}$.

The scheme
$\widetilde{\mathscr{S}}_{M_{p,F}}(\mathsf{G}_{V,\mathsf{F}},\mathsf{H}_{V,\mathsf{F}})$
classifies quintuples $(A,\lambda,i,\eta,\tau)$ where $(A,\lambda,i,\eta)$ are
as above and
$\tau:\Lambda^{\vee}_{V,F}\xrightarrow{\sim}R^{1}h_{*}\Omega_{A/S}^{*}$ is
compatible with the symplectic pairing and $\mathcal{O}_{F}$ actions on the
source and target. The $\Gamma$-action on the moduli problem
$\widetilde{\mathscr{S}_{M_{p,F}}}(\mathsf{G}_{V,\mathsf{F}},\mathsf{H}_{V,\mathsf{F}})$
sends a tuple $(A,\lambda,i,\eta,\tau)$ to the tuple
$(A,\lambda,i\otimes_{\mathcal{O}_{\mathsf{F},(p)},\gamma^{-1}}\mathcal{O}_{\mathsf{F},(p)},\eta,\gamma(\tau))$,
here
$\tau:\Lambda^{\vee}_{V,F}\otimes\mathcal{O}_{S}\xrightarrow{\sim}R^{1}h_{*}\Omega^{*}_{A/S}$
and we define $\gamma(\tau)$ to be the composition
$\tau\circ(\iota_{\gamma}\otimes\mathcal{O}_{S})$ where
$\iota_{\gamma}:\Lambda^{\vee}_{V,F}\otimes_{\mathcal{O}_{\mathsf{F},(p)},\gamma^{-1}}\mathcal{O}_{F}\xrightarrow{\sim}\Lambda^{\vee}_{V,F}$
is the natural isomorphism arising from the trivial descent datum on
$\Lambda_{V,F}\xrightarrow{\sim}\Lambda\otimes\mathcal{O}_{F}$. By definition,
this lifts the $\Gamma$ action on the moduli problem
$\mathscr{S}_{M_{p,F}}(\mathsf{G}_{V,\mathsf{F}},\mathsf{H}_{V,\mathsf{F}})$.
One is left to check that the map $\pi_{F,V}$ is $\Gamma$-equivariant, and
that this moduli interpretation for the $\Gamma$ action agrees with the naive
description of the $\Gamma$-action on the Hilbert–Siegel modular varieties we
consider by uniformization. We check the former, the latter is left to the
reader. By flatness and normality of
$\widetilde{\mathscr{S}}_{M_{p,F}}(\mathsf{G}_{V,\mathsf{F}},\mathsf{H}_{V,\mathsf{F}})$
and its local model it suffices to check this over the generic fiber, but here
we see that

$\pi_{F,V,\mathbb{Q}_{p}}(\gamma(A,\lambda,i,\eta,\tau))=\iota_{\gamma}^{-1}\tau^{-1}(\mathcal{F}^{1}(R^{1}h_{*}\Omega^{*}_{A/S}))$

which was exactly our definition of the $\Gamma$-action on the local model
$\mathbb{M}_{\mathcal{G}_{V,F},\mu,\mathbb{Q}_{p}}$, see Remark 4.3.9. ∎

######  Remark 4.3.12.

We had originally hoped to prove the $\Gamma$-equivariance in Proposition
4.3.11 using the $\Gamma$-equivariance of shtukas and the interpretation of
the local model diagram in terms of shtukas, see [PappasRapoportShtukas,
Section 4.9.1]. This does not seem to be possible however, since the scheme-
theoretic local model diagram is not uniquely pinned down by the v-sheaf
theoretic local model diagram, as discussed in [PappasRapoportShtukas, Section
4.9.1].

##  5\. Fixed points of integral models of Shimura varieties

In this section we prove Theorem 1 from the introduction.

###  5.1. Fixed points of integral models of Shimura varieties of Hodge type

Let ${(\mathsf{G},\mathsf{X})}$ be a Shimura datum of Hodge type, let
$\mathsf{F}$ be a totally real Galois extension of $\mathbb{Q}$ with Galois
group $\Gamma$. Let
${(\mathsf{G},\mathsf{X})}\to{(\mathsf{H}_{1},\mathsf{Y}_{1})}\to{(\mathsf{H},\mathsf{Y})}$
be as in Section 3.6. We let $\mathcal{G}^{\circ}$ be a parahoric model of $G$
that is the identity component of a Bruhat–Tits stabilizer group scheme
$\mathcal{G}$ corresponding to some point $x$ in the Bruhat–Tits building of
$G(\mathbb{Q}_{p})$. We let $\mathcal{H}^{\circ}\subset\mathcal{H}$ be the
corresponding objects for $H$, and we let $K_{p}=\mathcal{H}(\mathbb{Z}_{p})$
and $K_{p}^{\circ}=\mathcal{H}^{\circ}(\mathbb{Z}_{p})$. We let
$\mathcal{H}_{1}$ be the Bruhat–Tits stabilizer group scheme for $H_{1}$ and
we let $\mathcal{H}_{1}^{\prime}\subset\mathcal{H}_{1}$ be the inverse image
of $\mathcal{H}^{\circ}\subset\mathcal{H}$. We let $\mathcal{G}^{\mathrm{ad}}$
be the Bruhat–Tits stabilizer group scheme corresponding to the image of $x$
in the building of $G^{\mathrm{ad}}$, and similarly we define
$\mathcal{H}^{\mathrm{ad}}$. We also need the parahoric group schemes
$\mathcal{G}^{\mathrm{ad},\circ}$ and $\mathcal{H}^{\mathrm{ad},\circ}$, and
we observe that

(5.1.1)
$\displaystyle\mathcal{H}^{\mathrm{ad}}\simeq\operatorname{Res}_{\mathcal{O}_{F}/\mathbb{Z}_{p}}\mathcal{G}^{\mathrm{ad}}_{\mathcal{O}_{F}},\qquad\mathcal{H}^{\mathrm{ad},\circ}\simeq\operatorname{Res}_{\mathcal{O}_{F}/\mathbb{Z}_{p}}\mathcal{G}^{\mathrm{ad},\circ}_{\mathcal{O}_{F}}$

We set $K_{1,p}=\mathcal{H}_{1}(\mathbb{Z}_{p})=K_{p}\cap
H_{1}(\mathbb{Q}_{p})$ and
$K_{1,p}^{\prime}=\mathcal{H}_{1}(\mathbb{Z}_{p})=K_{p}^{\prime}\cap
H_{1}(\mathbb{Q}_{p})$. We also have
$K_{p}^{\Gamma}=\mathcal{G}(\mathbb{Z}_{p})$ and
$K_{p}^{\circ,\Gamma}=\mathcal{G}(\mathbb{Z}_{p})$. In what follows we let
$K^{p}\subset\mathsf{H}(\mathbb{A}_{f}^{p})$ denote a neat good compact open
subgroup such that there is a prime number $\ell\not=p$ coprime to the order
of $\Gamma$ such that $K^{p}=K^{p,\ell}K_{\ell}$ with $K_{\ell}$ a pro-$\ell$
group. For such $K^{p}$ we have
$K_{1},K_{1}^{\prime},K^{\Gamma},K^{\circ,\Gamma}$ defined in the obvious way
using $K_{1}^{p}$ and $K^{p,\Gamma}$. Unfortunately, the notation just
introduced conflicts with that of the introduction, but we will need the
Bruhat–Tits stabilizer group schemes in the proof of part (2) of Theorem 5.1.2
below.

Note that if $\mathsf{F}$ is tamely ramified over $\mathbb{Q}$, then such
$K^{p}$ form a cofinal collection of compact open subgroups by Proposition
3.4.7.

#### 5.1.1.

By [DanielsVHKimZhangII, Corollary 4.1.3] there is an integral model
$\mathscr{S}_{K_{1}^{\prime}}{(\mathsf{G},\mathsf{X})}$ equipped with a shtuka
(in the sense of [DanielsVHKimZhangII, Section 3.1.2])

(5.1.2)
$\displaystyle\mathscr{S}_{K_{1}^{\prime}}{(\mathsf{H}_{1},\mathsf{Y}_{1})}^{\lozenge/}\to\operatorname{Sht}_{\mathcal{H}_{1},\mu}.$

We note that these integral models agree, by construction, with the ones of
[KisinZhou] that we used in Section 4.3. We have similar integral models for
$K,K^{\circ},K_{1},K_{1}^{\prime},K^{\Gamma},K^{\circ,\Gamma}$.

######  Theorem 5.1.2.

Assume that $p$ is unramified in $\mathsf{F}$, that $p>2$ and that
${}^{1}(\mathbb{Q},\mathsf{G})\to{}^{1}(\mathsf{F},G)$.

  1. (1)

The natural map

(5.1.3)
$\displaystyle\mathscr{S}_{K^{\circ,\Gamma}}{(\mathsf{G},\mathsf{X})}\to\mathscr{S}_{K_{1}^{\prime}}{(\mathsf{H}_{1},\mathsf{Y}_{1})}^{\Gamma}.$

is a universal homeomorphism.

  2. (2)

If $p$ is coprime to $|\Gamma|\cdot|\pi_{1}(G^{\mathrm{der}})|$ and
${(\mathsf{G},\mathsf{X})}$ satisfies Hypothesis 4.3.2, then the natural map
of (5.1.3) is an isomorphism.

  3. (3)

If $K_{p}$ is hyperspecial, then the natural map of (5.1.3) is an isomorphism.

###### Proof.

Part (1): We first observe that the natural map
$\mathcal{H}_{1}^{\prime}\to\mathcal{H}^{\mathrm{ad}}$ factors through
$\mathcal{H}^{\mathrm{ad},\circ}$ because
$\mathcal{H}_{1}^{\prime}\to\mathcal{H}^{\mathrm{ad}}$ factors through
$\mathcal{H}$, and $\mathcal{H}_{1}^{\prime}$ maps to $\mathcal{H}^{\circ}$.
Now consider the corresponding map
$\mathscr{S}_{K_{1}^{\prime}}{(\mathsf{H}_{1},\mathsf{Y}_{1})}^{\lozenge/}\to\operatorname{Sht}_{\mathcal{H}_{1}^{\prime},\mu}\to\operatorname{Sht}_{\mathcal{H}^{\mathrm{ad},\circ},\mu}$;
we claim that it is $\Gamma$-equivariant in the $2$-categorical sense. It
suffices to prove this on the generic fiber as in the proof of Corollary
4.2.2. But then the map factors through a $\Gamma$-equivariant map to Shimura
varieties for ${(\mathsf{G}^{\mathrm{ad}},\mathsf{X}^{\mathrm{ad}})}$, and we
may apply 4.1.4 to ${(\mathsf{G}^{\mathrm{ad}},\mathsf{X}^{\mathrm{ad}})}$ to
conclude. It thus follows from Lemma 2.6.6, using the fact that $p$ is
unramified in $\mathsf{F}$, that there is an isomorphism

(5.1.4)
$\displaystyle\operatorname{Sht}_{\mathcal{G}^{\mathrm{ad},\circ},\mu}\to\left(\operatorname{Sht}_{\mathcal{H}^{\mathrm{ad},\circ},\mu}\right)^{h\Gamma}.$

This induces a map

(5.1.5)
$\displaystyle\mathscr{S}_{K_{1}^{\prime}}{(\mathsf{H}_{1},\mathsf{Y}_{1})}^{\lozenge/,\Gamma}\to\operatorname{Sht}_{\mathcal{G}^{\mathrm{ad},\circ},\mu}$

compatible with the maps
$\mathscr{S}_{K^{\circ,\Gamma}}{(\mathsf{G},\mathsf{X})}^{\lozenge/}\to\operatorname{Sht}_{\mathcal{G}^{\circ},\mu}\to\operatorname{Sht}_{\mathcal{G}^{\mathrm{ad},\circ},\mu}$
and
$\mathscr{S}_{K^{\circ,\Gamma}}{(\mathsf{G},\mathsf{X})}^{\lozenge/}\to\mathscr{S}_{K_{1}^{\prime}}{(\mathsf{H}_{1},\mathsf{Y}_{1})}^{\lozenge/,\Gamma}$.
For
$x\in\mathscr{S}_{K_{1}^{\prime}}{(\mathsf{H}_{1},\mathsf{Y}_{1})}^{\Gamma}(\overline{\mathbb{F}}_{p})$
we let
$b_{x}:\operatorname{Spd}\overline{\mathbb{F}}_{p}\to\operatorname{Sht}_{\mathcal{G}^{\mathrm{ad},\circ},\mu}$
be the corresponding $\mathcal{G}^{\mathrm{ad},\circ}$-shtuka. Then by the
proof of Lemma 4.2.4, there is a morphism

(5.1.6)
$\displaystyle\Psi_{b_{x}}:\left(\widehat{\mathscr{S}_{K_{1}^{\prime}}}{(\mathsf{H}_{1},\mathsf{Y}_{1})}^{\Gamma}\right)_{/x}^{\lozenge}\to\mathcal{M}^{\mathrm{int}}_{\mathcal{G}^{\mathrm{ad},\circ},b_{x},\mu,/x_{0}},$

and an isomorphism from the pullback under $\Psi_{b_{x}}$ of the tautological
$\mathcal{G}^{\mathrm{ad},\circ}$-shtuka on
$\mathcal{M}^{\mathrm{int}}_{\mathcal{G}^{\mathrm{ad},\circ},b_{x},\mu,/x_{0}}$
to the universal $\mathcal{G}^{\mathrm{ad},\circ}$-shtuka pulled back from
$\mathscr{S}_{K_{1}^{\prime}}{(\mathsf{H}_{1},\mathsf{Y}_{1})}^{\Gamma,\lozenge/}$.
If we let $d_{x}$ denote the induced map
$\operatorname{Spd}\overline{\mathbb{F}}_{p}\to\operatorname{Sht}_{\mathcal{H}^{\mathrm{ad},\circ},\mu}$,
then by the uniqueness proved in Lemma 4.2.4, there is a commutative diagram

(5.1.7)
${\left(\widehat{\mathscr{S}_{K_{1}^{\prime}}}{(\mathsf{H}_{1},\mathsf{Y}_{1})}^{\Gamma}\right)_{/x}^{\lozenge}}$${\mathcal{M}^{\mathrm{int}}_{\mathcal{G}^{\mathrm{ad},\circ},b_{x},\mu,/x_{0}}}$${\left(\widehat{\mathscr{S}_{K_{1}^{\prime}}}{(\mathsf{H}_{1},\mathsf{Y}_{1})}\right)_{/x}^{\lozenge}}$${\mathcal{M}^{\mathrm{int}}_{\mathcal{H}^{\mathrm{ad},\circ},b_{x},\mu,/x_{0}},}$$\scriptstyle{\Psi_{b_{x}}}$$\scriptstyle{\Psi_{d_{x}}}$

where the vertical arrows are the natural closed immersions. We note that the
map $\Psi_{d_{x}}$ is an isomorphism by Corollary 4.2.5 and the fact that

(5.1.8)
$\displaystyle\mathcal{M}^{\mathrm{int}}_{\mathcal{H}_{1}^{\prime},b_{x},\mu,/x_{0}}\to\mathcal{M}^{\mathrm{int}}_{\mathcal{H}^{\mathrm{ad},\circ},b_{x},\mu,/x_{0}}$

is an isomorphism, see [PappasRapoportRZ, Theorem 5.1.2]. It is moreover a
direct consequence of the uniqueness proved in Lemma 4.2.4 that the morphism
$\Psi_{d_{x}}$ is $\Gamma$-equivariant. Since taking $\Gamma$-invariants
commutes with taking diamond functors and formal completions, we see using
Proposition 2.6.10 that $\Psi_{b_{x}}$ is the $\Gamma$-fixed points of
$\Psi_{d_{x}}$. In particular, $\Psi_{b_{x}}$ is an isomorphism.

By [PappasRapoportRZ, Theorem 2.5.4, Theorem 2.5.5] in combination with
[GleasonLourenco, Corollary 1.4], using our assumption that $p>2$, there is a
flat normal formal scheme $\mathcal{N}$ such that

(5.1.9)
$\displaystyle\mathcal{N}^{\lozenge}\simeq\mathcal{M}^{\mathrm{int}}_{\mathcal{G}^{\mathrm{ad},\circ},b_{x},\mu,/x_{0}}.$

In particular, the unique $\overline{\mathbb{F}}_{p}$-point of
$\mathcal{M}^{\mathrm{int}}_{\mathcal{G}^{\mathrm{ad},\circ},b_{x},\mu,/x_{0}}$
(and thus of
$\left(\widehat{\mathscr{S}_{K_{1}^{\prime}}}{(\mathsf{H}_{1},\mathsf{Y}_{1})}^{\Gamma}\right)_{/x}$
) lifts to a $\operatorname{Spf}L$ point for a finite extension $L$ of
$\breve{\mathbb{Z}}_{p}$. Since this is true for all $x$, this implies that
$\mathscr{S}_{K_{1}^{\prime}}{(\mathsf{H}_{1},\mathsf{Y}_{1})}^{\Gamma}$ is
topologically flat over $\mathcal{O}_{E}$, i.e., that its generic fiber is
dense. Let
$\mathscr{S}_{K_{1}^{\prime}}{(\mathsf{H}_{1},\mathsf{Y}_{1})}^{\Gamma,\mathrm{awn}}\to\mathscr{S}_{K_{1}^{\prime}}{(\mathsf{H}_{1},\mathsf{Y}_{1})}^{\Gamma}$
be the absolute weak normalization, see [stacks-project, Lemma 0EUS]; this is
a universal homeomorphism and so
$\mathscr{S}_{K_{1}^{\prime}}{(\mathsf{H}_{1},\mathsf{Y}_{1})}^{\Gamma,\mathrm{awn}}$
is again topologically flat over $\mathcal{O}_{E}$. Then by [AGLR, Lemma
2.13], the natural map

(5.1.10)
$\displaystyle\left(\widehat{\mathscr{S}_{K_{1}^{\prime}}}{(\mathsf{H}_{1},\mathsf{Y}_{1})}^{\Gamma,\mathrm{awn}}\right)_{/x}^{\lozenge}\to\left(\widehat{\mathscr{S}_{K_{1}^{\prime}}}{(\mathsf{H}_{1},\mathsf{Y}_{1})}^{\Gamma}\right)_{/x}^{\lozenge}$

is an isomorphism. By the fully faithfulness of the $\lozenge$ functor on
absolute weakly normal formal schemes flat separated and topologically of
finite type over $\mathbb{Z}_{p}$, see [AGLR, Theorem 2.16], we find that

(5.1.11)
$\displaystyle\left(\widehat{\mathscr{S}_{K_{1}^{\prime}}}{(\mathsf{H}_{1},\mathsf{Y}_{1})}^{\Gamma,\mathrm{awn}}\right)_{/x}\simeq\mathcal{N}.$

This implies that the complete local rings of
$\mathscr{S}_{K_{1}^{\prime}}{(\mathsf{H}_{1},\mathsf{Y}_{1})}^{\Gamma,\mathrm{awn}}$
at $\overline{\mathbb{F}}_{p}$ points are normal, and thus that the local
rings are normal by [stacks-project, Lemma 0FIZ]. This shows that
$\mathscr{S}_{K_{1}^{\prime}}{(\mathsf{H}_{1},\mathsf{Y}_{1})}^{\Gamma,\mathrm{awn}}$
is normal, because normality (of a quasicompact scheme) can be checked at
closed points. By the universal property of the absolute weak normalization,
the natural map
$\mathscr{S}_{K^{\circ,\Gamma}}{(\mathsf{G},\mathsf{X})}\to\mathscr{S}_{K_{1}^{\prime}}{(\mathsf{H}_{1},\mathsf{Y}_{1})}^{\Gamma}$
lifts to a finite map

(5.1.12)
$\displaystyle\mathscr{S}_{K^{\circ,\Gamma}}{(\mathsf{G},\mathsf{X})}\to\mathscr{S}_{K_{1}^{\prime}}{(\mathsf{H}_{1},\mathsf{Y}_{1})}^{\Gamma,\mathrm{awn}}.$

This map is an isomorphism on the generic fiber by Theorem 3.6.4 and thus an
isomorphism since the target is normal and source and target are flat over
$\mathbb{Z}_{p}$, see [stacks-project, Lemma 0AB1]. The theorem now follows
from the fact that
$\mathscr{S}_{K_{1}^{\prime}}{(\mathsf{H}_{1},\mathsf{Y}_{1})}^{\Gamma,\mathrm{awn}}\to\mathscr{S}_{K_{1}}{(\mathsf{H}_{1},\mathsf{Y}_{1})}^{\Gamma}$
is a universal homeomorphism.

Part (2): We assume from now on that $p$ is coprime to the order of $\Gamma$
and to the order of $\pi_{1}(G^{\mathrm{der}})$, and that
${(\mathsf{G},\mathsf{X},\mathcal{G})}$ satisfies Assumption 4.3.2. It then
follows from Proposition 4.3.11 that there is a $\Gamma$-equivariant and
smooth map

(5.1.13)
$\displaystyle\mathscr{S}_{K_{1}}{(\mathsf{H}_{1},\mathsf{Y}_{1})}\to\left[\mathbb{M}_{\mathcal{H}^{\mathrm{ad}},\mu}/\mathcal{H}^{\mathrm{ad}}\right].$

It follows from [Edixhoven, Proposition 4.2], using the fact that $\Gamma$ is
of order prime-to-$p$, that the induced map

(5.1.14)
$\displaystyle\mathscr{S}_{K_{1}}{(\mathsf{H}_{1},\mathsf{Y}_{1})}^{\Gamma}\to\left[\mathbb{M}_{\mathcal{H}^{\mathrm{ad}},\mu}/\mathcal{H}^{\mathrm{ad}}\right]^{h\Gamma}.$

is also smooth. Recall that Corollary 2.4.8 tells us that the natural map

(5.1.15)
$\displaystyle\left[\mathbb{M}_{\mathcal{G}^{\mathrm{ad}},\mu}/\mathcal{G}^{\mathrm{ad}}\right]\to\left[\mathbb{M}_{\mathcal{H}^{\mathrm{ad}},\mu}/\mathcal{H}^{\mathrm{ad}}\right]^{h\Gamma}$

is an isomorphism. Thus
$\mathscr{S}_{K_{1}}{(\mathsf{H}_{1},\mathsf{Y}_{1})}^{\Gamma}$ has a smooth
local model diagram to $\mathbb{M}_{\mathcal{G}^{\mathrm{ad}},\mu}$ and is
thus flat over $\mathbb{Z}_{p}$ and normal since
$\mathbb{M}_{\mathcal{G}^{\mathrm{ad}},\mu}$ is, see [GleasonLourenco,
Corollary 1.4]. The natural map
$\mathscr{S}_{K_{1}^{\prime}}{(\mathsf{H}_{1},\mathsf{Y}_{1})}\to\mathscr{S}_{K_{1}}{(\mathsf{H}_{1},\mathsf{Y}_{1})}$
is $\Gamma$-equivariant (by [DanielsVHKimZhangII, Corollary 4.0.11]) and
finite étale (by the proof of [DanielsVHKimZhangII, Theorem 4.0.13]) and thus
the induced map on fixed points

(5.1.16)
$\displaystyle\mathscr{S}_{K_{1}^{\prime}}{(\mathsf{H}_{1},\mathsf{Y}_{1})}^{\Gamma}\to\mathscr{S}_{K_{1}}{(\mathsf{H}_{1},\mathsf{Y}_{1})}^{\Gamma}$

is also finite étale. It follows that
$\mathscr{S}_{K_{1}^{\prime}}{(\mathsf{H}_{1},\mathsf{Y}_{1})}^{\Gamma}$ is
flat over $\mathbb{Z}_{p}$ and normal since
$\mathscr{S}_{K_{1}}{(\mathsf{H}_{1},\mathsf{Y}_{1})}^{\Gamma}$ is. Now the
natural map of the theorem is a finite morphism that induces an isomorphism on
generic fibers by Theorem 3.6.4. Since the target is flat over
$\mathbb{Z}_{p}$ and normal, it follows from [stacks-project, Lemma 0AB1] that
the map is an isomorphism.

Part (3): We have a diagram

$\mathscr{S}_{K^{\circ,\Gamma}}{(\mathsf{G},\mathsf{X})}\to\mathscr{S}_{K_{1}^{\prime}}{(\mathsf{H}_{1},\mathsf{Y}_{1})}^{\Gamma}\to\mathscr{S}_{K_{1}^{\prime}}{(\mathsf{H}_{1},\mathsf{Y}_{1})}.$

We first observe that the composite is a closed immersion by [Xunormalization,
Theorem 1.1.1]. We now study the map on complete local rings, let us write

(5.1.17)
$\displaystyle\widehat{U}_{x}:=\widehat{\mathscr{S}}_{K_{1}^{\prime}}{(\mathsf{H}_{1},\mathsf{Y}_{1})}_{/x},\widehat{V}_{y}:=\widehat{\mathscr{S}}_{M_{p,F}}(\mathsf{G}_{V,\mathsf{F}},\mathsf{H}_{V,\mathsf{F}})_{/y}$

for $x\in\widehat{\mathscr{S}}_{K_{1}^{\prime}}(\overline{\mathbb{F}}_{p})$
and
$y\in\mathscr{S}_{M_{p,F}}(\mathsf{G}_{V,\mathsf{F}},\mathsf{H}_{V,\mathsf{F}})(\overline{\mathbb{F}}_{p})$.
Similarly for
$x\in\mathscr{S}_{K_{1}^{\prime}}{(\mathsf{H}_{1},\mathsf{Y}_{1})}^{\Gamma}(\overline{\mathbb{F}}_{p})$
we write
$\widehat{W}_{/x}:=\widehat{\mathscr{S}}_{K_{1}^{\prime}}{(\mathsf{H}_{1},\mathsf{Y}_{1})}^{\Gamma}_{/x}$,
and we let $x_{0}$ be the corresponding $\overline{\mathbb{F}}_{p}$ point of
$\mathscr{S}_{K^{\circ,\Gamma}}{(\mathsf{G},\mathsf{X})}$ (which exists by
part (1)) and $\widehat{T}_{x_{0}}$ the corresponding completion. There is a
chain of maps

$\widehat{T}_{x_{0}}\to\widehat{W}_{/x}\to\widehat{U}_{x}\to\widehat{V}_{/y}\to\widehat{\mathbb{M}}_{\mathcal{G}_{V,F},\mu,/\operatorname{GM}(y)},$

where all maps in this chain are closed immersions, and the final map is an
isomorphism to the local model for $(\mathcal{G}_{V,F},\mu)$, see
[KisinPappas, Proposition 4.2.2]. We wish to show that the final map
$\theta:\widehat{V}_{y}\to\widehat{\mathbb{M}}_{\mathcal{G}_{V,F},\mu,/\operatorname{GM}(y)}$
induces a $\Gamma$-equivariant map on tangent spaces. To do this we use that
this map has an explicit moduli theoretic interpretation on schemes that admit
divided powers. Let $(R_{y},m_{y})$ be the complete local ring associated to
the special fiber of $\widehat{V}_{y}$, and let $A_{y}=R_{y}/m_{y}^{p}$. Then
$R_{y}/m_{y}^{p}$ admits trivially a set of divided powers, and also caries an
action of the group $\Gamma$. If we let $(B,m_{B})$ denote the special fiber
of the complete local ring associated to
$\widehat{\mathbb{M}}_{\mathcal{G}_{V,F},\mu,/\operatorname{GM}(y)}$, then the
isomorphism $\bar{\theta}:B/m_{B}^{p}\to A_{y}/m_{y}^{p}$ has a classical
interpretation due to Grothendieck and Messing [Messing]: $A_{y}/m_{y}^{p}$ is
a quotient of the universal deformation ring of the universal principally
polarized abelian variety $\mathcal{A}$ with $\mathcal{O}_{F,p}$ action
$\iota_{A}$ living over $A_{y}$, and the map $\bar{\theta}$ is induced by the
map sending $\mathcal{A}$ to the image of the Hodge filtration of
$H^{1}_{dR}(\mathcal{A}/\operatorname{Spec}(A_{y}/m_{y}^{p}))$ inside of the
crystalline cohomology
$H^{1}_{cris}(\mathcal{A}_{y})_{\operatorname{Spec}(A_{y}/m_{y}^{p})}$, under
the $\mathcal{O}_{F,p}$-equivariant identification
$H^{1}_{dR}(\mathcal{A}/\operatorname{Spec}(A_{y}/m_{y}^{p}))\xrightarrow{\sim}H^{1}_{cris}(\mathcal{A}_{y})_{\operatorname{Spec}(A_{y}/m_{y}^{p})}$
(coming from the divided powers and the crystal property of crystalline
cohomology). Here we have an action of $\gamma\in\Gamma$ which sends
$A_{y}\to\gamma^{*}\mathcal{A}_{y}\xrightarrow{\sim}\mathcal{A}_{y}$ and thus
induces an $\mathcal{O}_{F,p}$-semilinear and $A_{y}/m_{y}^{p}$ linear
automorphism of
$H^{1}_{cris}(\mathcal{A}_{y})_{\operatorname{Spec}(A_{y}/m_{y}^{p})}$.
Because the map
$H^{1}_{dR}(\mathcal{A}/\operatorname{Spec}(A_{y}/m_{y}^{p}))\xrightarrow{\sim}H^{1}_{cris}(\mathcal{A}_{y})_{\operatorname{Spec}(A_{y}/m_{y}^{p})}$
is $\mathcal{O}_{F,p}$-equivariant, it is $\Gamma$-equivariant, and as there
is a canonical isomorphism
$\gamma^{*}(H^{1}_{dR}(\mathcal{A}/\operatorname{Spec}(A_{y}/m_{y}^{p})))\xrightarrow{\sim}H^{1}_{dR}(\gamma^{*}\mathcal{A}/\operatorname{Spec}(A_{y}/m_{y}^{p}))$
as $\mathcal{O}_{F,p}\otimes\operatorname{Spec}(A_{y}/m_{y}^{p}))$-modules, we
see that the map $\bar{\theta}$ is $\Gamma$-equivariant.

Finally, we have a closed immersion $\widehat{T}_{x_{0}}\to\widehat{W}_{x}$
where the source is smooth, and we wish to show that this is an isomorphism on
tangent spaces. Using [KisinPappas, Proposition 4.22] we may choose a point
$\operatorname{GM}(x)\in\widehat{\mathbb{M}}_{\mathcal{H},\mu}$ with image
$\operatorname{GM}(y)\in\widehat{\mathbb{M}}_{\mathcal{G}_{V,F},\mu}$ such
that the composition
$\widehat{U}_{/x}\to\widehat{V}_{y}\to\widehat{\mathbb{M}}_{\mathcal{G}_{V,F},\mu,/\operatorname{GM}(y)}$
factors through an isomorphism from $\widehat{U}_{/x}$ to
$\widehat{\mathbb{M}}_{\mathcal{H},\mu}$. Thus we have a commutative diagram

${\widehat{U}_{/x}}$${\widehat{\mathbb{M}}_{\mathcal{H},\mu,/\operatorname{GM}(x)}}$${\widehat{V}_{y}}$${\widehat{\mathbb{M}}_{\mathcal{G}_{V,F},\mu,/\operatorname{GM}(y)}.}$$\scriptstyle{i_{H_{1}}^{\mathrm{loc}}}$

As the vertical maps in the diagram are $\Gamma$-equivariant by Proposition
4.3.11, and the bottom horizontal map is $\Gamma$-equivariant on tangent
spaces by the result of the previous paragraph, we see that the induced
isomorphism
$T_{x}(\widehat{U}_{x})\xrightarrow{\sim}T_{GM(x)}(\widehat{\mathbb{M}}_{\mathcal{H},\mu,GM(x)})$
is $\Gamma$-equivariant on tangent spaces. Finally we deduce from Proposition
2.4.7 that $T_{x}(\widehat{W}_{x_{0}})\simeq
T_{x}(\widehat{U}_{x})^{\Gamma}\simeq
T_{GM(x)}(\widehat{\mathbb{M}}_{\mathcal{H},\mu,GM(x)})^{\Gamma}\simeq
T_{GM(x)}(\widehat{\mathbb{M}}_{\mathcal{G},\mu,GM(x)})$. Thus the map
$\widehat{T}_{x_{0}}\to\widehat{W}_{x}$ is a closed immersion of local formal
schemes with smooth source, which induces an isomorphism on tangent spaces and
a universal homeomorphism on topological spaces by part (1). By Lemma 5.1.3
below, it is thus an isomorphism.

∎

######  Lemma 5.1.3.

Let $f:A\to B$ be a surjective local homomorphism of complete local Noetherian
$\breve{\mathbb{Z}}_{p}$-algebras with $B$ flat and formally smooth over
$\breve{\mathbb{Z}}_{p}$. Suppose that $f$ induces a universal homeomorphism
on spectra. If $f$ induces an isomorphism on tangent spaces, then it is an
isomorphism.

###### Proof.

The proof is no doubt well known to experts, but we were unable to find a
suitable reference. Because $B$ is formally smooth over
$\breve{\mathbb{Z}}_{p}$ we have that it is a complete local ring of the form
$\breve{\mathbb{Z}}_{p}[[X_{1},\dots,X_{n}]]$. The map $f:A\to B$ admits a
section $g:B\to A$, which we may construct by taking $g(X_{i})$ to be any
$Y\in A$ such that $f(Y)=X_{i}$. The maximal ideal $m_{A}$ is a finitely
generated $A$ module as $A$ is Noetherian, and the map $g\circ
f|_{m_{A}}:m_{A}\to m_{A}$ is a surjective modulo the maximal ideal $m_{A}$ by
assumption, whence it is a surjection by Nakayama’s lemma. Thus we have that
$f$ induces an isomorphism $m_{B}\cong m_{A}$, and so as $m_{B}$ contains no
nilpotents, $m_{A}$ contains no nilpotents as well and thus $A$ is reduced.
Thus $f$ is an isomorphism. ∎

######  Remark 5.1.4.

One can prove a version of Part (1) of Theorem 5.1.2 for integral models of
Shimura varieties of abelian type ${(\mathsf{G},\mathsf{X})}$, using Theorem
3.5.1 as input on the generic fiber (thus we require the assumption that the
$\mathbb{R}$-split rank of $Z_{\mathsf{G}}$ is zero). To do this, one needs
integral models satisfying the Pappas–Rapoport axioms, see [DanielsYoucis] for
this.888In an earlier version of this paper, we proved a weaker version of
these axioms for the same integral models of Shimura varieties in order to
prove a version of Theorem 5.1.2. Our weaker version involved constructing
shtukas for $\mathcal{G}^{\mathrm{ad}}$, which is easier than constructing
$\mathcal{G}$-shtukas, see [DanielsYoucis, Remark 1.1]. In fact, the proof of
such a theorem is easier than the proof of Theorem 5.1.2, because one can
directly work with ${(\mathsf{H},\mathsf{Y})}$ instead of
${(\mathsf{H}_{1},\mathsf{Y}_{1})}$.

######  Remark 5.1.5.

If $p$ is completely split in $\mathsf{F}$, then the proof of part (1) of
Theorem 5.1.2 can be adapted to show that the natural map is actually an
isomorphism. The key point is that if $p$ is completely split, then one can
prove a version of Lemma 2.4.3 for the integral local Shimura varieties for
$\mathcal{H}^{\mathrm{ad},\circ}$, which then implies that Proposition 2.6.10
holds on the level of formal schemes. We leave the details to the interested
reader.

##  6\. Fixed points of Igusa stacks

In this section we discuss the fixed points of the Igusa stacks of
[DanielsVHKimZhang]. We use this to prove a version of Theorem 5.1.2 only
assuming the injectivity of
${}^{1}(\mathbb{Q},\mathsf{G})\to{}^{1}(\mathsf{F},\mathsf{G})$, see Theorem
6.3.1.

###  6.1. Igusa stacks

Fix an isomorphism $\overline{\mathbb{Q}}_{p}\to\mathbb{C}$. For
$L\subset\overline{\mathbb{Q}}_{p}$ let us denote by
$\operatorname{ShTrp}_{L,\operatorname{Hdg}}$ the category whose objects are
triples ${(\mathsf{G},\mathsf{X},\mathcal{G})}$ such that
$\mathsf{E}\to\mathbb{C}\to\overline{\mathbb{Q}}_{p}$ factors through $L$ and
such that ${(\mathsf{G},\mathsf{X},\mathcal{G})}$ is of Hodge type, and whose
morphisms are morphisms
$f:{(\mathsf{G},\mathsf{X})}\to(\mathsf{G}^{\prime},\mathsf{X}^{\prime})$ such
that $f$ extends (necessarily uniquely) to a morphism
$\mathcal{G}\to\mathcal{G}^{\prime}$. By [DanielsVHKimZhangII, Theorem I,
Corollary 4.0.11], there is a functor
$\mathscr{S}^{\diamond}:\operatorname{ShTrp}_{L,\operatorname{Hdg}}\to\mathcal{D}^{\circ}_{\mathcal{O}_{L}}$
sending ${(\mathsf{G},\mathsf{X},\mathcal{G})}$ to
$\mathscr{S}_{K_{p}}{(\mathsf{G},\mathsf{X})}^{\diamond}$, see Section 4.2.1.
Here $K_{p}=\mathcal{G}(\mathbb{Z}_{p})$ and
$\mathscr{S}_{K_{p}}{(\mathsf{G},\mathsf{X})}^{\diamond}$ is the diamond
associated to the formal scheme
$\widehat{\mathscr{S}}_{K_{p}}{(\mathsf{G},\mathsf{X})}$, which is the
$p$-adic completion of the integral model
$\mathscr{S}_{K_{p}}{(\mathsf{G},\mathsf{X})}$.

The following result is a consequence of [DanielsVHKimZhang, Theorem I,
Theorem II] as we will explain below. It was originally conjectured by
Scholze.

######  Theorem 6.1.1.

There is a functor
$\operatorname{Igs}:\operatorname{ShTrp}_{L,\operatorname{Hdg}}\to\mathcal{D}^{\circ}_{\mathcal{O}_{L}}$
together with a natural transformation
$\mathscr{S}^{\diamond}\to\operatorname{Igs}$ and a weak natural
transformations $\operatorname{Igs}\to\operatorname{Bun}$, and a strictly
commutative diagram of weak natural transformations

(6.1.1)
${\mathscr{S}^{\diamond}}$${\operatorname{Sht}}$${\operatorname{Igs}}$${\operatorname{Bun},}$

such that the resulting $2$-commutative diagrams of stacks on
$\operatorname{Perf}$

(6.1.2)
${\mathscr{S}_{K_{p}}{(\mathsf{G},\mathsf{X})}^{\diamond}}$${\operatorname{Sht}_{\mathcal{G},\mu}}$${\operatorname{Igs}_{K_{p}}{(\mathsf{G},\mathsf{X})}}$${\operatorname{Bun}_{G},}$

are $2$-Cartesian for all
${(\mathsf{G},\mathsf{X},\mathcal{G})}\in\operatorname{ShTrp}_{L,\operatorname{Hdg}}$,
and such that
$\operatorname{Igs}_{K_{p}}{(\mathsf{G},\mathsf{X})}\to\operatorname{Bun}_{G}$
factors through the open substack
$\operatorname{Bun}_{G,\mu^{-1}}\to\operatorname{Bun}_{G}$.

###### Proof.

The existence of $\operatorname{Igs}_{K_{p}}{(\mathsf{G},\mathsf{X})}$ is
[DanielsVHKimZhang, Theorem II], the existence of the strict functor
$\operatorname{Igs}$ and the strict natural transformation
$\mathscr{S}^{\diamond}\to\operatorname{Igs}$ is [DanielsVHKimZhang, Theorem
I]. The weak natural transformation $\operatorname{Igs}\to\operatorname{Bun}$
is constructed at the end of the proof of [DanielsVHKimZhang, Proposition
7.1.6], see the commutative cube there. By construction of the Igusa stack,
the diagram

(6.1.3)
${\mathscr{S}_{K_{p}}{(\mathsf{G},\mathsf{X})}^{\diamond}}$${\operatorname{Sht}_{\mathcal{G},\mu}}$${\operatorname{Igs}_{K_{p}}{(\mathsf{G},\mathsf{X})}}$${\operatorname{Bun}_{G},}$

is strictly commutative for all ${(\mathsf{G},\mathsf{X},\mathcal{G})}$, and
thus we see that the result follows.

∎

###  6.2. Fixed points of Igusa stacks

Let the notation be as in Section 5.1 and let $K_{p,1}^{\circ}\subset
K_{p}\cap H_{1}(\mathbb{Q}_{p})$ be the unique parahoric subgroup,
corresponding to the identity component of $\mathcal{H}_{1}$.

######  Theorem 6.2.1.

If ${}^{1}(\mathbb{Q},\mathsf{G})\to{}^{1}(\mathsf{F},G)$ is injective, then
the natural map

(6.2.1)
$\displaystyle\operatorname{Igs}_{K_{p}^{\circ,\Gamma}}{(\mathsf{G},\mathsf{X})}\to\operatorname{Igs}_{K_{p,1}^{\circ}}{(\mathsf{H}_{1},\mathsf{Y}_{1})}^{\Gamma}\times_{\operatorname{Bun}_{H_{1}}^{h\Gamma}}\operatorname{Bun}_{G,\mu^{-1}}$

is an isomorphism.

To prove the theorem, we need to introduce potentially crystalline loci.

####  6.2.2. The potentially crystalline locus in Hodge type Shimura varieties

Recall that there is an open immersion
$\mathbf{Sh}_{K}{(\mathsf{G},\mathsf{X})}^{\circ,\mathrm{an}}\subset\mathbf{Sh}_{K}{(\mathsf{G},\mathsf{X})}^{\mathrm{an}}$
of rigid spaces over $E$ called the _potentially crystalline locus_ ,
constructed in [ImaiMieda], see [ImaiMieda, Theorem 5.17]. The formation of
$\mathbf{Sh}_{K}{(\mathsf{G},\mathsf{X})}^{\circ,\lozenge}$ is compatible with
changing $K$, cf. [ImaiMieda, Corollary 5.29], and we will also consider

(6.2.2)
$\displaystyle\mathbf{Sh}_{K_{p}}{(\mathsf{G},\mathsf{X})}^{\circ,\lozenge}$
$\displaystyle=\varprojlim_{K^{p}\subset\mathsf{G}(\mathbb{A}_{f}^{p})}\mathbf{Sh}_{K^{p}K_{p}}{(\mathsf{G},\mathsf{X})}^{\circ,\lozenge}$
(6.2.3) $\displaystyle\mathbf{Sh}{(\mathsf{G},\mathsf{X})}^{\circ,\lozenge}$
$\displaystyle=\varprojlim_{K\subset\mathsf{G}(\mathbb{A}_{f}^{p})}\mathbf{Sh}_{K}{(\mathsf{G},\mathsf{X})}^{\circ,\lozenge}.$

######  Lemma 6.2.3.

If ${(\mathsf{G},\mathsf{X})}\to(\mathsf{G}^{\prime},\mathsf{X}^{\prime})$ is
a closed immersion of Shimura data, then for $K_{p}^{\prime}\subset
G^{\prime}(\mathbb{Q}_{p})$ containing $K\subset G(\mathbb{Q}_{p})$ there are
equalities of open subdiamonds

(6.2.4)
$\displaystyle\mathbf{Sh}_{K_{p}}{(\mathsf{G},\mathsf{X})}^{\circ,\lozenge}$
$\displaystyle=\mathbf{Sh}_{K_{p}^{\prime}}(\mathsf{G}^{\prime},\mathsf{X}^{\prime})^{\circ,\lozenge}\times_{\mathbf{Sh}_{K_{p}^{\prime}}(\mathsf{G}^{\prime},\mathsf{X}^{\prime})^{\lozenge}}\mathbf{Sh}_{K_{p}}{(\mathsf{G},\mathsf{X})}^{\lozenge}$
(6.2.5) $\displaystyle\mathbf{Sh}{(\mathsf{G},\mathsf{X})}^{\circ,\lozenge}$
$\displaystyle=\mathbf{Sh}(\mathsf{G}^{\prime},\mathsf{X}^{\prime})^{\circ,\lozenge}\times_{\mathbf{Sh}(\mathsf{G}^{\prime},\mathsf{X}^{\prime})^{\lozenge}}\mathbf{Sh}{(\mathsf{G},\mathsf{X})}^{\lozenge}.$

###### Proof.

This is a direct consequence of [ImaiMieda, Lemma 2.2], cf.
[DanielsVHKimZhang, Lemma 5.1.6, Lemma 5.1.7]. ∎

The following lemma is [DanielsVHKimZhang, Lemma 5.1.6]

######  Lemma 6.2.4.

There is an equality

(6.2.6)
$\displaystyle\mathbf{Sh}_{K}{(\mathsf{G},\mathsf{X})}^{\circ,\lozenge}=\mathscr{S}_{K_{p}}{(\mathsf{G},\mathsf{X})}^{\diamond}\times_{\operatorname{Spd}\mathcal{O}_{E}}\operatorname{Spd}E$

of open subdiamonds of $\mathbf{Sh}_{K}{(\mathsf{G},\mathsf{X})}^{\lozenge}$.

######  Lemma 6.2.5.

If ${}^{1}(\mathbb{Q},\mathsf{G})\to{}^{1}(\mathsf{F},G)$ is injective, then
the following diagram is $2$-Cartesian

(6.2.7)
${\mathbf{Sh}_{K_{p}^{\circ,\Gamma}}{(\mathsf{G},\mathsf{X})}^{\circ,\lozenge}}$${\mathbf{Sh}_{K_{p,1}^{\circ}}{(\mathsf{H}_{1},\mathsf{Y}_{1})}^{\circ,\Gamma,\lozenge}}$${\operatorname{Sht}_{\mathcal{G},\mu,E}}$${\left(\operatorname{Sht}_{\mathcal{H}_{1}^{\circ},\mu,E}\right)^{h\Gamma}.}$

###### Proof.

It follows from Lemma 3.6.7 that the natural map (where the limit runs over
$\Gamma$-stable compact open subgroups of $\mathsf{H}(\mathbb{A}_{f}^{p})$)

(6.2.8)
$\displaystyle\varprojlim_{K\subset\mathsf{H}(\mathbb{A}_{f}^{p})}\mathbf{Sh}_{K^{\Gamma}}{(\mathsf{G},\mathsf{X})}^{\lozenge}\to\varprojlim_{K\subset\mathsf{H}(\mathbb{A}_{f}^{p})}\mathbf{Sh}_{K_{1}}{(\mathsf{H}_{1},\mathsf{Y}_{1})}^{\lozenge,\Gamma}$

is an isomorphism. We then invoke Lemma 6.2.4 to get an isomorphism

(6.2.9)
$\displaystyle\mathbf{Sh}{(\mathsf{G},\mathsf{X})}^{\circ,\lozenge}\to\mathbf{Sh}{(\mathsf{H}_{1},\mathsf{Y}_{1})}^{\circ,\lozenge}.$

The map
$\mathbf{Sh}{(\mathsf{H}_{1},\mathsf{Y}_{1})}^{\circ,\lozenge}\to\mathbf{Sh}_{K_{p,1}^{\circ}}{(\mathsf{H}_{1},\mathsf{Y}_{1})}^{\circ,\lozenge}$
is a torsor for the sheaf of groups $\underline{K_{p,1}^{\circ}}$ associated
to the topological group $K_{p,1}^{\circ}$. Now we observe that there is a
natural (in particular $\Gamma$-equivariant) isomorphism

(6.2.10)
$\displaystyle\operatorname{Sht}_{\mathcal{H}_{1}^{\circ},\mu,E}\simeq\left[\operatorname{Gr}_{H,\mu^{-1}}/\underline{K_{p,1}^{\circ}}\right],$

see [ZhangThesis, Proposition 11.17]. Moreover, by construction, the
$\underline{K_{p,1}^{\circ}}$-torsor over
$\mathbf{Sh}_{K_{p,1}^{\circ}}{(\mathsf{H}_{1},\mathsf{Y}_{1})}^{\circ,\lozenge}$
coming from the map

(6.2.11)
$\displaystyle\mathbf{Sh}_{K_{p,1}}{(\mathsf{H}_{1},\mathsf{Y}_{1})}^{\circ,\lozenge}\to\operatorname{Sht}_{\mathcal{H}_{1}^{\circ},\mu,E},$

is given by
$\mathbf{Sh}{(\mathsf{H}_{1},\mathsf{Y}_{1})}^{\circ,\lozenge}\to\mathbf{Sh}_{K_{p,1}^{\circ}}{(\mathsf{H}_{1},\mathsf{Y}_{1})}^{\circ,\lozenge}$.
Thus we have a $\Gamma$-equivariant $2$-Cartesian diagram

(6.2.12)
${\mathbf{Sh}{(\mathsf{H}_{1},\mathsf{Y}_{1})}^{\circ,\lozenge}}$${\operatorname{Gr}_{H,\mu^{-1}}}$${\mathbf{Sh}{(\mathsf{H}_{1},\mathsf{Y}_{1})}^{\circ,\lozenge}/\underline{K_{p,1}^{\circ}}}$${\left[\operatorname{Gr}_{H,\mu^{-1}}/\underline{K_{p,1}^{\circ}}\right],}$

whose homotopy fixed points are again $2$-Cartesian by Lemma A.1.11. The
homotopy fixed points diagram looks like (see the proof of Proposition 2.4.7)

(6.2.13)
${\mathbf{Sh}{(\mathsf{G},\mathsf{X})}^{\circ,\lozenge}}$${\operatorname{Gr}_{G,\mu^{-1}}}$${\left(\mathbf{Sh}{(\mathsf{H}_{1},\mathsf{Y}_{1})}^{\circ,\lozenge}/\underline{K_{p,1}^{\circ}}\right)^{\Gamma}}$${\left[\operatorname{Gr}_{H,\mu^{-1}}/\underline{K_{p,1}^{\circ}}\right]^{h\Gamma}.}$

If we base change the bottom row via
$\mathbb{B}\underline{K_{p}^{\circ,\Gamma}}\to\mathbb{B}\underline{K_{p,1}^{\circ}}$,
then by Proposition A.2.8 we get the Cartesian diagram

(6.2.14)
${\mathbf{Sh}{(\mathsf{G},\mathsf{X})}^{\circ,\lozenge}}$${\operatorname{Gr}_{G,\mu^{-1}}}$${\mathbf{Sh}{(\mathsf{G},\mathsf{X})}^{\circ,\lozenge}/\underline{K_{p}^{\circ,\Gamma}}}$${\left[\operatorname{Gr}_{G,\mu^{-1}}/\underline{K_{p}^{\circ,\Gamma}}\right].}$

By Proposition A.2.8, this proves that the following diagram is Cartesian

(6.2.15)
${\mathbf{Sh}_{K_{p}^{\circ,\Gamma}}{(\mathsf{G},\mathsf{X})}^{\circ,\lozenge}}$${\mathbf{Sh}_{K_{p,1}^{\circ}}{(\mathsf{H}_{1},\mathsf{Y}_{1})}^{\circ,\Gamma,\lozenge}}$${\left[\operatorname{Gr}_{G,\mu^{-1}}/\underline{K_{p}^{\circ,\Gamma}}\right]}$${\left[\operatorname{Gr}_{H,\mu^{-1}}/\underline{K_{p,1}^{\circ}}\right]^{h\Gamma},}$

proving the lemma. ∎

###### Proof of Theorem 6.2.1.

It follows from Theorem 6.1.1 that there is a $2$-commutative square

(6.2.16)
${\operatorname{Igs}_{K_{p}^{\circ,\Gamma}}{(\mathsf{G},\mathsf{X})}}$${\operatorname{Bun}_{G,\mu^{-1}}}$${\operatorname{Igs}_{K_{p,1}^{\circ}}{(\mathsf{H}_{1},\mathsf{Y}_{1})}^{h\Gamma}}$${\operatorname{Bun}_{H_{1}}^{h\Gamma}.}$

This induces a map

(6.2.17)
$\displaystyle\operatorname{Igs}_{K_{p}^{\circ,\Gamma}}{(\mathsf{G},\mathsf{X})}\to\operatorname{Igs}_{K_{p,1}^{\circ}}{(\mathsf{H}_{1},\mathsf{Y}_{1})}^{h\Gamma}\times_{\operatorname{Bun}_{H_{1}}^{h\Gamma}}\operatorname{Bun}_{G,\mu^{-1}}$

which we will show is an isomorphism. By v-descent we can do this after
basechanging via the v-cover

(6.2.18)
$\displaystyle\operatorname{Sht}_{\mathcal{G}^{\circ},\mu,E}\to\operatorname{Bun}_{G,\mu^{-1}}$

from [DanielsVHKimZhang, Corollary 6.4.2]. Using Theorem 6.1.1, Lemma 6.2.4
and Lemma A.1.11, we can identify the basechanged map with the natural map

(6.2.19)
$\displaystyle\mathbf{Sh}_{K_{p}^{\circ,\Gamma}}{(\mathsf{G},\mathsf{X})}^{\circ,\lozenge}\to\mathbf{Sh}_{K_{p,1}^{\circ}}{(\mathsf{H}_{1},\mathsf{Y}_{1})}^{\circ,\lozenge,\Gamma}\times_{\operatorname{Sht}_{\mathcal{H}_{1}^{\circ},\mu,E}^{h\Gamma}}\operatorname{Sht}_{\mathcal{G}^{\circ},\mu,E}.$

This natural map is an isomorphism by Lemma 6.2.5. ∎

###  6.3. Fixed points of integral models of Shimura varieties of Hodge type
II

Let the notation be as in Section 6.2 above; the following result seems
essentially optimal.

######  Theorem 6.3.1.

If ${}^{1}(\mathbb{Q},\mathsf{G})\to{}^{1}(\mathsf{F},G)$ is injective, then
the following diagram of v-stacks is $2$-Cartesian

(6.3.1)
${\mathscr{S}_{K_{p}^{\circ,\Gamma}}{(\mathsf{G},\mathsf{X})}^{\lozenge/}}$${\mathscr{S}_{K_{p,1}^{\circ}}{(\mathsf{H}_{1},\mathsf{Y}_{1})}^{\Gamma,\lozenge/}}$${\operatorname{Sht}_{\mathcal{G}^{\circ},\mu}}$${\operatorname{Sht}_{\mathcal{H}_{1}^{\circ},\mu}^{h\Gamma}.}$

###### Proof.

If we basechange the isomorphism

(6.3.2)
$\displaystyle\operatorname{Igs}_{K_{p}^{\circ,\Gamma}}{(\mathsf{G},\mathsf{X})}\to\operatorname{Igs}_{K_{p,1}^{\circ}}{(\mathsf{H}_{1},\mathsf{Y}_{1})}^{h\Gamma}\times_{\operatorname{Bun}_{H_{1}}^{h\Gamma}}\operatorname{Bun}_{G,\mu^{-1}}$

of Theorem 6.2.1 via
$\operatorname{Sht}_{\mathcal{G}^{\circ},\mu}\to\operatorname{Bun}_{G,\mu^{-1}}$,
then by Theorem 6.1.1 we get the natural map

(6.3.3)
$\displaystyle\mathscr{S}_{K_{p}^{\circ,\Gamma}}{(\mathsf{G},\mathsf{X})}^{\diamond}\to\mathscr{S}_{K_{1,p}^{\circ}}{(\mathsf{H}_{1},\mathsf{Y}_{1})}^{\Gamma,\diamond}\times_{\operatorname{Sht}_{\mathcal{H}_{1}^{\circ},\mu}^{h\Gamma}}\operatorname{Sht}_{\mathcal{G}^{\circ},\mu},$

which is therefore an isomorphism of v-sheaves. If we combine this isomorphism
with Lemma 6.2.5, then we see that the natural map

(6.3.4)
$\displaystyle\mathscr{S}_{K_{p}^{\circ,\Gamma}}{(\mathsf{G},\mathsf{X})}^{\lozenge/}\to\mathscr{S}_{K_{p,1}^{\circ}}{(\mathsf{H}_{1},\mathsf{Y}_{1})}^{\Gamma,\lozenge/}\times_{\operatorname{Sht}_{\mathcal{H}_{1}^{\circ},\mu}^{h\Gamma}}\operatorname{Sht}_{\mathcal{G}^{\circ},\mu},$

is an isomorphism (this follows from the definition of $\lozenge/$, see
Section 2.3.3). This concludes the proof. ∎

######  Remark 6.3.2.

If $\mathsf{F}$ is tamely ramified over $\mathbb{Q}$, then one can also use
Theorem 3.6.4 to prove a version of Lemma 6.2.5 for well chosen
$K^{p}\subset\mathsf{H}(\mathbb{A}_{f}^{p})$. This will then lead to an
analogue of Theorem 6.2.1 for the morphism of stacks999We are now applying
Theorem 6.1.1 with quasi-parahoric level, which is also proved in
[DanielsVHKimZhang].

(6.3.5)
$\displaystyle\left[\operatorname{Igs}_{K_{p}^{\circ,\Gamma}}{(\mathsf{G},\mathsf{X})}/\underline{K^{p,\Gamma}}\right]\to\left[\operatorname{Igs}_{K_{p,1}^{\prime}}{(\mathsf{G},\mathsf{X})}/\underline{K_{1}^{p}}\right]^{h\Gamma}$

and to an analogue of Theorem 6.3.1 for the Shimura varieties of level
$K^{\circ,\Gamma}$ and $K_{1}^{\prime}$.

##  Appendix A Some $(2,1)$-category theory

###  A.1. Strict $(2,1)$-categories and weak functors

Recall from [stacks-project, Section 02X8] the definition of a strict
$(2,1)$-category. We will use $\operatorname{Mor}^{1}(x,y)$ to refer to the
category of $1$-morphisms between $x$ and $y$ in such a category, and
sometimes abusively to refer to the class of objects of this category using
the same notation. We will sometimes refer to isomorphisms in the category
$\operatorname{Mor}^{1}(x,y)$ as natural transformations.

######  Definition A.1.1.

Let $\mathcal{C}$ be a 1-category and $\mathcal{D}$ a strict $(2,1)$-category.
We define a _weak functor_ $\mathcal{F}:\mathcal{C}\to\mathcal{D}$ to be a
pair $(\mathcal{F},\eta_{\mathcal{F}})$ consisting of:

  1. (1)

An assignment

$\mathcal{F}:\operatorname{Ob}(\mathcal{C})\to\operatorname{Ob}(\mathcal{D})$

and a map
$\mathcal{F}:\operatorname{Mor}(x,y)\to\operatorname{Ob}(\operatorname{Mor}^{1}(\mathcal{F}(x),\mathcal{F}(y)))$.

  2. (2)

For every $x\in\operatorname{Ob}(\mathcal{C})$ a 2-morphism
$\eta_{\mathcal{F},x}:\text{Id}_{\mathcal{F}(x)}\to\mathcal{F}(\text{Id}_{x})$.

  3. (3)

For every composable pair $f:x\to y$, $g:y\to z$ in $\mathcal{C}$ a 2-morphism
$\eta_{\mathcal{F},f,g}:\mathcal{F}(g\circ
f)\to\mathcal{F}(g)\circ\mathcal{F}(f)$.

Such that:

  1. (1)

For any morphism $f:x\to y$ in $\mathcal{C}$ we have that

$\eta_{\mathcal{F},f,\text{Id}_{y}}=\eta_{\mathcal{F},y}\circ\text{Id}_{\mathcal{F}(f)}$

and

$\eta_{\mathcal{F},\text{Id}_{x},f}=\text{Id}_{\mathcal{F}(f)}\circ\eta_{\mathcal{F},x}.$

  2. (2)

For any composable triple $f:x\to y,g:y\to z,h:z\to t$ we have

$(\text{Id}_{\mathcal{F}(h)}\circ\eta_{\mathcal{F},f,g})\circ\eta_{\mathcal{F},g\circ
f,h}=(\eta_{\mathcal{F},g,h}\circ\text{Id}_{\mathcal{F}(f)})\circ\eta_{\mathcal{F},f,h\circ
g}.$

######  Definition A.1.2.

A weak natural transformation $\epsilon:\mathcal{F}\to\mathcal{G}$ of weak
functors $\mathcal{F},\mathcal{G}:\mathcal{C}\to\mathcal{D}$ is:

  1. (1)

A collection for $x\in\operatorname{Ob}(\mathcal{C})$ of
$\epsilon_{x}:\mathcal{F}(x)\to\mathcal{G}(x)$ of 1-morphisms in
$\mathcal{D}$.

  2. (2)

For each morphism $f:x\to y$ in $\mathcal{C}$, a natural transformation
$\epsilon(f):\epsilon_{y}\circ\mathcal{F}(f)\to\mathcal{G}(f)\circ\epsilon_{x}$
in the category $\operatorname{Mor}^{1}(\mathcal{F}(x),\mathcal{G}(y))$, we
will denote this in diagram form by

${\mathcal{F}(x)}$${\mathcal{F}(y)}$${\mathcal{G}(x)}$${\mathcal{G}(y),}$$\scriptstyle{\mathcal{F}(f)}$$\scriptstyle{\epsilon_{x}}$$\scriptstyle{\epsilon_{y}}$$\scriptstyle{\epsilon(f)}$$\scriptstyle{\mathcal{G}(f)}$

satisfying the following conditions:

  1. (1)

We have an equality of natural transformations for every pair of morphisms
$f:x\to y,g:y\to z$:

$\epsilon(g\circ
f)\circ\eta_{\mathcal{F},f,g}=\eta_{\mathcal{G},f,g}\circ\mathcal{G}(g)_{*}(\epsilon(f))\circ\mathcal{F}(f)^{*}(\epsilon(g)).$

  2. (2)

For every object $x\in\operatorname{Ob}(\mathcal{C})$ we ask for commutativity
of the diagram

${\epsilon_{x}}$${\epsilon_{x}\circ\text{Id}_{\mathcal{F}(x)}}$${\text{Id}_{\mathcal{G}(x)}\circ\epsilon_{x}}$${\epsilon_{x}\circ\mathcal{F}(\text{Id}_{x})}$${\mathcal{G}(\text{Id}_{x})\circ\epsilon_{x}}$$\scriptstyle{=}$$\scriptstyle{\eta_{\mathcal{F},x}}$$\scriptstyle{=}$$\scriptstyle{\eta_{\mathcal{G},x}}$$\scriptstyle{\epsilon(\text{Id}_{x})}$

inside of the category $\operatorname{Mor}(\mathcal{F}(x),\mathcal{G}(x))$. We
note that because we work with strict $(2,1)$-categories the commutativity of
this diagram is well-posed.

Let $\Gamma$ be an abstract group. We define the category $B\Gamma$ to be the
classifying category of the abstract group $\Gamma$ (that is, the category
with one object $\ast$ with automorphism group $\Gamma$). Suppose
$\mathcal{D}$ is a strict $(2,1)$-category.

######  Definition A.1.3.

A $\Gamma$-object in $\mathcal{D}$ is a weak functor
$\mathcal{F}:B\Gamma\to\mathcal{D}$. A $\Gamma$-equivariant morphism is a weak
natural transformation of such functors.

Let $\mathcal{D}$ be a strict $(2,1)$-category and let $x$ be an object of
$\mathcal{D}$. Then a weak functor $\mathcal{F}:B\Gamma\to\mathcal{D}$ sending
$\ast$ to $x$ is called a _weak $\Gamma$-action on $x$_.

######  Example A.1.4.

Let $\mathcal{D}$ be a strict $(2,1)$-category and let $x$ be an object of
$\mathcal{D}$. Then the trivial $\Gamma$-action on $x$ is the following weak
functor $\mathcal{F}:B\Gamma\to\mathcal{D}$: On objects it sends $\ast$ to $x$
and on morphisms it sends all morphisms to the identity $x\to x$. Moreover the
$2$-morphisms $\eta_{\mathcal{F},x}$ and $\eta_{\mathcal{F},f,g}$ are all
taken to be the identity $2$-morphism.

######  Example A.1.5.

Let $\mathcal{D}$ be a strict $(2,1)$-category and let $x$ be an object of
$\mathcal{D}$ such that $\operatorname{Mor}^{1}(x,x)$ is a discrete
category.101010A discrete category is a category where the only morphisms are
the identity morphisms. Then $\operatorname{Ob}(\operatorname{Mor}^{1}(x,x)$
is a group under composition of morphisms and a weak $\Gamma$-action on $x$ is
the same as a group homomorphism
$\Gamma\to\operatorname{Ob}(\operatorname{Mor}^{1}(x,x)$.

#### A.1.6.

Recall [stacks-project, Tag 003R] that fiber products in the $(2,1)$-category
of categories have the following description. Let
$\mathcal{A},\mathcal{B},\mathcal{C}$ be $(2,1)$-categories and let
$\mathcal{F}:\mathcal{A}\to\mathcal{C},\mathcal{G}:\mathcal{B}\to\mathcal{C}$
be functors. Then there is the following canonical presentation of the fiber
product:

######  Definition A.1.7.

The fiber product $\mathcal{A}\times_{\mathcal{C}}\mathcal{B}$ identifies with
the strict $(2,1)$-category whose objects consist of triples $(a,b,f)$ where
$a\in\operatorname{Ob}(\mathcal{A}),b\in\operatorname{Ob}(\mathcal{B})$ and
$f:\mathcal{F}(a)\to\mathcal{G}(b)$ is an isomorphism in $\mathcal{C}$. The
morphisms $\phi:(a,b,f)\to(c,d,g)$ consist of pairs $(X,Y)$ where $X:a\to
c,Y:b\to d$ are such that the diagram

${\mathcal{F}(a)}$${\mathcal{G}(b)}$${\mathcal{F}(c)}$${\mathcal{G}(d)}$$\scriptstyle{f}$$\scriptstyle{X}$$\scriptstyle{Y}$$\scriptstyle{g}$

commutes.

######  Definition A.1.8.

Let $X$ be a $\Gamma$-object in the strict $(2,1)$-category of categories,
defined by some functor $\mathcal{F}$. Then we define the $\Gamma$-_homotopy
fixed points_ $X^{h\Gamma}$ of $X$ to be the following category: The objects
of $X^{h\Gamma}$ are tuples $(x,\\{\tau_{\gamma}\\}_{\gamma\in\Gamma})$, where
$x\in\operatorname{Ob}(X)$ is an object and
$\tau_{\gamma}:x\to\mathcal{F}(\gamma)(x)$ for each $\gamma\in\Gamma$ is an
isomorphism such that for all $\gamma,\gamma^{\prime}$ the diagram

${x}$${\mathcal{F}(\gamma^{\prime})(x)}$${\mathcal{F}(\gamma^{\prime}\gamma)(x)}$${\mathcal{F}(\gamma^{\prime})(\mathcal{F}(\gamma)(x))}$$\scriptstyle{\tau_{\gamma^{\prime}}}$$\scriptstyle{\tau_{\gamma^{\prime}\gamma}}$$\scriptstyle{\mathcal{F}(\gamma^{\prime})(\tau_{\gamma})}$$\scriptstyle{\eta_{\mathcal{F},\gamma,\gamma^{\prime}}}$

is commutative. A morphism $f:(x,\tau)\to(x^{\prime},\tau^{\prime})$ of such
objects is a morphism $f:x\to x^{\prime}$ such that the following diagram
commutes for all $\gamma\in\Gamma$

(A.1.1)
${x}$${x^{\prime}}$${\mathcal{F}(\gamma)(x)}$${\mathcal{F}(\gamma)(x^{\prime}).}$$\scriptstyle{f}$$\scriptstyle{\tau_{\gamma}}$$\scriptstyle{\tau^{\prime}_{\gamma}}$$\scriptstyle{\mathcal{F}(f)}$

######  Lemma A.1.9.

A $\Gamma$-equivariant morphism $\alpha:X\to Y$ of $\Gamma$-objects in the
strict $(2,1)$-category of categories defines a natural functor

(A.1.2) $\displaystyle\alpha^{h\Gamma}:X^{h\Gamma}\to Y^{h\Gamma}.$

Furthermore, this functor is an equivalence if $\alpha$ is an equivalence.

Let $\mathcal{CAT}$ be the strict $(2,1)$-category of categories and suppose
that $X$ and $Y$ are given by weak functors
$\mathcal{F},\mathcal{G}:B\Gamma\to\mathcal{CAT}$. Then the morphism $\alpha$
is precisely a weak natural transformation
$\epsilon:\mathcal{F}\to\mathcal{G}$. In particular, for each
$\gamma\in\Gamma$ we have a natural transformation

(A.1.3)
${X}$${X}$${Y}$${Y.}$$\scriptstyle{\mathcal{F}(\gamma)}$$\scriptstyle{\alpha}$$\scriptstyle{\alpha}$$\scriptstyle{\epsilon(f)}$$\scriptstyle{\mathcal{G}(\gamma)}$

Given a tuple $(x,\\{\tau_{\gamma}\\}_{\gamma\in\Gamma})\in X^{h\Gamma}$ we
would like to define its image under $\alpha^{h\Gamma}$ to be the tuple
$(\alpha(x),\\{\epsilon_{\gamma}\circ\alpha(\tau_{\gamma})\\}_{\gamma\in\Gamma})$.
To check that this tuple defines an object of $Y^{h\Gamma}$, we need to check
that for $\gamma,\gamma^{\prime}\in\Gamma$ the diagram

(A.1.4)
${\alpha(x)}$${\mathcal{G}(\gamma^{\prime})(\alpha(x))}$${\mathcal{G}(\gamma^{\prime}\gamma)(\alpha(x))}$${\mathcal{G}(\gamma^{\prime})(\mathcal{G}(\gamma)(\alpha(x))}$$\scriptstyle{\epsilon_{\gamma^{\prime}}\circ\alpha(\tau_{\gamma^{\prime}})}$$\scriptstyle{\epsilon_{\gamma^{\prime}\gamma}\circ\alpha(\tau_{\gamma^{\prime}\gamma})}$$\scriptstyle{\mathcal{G}(\gamma^{\prime})(\epsilon_{\gamma}\circ\alpha(\tau_{\gamma}))}$$\scriptstyle{\eta_{\mathcal{G},\gamma,\gamma^{\prime}}}$

commutes. But this is a direct consequence of the fact that $\epsilon$ is a
weak natural transformation. Thus we can define $\alpha^{h\Gamma}$ on the
level of objects by sending $(x,\\{\tau_{\gamma}\\}_{\gamma\in\Gamma})\in
X^{h\Gamma}$ to the tuple
$(\alpha(x),\\{\epsilon_{\gamma}\circ\alpha(\tau_{\gamma})\\}_{\gamma\in\Gamma})\in
Y^{h\Gamma}$.

Given a morphism
$f:(x,\\{\tau_{\gamma}\\}_{\gamma\in\Gamma})\to(x^{\prime},\\{\tau^{\prime}_{\gamma}\\}_{\gamma\in\Gamma})$
in $X^{h\Gamma}$, one can check that

(A.1.5) $\displaystyle\alpha(f):\alpha(x)\to\alpha(x^{\prime})$

is a morphism in $Y^{h\Gamma}$. The association $f\mapsto\alpha(f)$ is
compatible with compositions, since $\alpha$ is a functor from $X$ to $Y$.
Thus we have constructed the desired functor $\alpha^{h\Gamma}$.

If $\alpha$ is fully faithful, then it is immediate that $\alpha^{h\Gamma}$ is
also fully faithful. If $\alpha$ is essentially surjective and fully faithful,
then $\alpha^{h\Gamma}$ is also essentially surjective. Indeed, given
$(y,\\{\kappa_{\gamma}\\}_{\gamma\in\Gamma})\in Y^{h\Gamma}$ we may choose an
object $x\in X$ and an isomorphism $\xi:\alpha(x)\to y$. For each
$\gamma\in\Gamma$ there is an induced isomorphism

(A.1.6) $\displaystyle\alpha(\mathcal{F}(\gamma))\to\mathcal{G}(\gamma)(y)$

given by $\mathcal{G}(\gamma)(\xi)\circ\epsilon_{\gamma}$. Using the fully
faithfulness, the isomorphism $\kappa_{\gamma}:y\to\mathcal{G}(\gamma)(y)$
induces a unique isomorphism $\tau_{\gamma}:x\to\alpha(x)$. Using this
uniqueness, it is not hard to check that the tuple
$(x,\\{\tau_{\gamma}\\}_{\gamma\in\Gamma})$ defines an object in
$X^{h\Gamma}$. Its image under $\alpha^{h\Gamma}$ is moreover isomorphic to
$(y,\\{\kappa_{\gamma}\\}_{\gamma\in\Gamma})$ by construction.

######  Lemma A.1.10.

Let $\mathcal{F},\mathcal{G},\mathcal{H}:\mathcal{C}\to\mathcal{CAT}$ where
$\mathcal{C}$ is a 1-category and $\mathcal{CAT}$ is the strict
$(2,1)$-category of categories. Let
$\delta:\mathcal{F}\to\mathcal{H},\epsilon:\mathcal{G}\to\mathcal{H}$ be weak
natural transformations. Then there exists a canonical weak functor
$\mathcal{F}\times_{\mathcal{H}}\mathcal{G}:\mathcal{C}\to\mathcal{CAT}$ such
that
$(\mathcal{F}\times_{\mathcal{H}}\mathcal{G})(x)=\mathcal{F}(x)\times_{\mathcal{H}(x)}\mathcal{G}(x)$
for all $x\in\operatorname{Ob}(\mathcal{C})$.

###### Proof.

Step 1: Objects. We define
$\mathcal{P}:=(\mathcal{F}\times_{\mathcal{H}}\mathcal{G})$ on objects by
sending $x\mapsto\mathcal{F}(x)\times_{\mathcal{H}(x)}\mathcal{G}(x)$. For
$x,y\in\mathcal{C}$ we define

(A.1.7)
$\displaystyle\mathcal{P}:\operatorname{Mor}_{\mathcal{C}}(x,y)\to\operatorname{Ob}\left(\operatorname{Mor}(\mathcal{F}(x)\times_{\mathcal{H}(x)}\mathcal{G}(x),\mathcal{F}(y)\times_{\mathcal{H}(y)}\mathcal{G}(y)\right)$

to be the assignment taking $g:x\to y$ to the functor

(A.1.8)
$\displaystyle\mathcal{F}(g)\times_{\mathcal{H}(g)}\mathcal{G}(g):\mathcal{F}(x)\times_{\mathcal{H}(x)}\mathcal{G}(x)\to\mathcal{F}(y)\times_{\mathcal{H}(y)}\mathcal{G}(y)$

taking $(a,b,f)\in\mathcal{F}(x)\times_{\mathcal{H}(x)}\mathcal{G}(x)$ to the
triple $(\mathcal{F}(g)(a),\mathcal{G}(g)(b),\mathcal{P}(g)(f))$, where
$\mathcal{P}(g)(f))$ is defined as

(A.1.9)
${\delta_{y}(\mathcal{F}(g)(a))}$${\mathcal{H}(g)(\delta_{x}(a))}$${\mathcal{H}(g)(\delta_{x}(b))}$${\epsilon_{y}(\mathcal{G}(g)(b)).}$$\scriptstyle{\mathcal{P}(g)(f))}$$\scriptstyle{\delta(g)}$$\scriptstyle{\mathcal{H}(g)(f)}$$\scriptstyle{\epsilon(g)^{-1}}$

Step 2: Morphisms. A morphism $p:(a,b,f)\to(a^{\prime},b^{\prime},f^{\prime})$
is sent to the morphism

(A.1.10) $\displaystyle(\mathcal{F}(g)(p),\mathcal{G}(g)(p))$

which one checks is a morphism in
$\mathcal{F}(y)\times_{\mathcal{H}(y)}\mathcal{G}(y)$. To check that
$\mathcal{F}(g)\times_{\mathcal{H}(g)}\mathcal{G}(g)$ is a functor, one uses
the fact that $\delta$ and $\epsilon$ are weak natural transformations (and
thus have compatibility with composition).

Step 3: Identity natural transformations. Next, for each
$x\in\operatorname{Ob}(\mathcal{C})$ we define a natural transformation
$\eta_{x}:\text{Id}_{\mathcal{P}(x)}\to(\mathcal{P})(\text{Id}_{x})$ given by
the morphism

(A.1.11)
$\displaystyle\eta_{x,(a,b,f)}:(a,b,f)\to(\mathcal{F}(\operatorname{Id}_{x})(a),\mathcal{G}(\operatorname{Id}_{x})(b),\mathcal{P}(\operatorname{Id}_{x})(f))$

described by $(\eta_{\mathcal{F},x,a},\eta_{\mathcal{G},x,a})$. Naturality
follows from the naturality of $\eta_{\mathcal{F}}$ and $\eta_{\mathcal{G}}$,
and we must simply check that the morphisms
$(\eta_{\mathcal{F},x,a},\eta_{\mathcal{G},x,b})$ are morphisms in the
category $\mathcal{F}(x)\times_{\mathcal{H}(x)}\mathcal{G}(x)$. We recall that
this comes down to showing that the top square in the following diagram is
commutative for every
$(a,b,f)\in\operatorname{Ob}(\mathcal{F}(x)\times_{\mathcal{H}(x)}\mathcal{G}(x))$

(A.1.12)
${\delta_{x}(a)}$${\epsilon_{x}(b)}$${\delta_{x}(\mathcal{F}(\text{Id}_{x})(a))}$${\epsilon_{x}(\mathcal{G}(\text{Id}_{x})(b))}$${\mathcal{H}(\text{Id}_{x})(\delta_{x}(a))}$${\mathcal{H}(\text{Id}_{x})(\epsilon_{x}(b)).}$$\scriptstyle{f}$$\scriptstyle{\delta_{x}(\eta_{\mathcal{F},x,a})}$$\scriptstyle{\epsilon_{x}(\eta_{\mathcal{G},x,b})}$$\scriptstyle{\delta(\text{Id}_{x})}$$\scriptstyle{\mathcal{P}(\operatorname{Id}_{x})(f)}$$\scriptstyle{\epsilon(\text{Id}_{x})}$$\scriptstyle{\mathcal{H}(\text{Id}_{x})(f)}$

But we know that the bottom square is commutative by the definition of
$\mathcal{P}(\operatorname{Id}_{x})(f)$. Moreover, because $\delta$ is a weak
natural transformation from $\mathcal{F}$ to $\mathcal{H}$, we know that that
$\delta(\text{Id}_{x})\circ\delta_{x}(\eta_{\mathcal{F},x,a})=\eta_{\mathcal{H},x}\circ\delta_{x}$
as morphisms between $\delta_{x}(a)$ and
$\mathcal{H}(\text{Id}_{x})(\delta_{x}(a))$ in the category $\mathcal{H}(x)$.
In addition
$\epsilon(\text{Id}_{x})\circ\epsilon_{x}(\eta_{\mathcal{G},x,b})=\eta_{\mathcal{H},x,b}\circ\epsilon_{x}$
as morphisms from $\epsilon_{x}(b)$ to
$\mathcal{H}(\text{Id}_{x})(\epsilon_{x}(b))$. The commutativity of the outer
square of (A.1.12) follows these last two observations. Since we also have
commutativity of the bottom diagram, the commutativity of the top diagram
follows since the vertical arrows are all isomorphisms.

Step 4: Natural transformations for compositions of morphisms. Finally, for a
pair of morphisms $\gamma^{\prime}:x\to y$ and $\gamma:y\to z\in\mathcal{C}$
we define a natural transformation
$\eta_{\mathcal{P},\gamma^{\prime},\gamma}:\mathcal{P}(\gamma^{\prime}\circ\gamma)\to\mathcal{P}(\gamma^{\prime})\circ\mathcal{P}(\gamma)$
of functors $\mathcal{P}(x)\to\mathcal{P}(z)$ which is defined on objects
$(a,b,f)\in\mathcal{P}(x)$ by

$(\eta_{\mathcal{F},\gamma,\gamma^{\prime}}(a),\eta_{\mathcal{G},\gamma,\gamma^{\prime}}(b)).$

Once again, the naturality is straightforward if we can check that this is an
isomorphism in $\mathcal{P}(z)$. For this, we consider the following diagram
of isomorphisms in $\mathcal{H}(z)$

(A.1.13)
${\mathcal{H}(\gamma^{\prime}\circ\gamma)(\delta_{x}(a))}$${\mathcal{H}(\gamma^{\prime}\circ\gamma)(\epsilon_{x}(a))}$${\delta_{z}(\mathcal{F}(\gamma^{\prime}\circ\gamma)(a))}$${\epsilon_{z}(\mathcal{G}(\gamma^{\prime}\circ\gamma)(b))}$${\delta_{z}(\mathcal{F}(\gamma^{\prime})(\mathcal{F}(\gamma)(a)))}$${\epsilon_{z}(\mathcal{G}(\gamma^{\prime})(\mathcal{G}(\gamma)(b)))}$${\mathcal{H}(\gamma^{\prime})(\delta_{y}(\mathcal{F}(\gamma)(a)))}$${\mathcal{H}(\gamma^{\prime})(\epsilon_{y}(\mathcal{G}(\gamma)(a)))}$${\mathcal{H}(\gamma^{\prime})(\mathcal{H}(\gamma)(\delta_{x}(a)))}$${\mathcal{H}(\gamma^{\prime})(\mathcal{H}(\gamma)(\epsilon_{x}(b))).}$$\scriptstyle{\mathcal{H}(\gamma^{\prime}\circ\gamma)(f)}$$\scriptstyle{\epsilon(\gamma^{\prime}\gamma)^{-1})}$$\scriptstyle{\delta(\gamma^{\prime}\gamma)}$$\scriptstyle{\delta_{z}(\eta_{\mathcal{F},\gamma,\gamma^{\prime}}(a))}$$\scriptstyle{\mathcal{P}(\gamma^{\prime}\circ\gamma)(f)}$$\scriptstyle{\epsilon_{z}(\eta_{\mathcal{G},\gamma,\gamma^{\prime}}(b))}$$\scriptstyle{\delta(\gamma^{\prime})}$$\scriptstyle{\mathcal{P}(\gamma^{\prime})(\mathcal{P}(\gamma)(f))}$$\scriptstyle{\mathcal{H}(\gamma^{\prime})(\mathcal{P}(\gamma)(f))}$$\scriptstyle{\mathcal{H}(\gamma^{\prime})(\delta(\gamma))}$$\scriptstyle{\epsilon(\gamma^{\prime})^{-1}}$$\scriptstyle{\mathcal{H}(\gamma^{\prime})(\mathcal{H}(\gamma)(f))}$$\scriptstyle{(\mathcal{H}(\gamma^{\prime})(\epsilon(\gamma))^{-1}}$

We are asked to show that the second square from the top commutes, and we see
that this reduces to checking commutativity of the outer square in the
diagram. But this again follows from the fact that $\delta$ and $\epsilon$ are
weak natural transformations.

Step 5: End of the proof. We have now specified the data required to define a
weak functor $\mathcal{C}\to\mathcal{CAT}$. To check that this is a weak
functor, we need to check certain identities of natural transformations hold,
see Definition A.1.1. Since all natural transformations $\eta_{\mathcal{P}}$
are defined as pairs $(\eta_{\mathcal{F}},\eta_{\mathcal{G}})$, the identities
for $\eta_{\mathcal{P}}$ follow from those for $\eta_{\mathcal{F}}$ and
$\eta_{\mathcal{G}}$. ∎

We have the following key lemma. Let
$\mathcal{F},\mathcal{G},\mathcal{H}:B\Gamma\to\mathcal{CAT}$ be weak functors
and let $\delta:\mathcal{F}\to\mathcal{H},\epsilon:\mathcal{G}\to\mathcal{H}$
be weak natural transformations. Recall the fiber product weak functor
$\mathcal{F}\times_{\mathcal{H}}\mathcal{G}$ from Lemma A.1.10.

######  Lemma A.1.11.

There is an equivalence of categories

$\beta:\mathcal{F}^{h\Gamma}(\ast)\times_{\mathcal{H}^{h\Gamma}(\ast)}\mathcal{G}^{h\Gamma}(\ast)\to(\mathcal{F}\times_{\mathcal{H}}\mathcal{G})(\ast)^{h\Gamma}.$

###### Proof.

Let us write $\mathcal{P}=\mathcal{F}\times_{\mathcal{H}}\mathcal{G}$ for
simplicity.

Step 1: Defining the functor on objects. The functor $\beta$ has the following
construction on objects: Let $(a,\\{\tau_{\gamma}\\}_{\gamma\in\Gamma})$ be an
object in $\mathcal{F}^{h\Gamma}(\ast)$, let
$(b,\\{\sigma_{\gamma}\\}_{\gamma\in\Gamma})$ be an object in
$\mathcal{G}^{h\Gamma}(\ast)$ and let
$f:\delta^{h\Gamma}(a,\\{\tau_{\gamma}\\}_{\gamma\in\Gamma})\to\epsilon^{h\Gamma}(b,\\{\sigma_{\gamma}\\}_{\gamma\in\Gamma})$
be an isomorphism in $\mathcal{H}^{h\Gamma}(\ast)$. Observe that $f$ is per
definition an automorphism $\delta(a)\to\epsilon(b)$ in $\mathcal{H}$
satisfying certain extra properties.

We define
$\beta((a,\\{\tau_{\gamma}\\}_{\gamma\in\Gamma}),(b,\\{\sigma_{\gamma}\\}_{\gamma\in\Gamma}),f)$
to be the triple $(a,b,f,\\{\rho_{\gamma}\\}_{\gamma\in\Gamma})$, where
$\rho_{\gamma}$ is the pair $(\tau_{\gamma},\rho_{\gamma})$. To check that our
triple defines an element of $\mathcal{P}(\ast)^{h\Gamma}$, we need to check
that for all $\gamma,\gamma^{\prime}$ the diagram

(A.1.14)
${(a,b,f)}$${\mathcal{P}(\gamma^{\prime})(a,b,f)}$${\mathcal{P}(\gamma^{\prime}\gamma)(a,b,f)}$${\mathcal{P}(\gamma^{\prime})(\mathcal{P}(\gamma)(a,b,f))}$$\scriptstyle{\rho_{\gamma}}$$\scriptstyle{\tau_{\gamma^{\prime}\gamma}}$$\scriptstyle{\mathcal{P}(\gamma)(\rho_{\gamma})}$$\scriptstyle{\eta_{\mathcal{P},\gamma,\gamma^{\prime}}}$

commutes. But morphisms in $\mathcal{P}(\ast)^{h\Gamma}$ are determined by the
corresponding morphisms in $\mathcal{P}(\ast)$, which are in turn determined
by a pair consisting of a morphism in $\mathcal{F}(\ast)$ and a morphism in
$\mathcal{G}(\ast)$. The commutativity of the diagram (A.1.14) then follows
from the definition of $\eta_{\mathcal{P},\gamma,\gamma^{\prime}}$ and
$\mathcal{P}$, in combination with the fact that
$(a,\\{\tau_{\gamma}\\}_{\gamma\in\Gamma})$ is an object in
$\mathcal{F}^{h\Gamma}(\ast)$ and the fact that
$(b,\\{\sigma_{\gamma}\\}_{\gamma\in\Gamma})$ is an object of
$\mathcal{G}^{h\Gamma}(\ast)$.

Step 2: Defining the functor on morphisms. A morphism

$g:((a,\\{\tau_{\gamma}\\}_{\gamma\in\Gamma}),(b,\\{\sigma_{\gamma}\\}_{\gamma\in\Gamma}),f)\to((a^{\prime},\\{\tau^{\prime}_{\gamma}\\}_{\gamma\in\Gamma}),(b^{\prime},\\{\sigma^{\prime}_{\gamma}\\}_{\gamma\in\Gamma}),f^{\prime})$

in
$\mathcal{F}^{h\Gamma}(\ast)\times_{\mathcal{H}^{h\Gamma}(\ast)}\mathcal{G}^{h\Gamma}(\ast)$
corresponds to a pair of morphisms $(g_{1},g_{2})$. Concretely, we have
$g_{1}:a\to a^{\prime}$ and $g_{2}:b\to b^{\prime}$ such that the following
diagrams commute for all $\gamma\in\Gamma$:

(A.1.15)
${a}$${a^{\prime}}$${\mathcal{F}(\gamma)(a)}$${\mathcal{F}(\gamma)(a^{\prime})}$$\scriptstyle{g_{1}}$$\scriptstyle{\tau_{\gamma}}$$\scriptstyle{\tau^{\prime}_{\gamma}}$$\scriptstyle{\mathcal{F}(g_{1})}$${b}$${b^{\prime}}$${\mathcal{G}(\gamma)(b)}$${\mathcal{G}(\gamma)(b^{\prime})}$$\scriptstyle{g_{2}}$$\scriptstyle{\sigma_{\gamma}}$$\scriptstyle{\sigma^{\prime}_{\gamma}}$$\scriptstyle{\mathcal{F}(g_{2})}$${\delta(a)}$${\epsilon(b)}$${\delta(a^{\prime})}$${\epsilon(b^{\prime}).}$$\scriptstyle{f}$$\scriptstyle{\delta(g_{1})}$$\scriptstyle{\epsilon(g_{2})}$$\scriptstyle{f^{\prime}}$

We will define $\beta(g)$ to be the morphism

(A.1.16)
$\displaystyle(a,b,f,\\{\rho_{\gamma}\\}_{\gamma\in\Gamma})\to(a^{\prime},b^{\prime},f^{\prime},\\{\rho^{\prime}_{\gamma}\\}_{\gamma\in\Gamma})$

corresponding to $(g_{1},g_{2})$. This is a morphism in $\mathcal{P}(\ast)$
between $(a,b,f)$ and $(a^{\prime},b^{\prime},f)$ by the commutativity of the
third diagram in (A.1.15). To show that this is a morphism in
$\mathcal{P}(\ast)^{h\Gamma}$, we have to show that for all $\gamma\in\Gamma$
the diagram

(A.1.17)
${(a,b,f)}$${(a^{\prime},b^{\prime},f^{\prime})}$${P(\gamma)(a,b,f)}$${P(\gamma)(a^{\prime},b^{\prime},f^{\prime})}$$\scriptstyle{g}$$\scriptstyle{\rho_{\gamma}}$$\scriptstyle{\rho^{\prime}_{\gamma}}$$\scriptstyle{\mathcal{P}(\gamma)(g)}$

commutes. But morphisms in $\mathcal{P}(\ast)$ are determined by pairs of
morphisms in $\mathcal{F}(\ast)$ and $\mathcal{H}(\ast)$. The commutativity
now follows from the commutativity of the first two diagrams of (A.1.15) and
the definition of $P(\gamma)$ and $\rho_{\gamma}$.

Step 3: Showing $\beta$ is fully faithful. Suppose we are given objects

$((a,\\{\tau_{\gamma}\\}_{\gamma\in\Gamma}),(b,\\{\sigma_{\gamma}\\}_{\gamma\in\Gamma}),f),((a^{\prime},\\{\tau^{\prime}_{\gamma}\\}_{\gamma\in\Gamma}),(b^{\prime},\\{\sigma^{\prime}_{\gamma}\\}_{\gamma\in\Gamma}),f^{\prime})\in\mathcal{F}^{h\Gamma}(\ast)\times_{\mathcal{H}^{h\Gamma}(\ast)}\mathcal{G}^{h\Gamma}(\ast).$

We have seen in Step 2 that a morphisms between these objects consists of a
pair of morphisms $g_{1}:a\to a^{\prime}$ and $g_{2}:b\to b^{\prime}$ such
that the diagrams in (A.1.15) commute. There are also conditions for the pair
$(g_{1},g_{2})$ to define a morphism between

(A.1.18)
$\displaystyle\beta((a,\\{\tau_{\gamma}\\}_{\gamma\in\Gamma}),(b,\\{\sigma_{\gamma}\\}_{\gamma\in\Gamma}),f)),\beta((a^{\prime},\\{\tau^{\prime}_{\gamma}\\}_{\gamma\in\Gamma}),(b^{\prime},\\{\sigma^{\prime}_{\gamma}\\}_{\gamma\in\Gamma}),f^{\prime})),$

namely the commutativity of the third diagram in (A.1.15) and the
commutativity of (A.1.17). But it is immediate that the latter condition is
equivalent to the commutativity of the first two diagrams in (A.1.15), proving
fully faithfulness.

Step 4: Essential surjectivity of $\beta$. Let
$(a,b,f,\\{\rho_{\gamma}\\}_{\gamma\in\Gamma})$ be an object of
$\mathcal{P}(\ast)^{h\Gamma}$. Then for each $\gamma\in\Gamma$ the map
$\rho_{\gamma}$ consists of a pair of an isomorphism
$\tau_{\gamma}:a\to\mathcal{F}(\gamma)(a)$ and an isomorphism
$\sigma_{\gamma}:b\to\mathcal{G}(\gamma)(b)$. It is straightforward to show
that $(a,b,f,\\{\rho_{\gamma}\\}_{\gamma\in\Gamma})$ is equal to (!!) the
image under $\beta$ of the triple
$(a,\\{\tau_{\gamma}\\}_{\gamma\in\Gamma}),(b,\\{\sigma_{\gamma}\\}_{\gamma\in\Gamma}),f)$.
∎

###  A.2. Quotient stacks and fixed points

In this section we will study the homotopy fixed points of quotients stacks
and in particular the interaction between the formation of quotient stacks and
homotopy fixed points.

Let $S$ be a small category with all fiber products, equipped with a
Grothendieck pretopology. We will consider the strict $(2,1)$-category
$\operatorname{Cat}_{/S}$ of categories fibered in groupoids over $S$, see
[stacks-project, Definition 02XS]. The $2$-fiber product of categories fibered
in groupoids as described in [stacks-project, Lemma 0040] is the same as the
one described in Definition A.1.7, thus Lemma A.1.11 holds for categories
fibered in groupoids.

Let $G$ be a sheaf of groups on $S$ and consider the category fibered in
groupoids $\mathbb{B}G\to S$: Its objects are pairs $(\mathcal{P},T)$ where
$T$ is an object of $S$ and where $\mathcal{P}\to T$ is a left $G$-torsor. A
morphism $(\mathcal{P},T)\to(\mathcal{P}^{\prime},T^{\prime})$ is a pair of
morphisms $a:T\to T^{\prime}$ and $b:\mathcal{P}\to\mathcal{P}^{\prime}$ such
that $b$ is $G$-equivariant and such that the following diagram commutes and
is Cartesian

(A.2.1)
${\mathcal{P}}$${\mathcal{P}^{\prime}}$${T}$${T^{\prime}.}$$\scriptstyle{b}$$\scriptstyle{a}$

The composition of morphisms is given by concatenating Cartesian diagrams.

Recall the notation of a stack in groupoids over $S$, see [stacks-project,
Definition 02ZI]. We observe that the proof of [stacks-project, Lemma 04UK]
can be repeated to show that $\mathbb{B}G\to S$ is a stack in groupoids over
$S$.

Given a morphism $f:G\to G^{\prime}$ of sheaves of groups, there is a functor
$\mathbb{B}f:\mathbb{B}G\to\mathbb{B}G^{\prime}$ sending

(A.2.2)
$\displaystyle(T,\mathcal{P})\mapsto(T,\mathcal{P}\times^{G}G^{\prime})$

and on morphisms sending $b:\mathcal{P}\to\mathcal{P}^{\prime}$ to the
morphism
$b^{\prime}:\mathcal{P}\times^{G}G^{\prime}\to\mathcal{P}\times^{G}G^{\prime}$
defined by $b^{\prime}(p,g^{\prime})=(b(p),g^{\prime})$. Here we consider
$\mathcal{P}\times^{G}G^{\prime}$ as the quotient (sheaf) of $P\times
G^{\prime}$ by the right action of $G$ given by $(p,h)\cdot g=(g^{-1}\cdot
p,hf(g))$. We let $G^{\prime}$ act on $P\times G^{\prime}$ by
$g^{\prime}\cdot(p,h)=(p,g^{\prime}h)$, which clearly descends to the quotient
$\mathcal{P}\times^{G}G^{\prime}$ turning that quotient into a left
$G^{\prime}$-torsor.

######  Lemma A.2.1.

Let $\operatorname{Grp}_{S}$ be the $1$-category of sheaves of groups in $S$.
There is a weak functor

(A.2.3)
$\displaystyle\mathbb{B}:\operatorname{Grp}_{S}\to\operatorname{Cat}_{/S}$

which on objects sends $G$ to $\mathbb{B}G$ and on morphisms sends $f:G\to
G^{\prime}$ to $\mathbb{B}f:\mathbb{B}G\to\mathbb{B}G^{\prime}$.

###### Proof.

Part (1) of Definition A.1.1 has been specified by the lemma. For part (2) we
have to specify a $2$-morphism

(A.2.4)
$\displaystyle\eta_{\mathbb{B},G}:\operatorname{Id}_{\mathbb{B}G}\to\mathbb{B}(\operatorname{Id}_{G}).$

We take the one which is given on objects by the map

(A.2.5) $\displaystyle\mathcal{P}$ $\displaystyle\to\mathcal{P}\times^{G}G$
(A.2.6) $\displaystyle p$ $\displaystyle\mapsto(p,1).$

Moreover, given $f_{1}:G_{1}\to G_{2}$ and $f_{2}:G_{2}\to G_{3}$ we have to
specify a $2$-morphism

(A.2.7) $\displaystyle\eta_{\mathbb{B},f_{1},f_{2}}:\mathbb{B}(f_{2}\circ
f_{1})\to\mathbb{B}(f_{2})\circ\mathbb{B}(f_{1}).$

We take the one given by the isomorphism

(A.2.8) $\displaystyle\mathcal{P}\times^{G_{1}}G_{3}$
$\displaystyle\to\left(\mathcal{P}^{\times
G_{1}}G_{2}\right)\times^{G_{2}}G_{3}$ (A.2.9) $\displaystyle(p,g_{3})$
$\displaystyle\mapsto((p,1),g_{3}).$

It is straightforward, if somewhat tedious, to check that these coherence data
satisfy Definition A.1.1 ∎

Now let $X$ be a sheaf on $S$ equipped with a left action of a sheaf of groups
$G$. Then we define a category fibered in groupoids $\left[X/G\right]\to S$
whose objects are triples $(T,\mathcal{P},\omega)$, where $T$ is an object of
$S$, where $\mathcal{P}\to T$ is a $G$-torsor and where $\omega:\mathcal{P}\to
X$ is a $G$-equivariant map of sheaves. A morphism
$(T,\mathcal{P},\omega)\to(T^{\prime},\mathcal{P}^{\prime},\omega^{\prime})$
is a pair $(a,b)$, where $a:T\to T^{\prime}$ is a morphism in $S$ and
$b:\mathcal{P}\to\mathcal{P}^{\prime}$ is a $G$-equivariant morphism such that
the following two diagrams commute

(A.2.10)
${\mathcal{P}}$${\mathcal{P}^{\prime}}$${T}$${T^{\prime},}$$\scriptstyle{b}$$\scriptstyle{a}$
${\mathcal{P}}$${\mathcal{P}^{\prime}}$${X}$${X,}$$\scriptstyle{b}$$\scriptstyle{\omega}$$\scriptstyle{\omega^{\prime}}$$\scriptstyle{\operatorname{Id}_{X}}$

and such that the first diagram is Cartesian. We observe that
$\left[X/G\right]$ is a stack in groupoids over $S$; indeed, the proof of
[stacks-project, Lemma 0370] goes through verbatim.

Note that there is a natural forgetful morphism
$\left[X/G\right]\to\mathbb{B}G$ of categories fibered in groupoids over $S$
which sends $(T,\mathcal{P},\omega)\mapsto(T,\mathcal{P})$.

#### A.2.2.

Let $\operatorname{GrpShv}_{S}$ be the category whose objects are triples
$(G,X,A)$, where $G$ is a sheaf of groups on $S$, where $X$ is a sheaf on $S$,
and where $A:G\times X\to X$ is a left action of $G$ on $X$. A morphism
$(G,X,A)\to(G^{\prime},X^{\prime},A^{\prime})$ consists of a homomorphism of
groups $h:G\to G^{\prime}$ and a morphism of sheaves $f:X\to X^{\prime}$, such
that $f$ is $G$-equivariant via $h$. There is an obvious forgetful functor
$\operatorname{GrpShv}_{S}\to\operatorname{Grp}_{S}$ sending $(G,X,A)$ to $G$.
We consider $\mathbb{B}$ as a weak functor on $\operatorname{GrpShv}_{S}$ via
this forgetful functor.

######  Lemma A.2.3.

There is a weak functor
$\mathscr{Q}:\operatorname{GrpShv}_{S}\to\operatorname{Cat}_{/S}$, which on
objects sends $(G,X,A)$ to $\left[X/G\right]$. There is moreover a weak
natural transformation $\mathscr{Q}\to\mathbb{B}$, which on objects induces
the natural forgetful functor

(A.2.11) $\displaystyle\left[X/G\right]\to\mathbb{B}G.$

###### Proof.

Given a morphism $(h,f):(G,X,A)\to(G^{\prime},X^{\prime},A^{\prime})$ we
define a morphism

(A.2.12)
$\displaystyle\mathscr{Q}(h,f):\left[X/G\right]\to\left[X^{\prime}/G^{\prime}\right]$

by sending $(T,\mathcal{P},\omega)\mapsto(T,\mathcal{P}^{\times
G}G^{\prime},\omega_{h})$, where

(A.2.13) $\displaystyle\omega_{h}:\mathcal{P}^{\times G}G^{\prime}$
$\displaystyle\to X^{\prime}$ (A.2.14) $\displaystyle(p,g^{\prime})$
$\displaystyle\mapsto A^{\prime}(g^{\prime},f(\omega(p)).$

For the identity natural transformation

(A.2.15)
$\displaystyle\eta_{\mathscr{Q},(G,X,A)}:\operatorname{Id}_{\mathscr{Q}(G,X,A)}\to\mathscr{Q}(\operatorname{Id}_{G},\operatorname{Id}_{X})$

we take the isomorphism $(T,\mathcal{P},\omega)\mapsto(T,\mathcal{P}^{\times
G}G,\omega_{\operatorname{Id}_{G}})$ given by

(A.2.16) $\displaystyle\mathcal{P}$ $\displaystyle\to\mathcal{P}^{\times G}G$
(A.2.17) $\displaystyle p$ $\displaystyle\mapsto(p,1),$

which is compatible with $\omega$ and $\omega_{\operatorname{Id}_{G}}$ since

(A.2.18)
$\displaystyle(\omega_{\operatorname{Id}_{G}})(p,1)=A(1,\operatorname{Id}_{X}(\omega(p))=\omega(p).$

Given a pair of composable morphisms
$(h,f):(G_{1},X_{1},A_{1})\to(G_{2},X_{2},A_{2})$ and
$(h^{\prime},f^{\prime}):(G_{2},X_{2},A_{2})\to(G_{3},X_{3},A_{3})$, we need
to specify a natural transformation

(A.2.19)
$\displaystyle\eta_{\mathscr{Q},(h,f),(h^{\prime},f^{\prime})}:\mathscr{Q}(h^{\prime}\circ
h,f^{\prime}\circ
f)\to\mathscr{Q}(h^{\prime},f^{\prime})\circ\mathscr{Q}(h,f).$

For this we take the isomorphism in $\left[X_{3}/G_{3}\right]$ given by

(A.2.20)
$\displaystyle(S,\mathcal{P}\times^{G_{1}}G_{3},\omega_{h^{\prime}\circ
h})\to(S,\left(\mathcal{P}\times^{G_{2}}G_{2}\right)\times^{G_{2}}G_{3}),\left(\omega_{h}\right)_{h^{\prime}})$

given by the isomorphism

(A.2.21) $\displaystyle\mathcal{P}\times^{G_{1}}G_{3}$
$\displaystyle\to\left(\mathcal{P}^{\times
G_{1}}G_{2}\right)\times^{G_{2}}G_{3}$ (A.2.22) $\displaystyle(p,g_{3})$
$\displaystyle\mapsto((p,1),g_{3}).$

from the proof of Lemma A.2.1. We have to check that this is an isomorphism in
$\left[X_{3}/G_{3}\right]$ which comes down to the equality

(A.2.23)
$\displaystyle\omega_{h^{\prime}h}(((p,1),g_{3}))=\omega_{h^{\prime}\circ
h}(p,g_{3}).$

To check this we compute that

(A.2.24) $\displaystyle\left(\omega_{h}\right)_{h^{\prime}}(((p,1),g_{3}))$
$\displaystyle=A^{\prime}(g_{3},f^{\prime}(\omega_{h}(p,1)))$ (A.2.25)
$\displaystyle=A_{3}(g_{3},f^{\prime}(A_{2}(1,f(p))))$ (A.2.26)
$\displaystyle=A_{3}(g_{3},f^{\prime}(f(p)))$ (A.2.27)
$\displaystyle=\omega_{h^{\prime}\circ h}(p,g_{3}).$

As in the proof of Lemma A.2.1, we omit the verification that these coherence
data satisfy Definition A.1.1.

The weak natural transformation $\mathscr{Q}\to\mathbb{B}$ is easy to specify
because it follows by construction that for
$(h,f):(G,X,A)\to(G^{\prime},X^{\prime},A^{\prime})$ the diagram

(A.2.28)
${\left[X/G\right]}$${\left[X^{\prime}/G^{\prime}\right]}$${\mathbb{B}G}$${\mathbb{B}G^{\prime}}$$\scriptstyle{\mathscr{Q}(h,f)}$$\scriptstyle{\mathbb{B}(h)}$

is strictly commutative. Thus we can specify the identity natural
transformation in (2) in Definition A.1.2, which clearly satisfies the
properties outlined in that definition. ∎

We will now spell out a particular case of Lemma A.2.3: Let $\Gamma$ be an
abstract group, let $(G,X,A)\in\operatorname{GrpShv}_{S}$ and let
$\Gamma\to\operatorname{Isom}_{\operatorname{GrpShv}_{S}}((G,X,A),(G,X,A))$ be
a group homomorphism. This gives us a functor

(A.2.29) $\displaystyle B\Gamma\to\operatorname{GrpShv}_{S},$

which we can compose with the weak functor of Lemma A.2.3 to get a
$\Gamma$-object in categories fibered in groupoids over $S$, see Example
A.1.5. Concretely, the group homomorphism

$\Gamma\to\operatorname{Isom}_{\operatorname{GrpShv}_{S}}((G,X,A),(G,X,A))$

comes down to the following data: We have a left action of $\Gamma$ on $G$
over $S$; let us write $\alpha_{\gamma}:G\to G$ for the induced isomorphism of
groups for each $\gamma\in\Gamma$. We have a left action of $\Gamma$ on $X$,
which we denote by $\beta_{\gamma}:X\to X$ for each $\gamma\in\Gamma$. These
morphisms are subject to the following commutative diagram for all
$\gamma\in\Gamma$

(A.2.30) ${G\times X}$${X}$${G\times
X}$${X.}$$\scriptstyle{A}$$\scriptstyle{\alpha_{\gamma}\times\beta_{\gamma}}$$\scriptstyle{\beta_{\gamma}}$$\scriptstyle{A}$

The induced action of $\Gamma$ on $\left[X/G\right]$ on objects takes a tuple
$(T,\mathcal{P},\omega)$ and sends it to the tuple
$(T,\mathcal{P}\times^{G,\gamma}G,\omega_{\gamma})$. Concretely, we have that

(A.2.31) $\displaystyle\omega_{\gamma}(p,g)$
$\displaystyle=A(g,\beta_{\gamma}(\omega(p)))$

which means that $\omega_{\gamma}(p,1)=\beta_{\gamma}(\omega(p))$.

#### A.2.4.

By Lemma A.2.3 there is a $\Gamma$-equivariant morphism

(A.2.32) $\displaystyle\left[X/G\right]\to\mathbb{B}G$

and thus by Lemma A.1.9 there is a functor

(A.2.33) $\displaystyle\left[X/G\right]^{h\Gamma}\to(\mathbb{B}G)^{h\Gamma}.$

Here $(\mathbb{B}G)^{h\Gamma}$ is the $\Gamma$-homotopy fixed points of
$(\mathbb{B}G)$ in the sense of definition A.1.8, which is a category fibered
in groupoids over $S$.

######  Lemma A.2.5.

The category fibered in groupoids $(\mathbb{B}G)^{h\Gamma}$ is a stack in
groupoids over $S$.

###### Proof.

Descent for morphisms in $(\mathbb{B}G)^{h\Gamma}$ follows directly from
descent of morphisms in $\mathbb{B}G$ since the forgetful map
$(\mathbb{B}G)^{h\Gamma}\to\mathbb{B}G$ is faithful by construction. Descent
for objects in $(\mathbb{B}G)^{h\Gamma}$ follows from descent for objects in
$\mathbb{B}G$ in combination with descent for morphisms in $\mathbb{B}G$, and
the weak functoriality of the $\Gamma$ action. ∎

#### A.2.6.

Now let $G^{\Gamma}$ be the fixed point sheaf of $G$ and let $X^{\Gamma}$ be
the fixed point sheaf of $X$. Then there is a natural morphism

(A.2.34) $\displaystyle\mathbb{B}G^{\Gamma}\to(\mathbb{B}G)^{h\Gamma}$

of categories fibered in groupoids over $S$ which sends $(T,\mathcal{P})$ to
$(T,\mathcal{P}\times^{G^{\Gamma}}G)$ equipped with the isomorphisms

(A.2.35)
$\displaystyle\tau_{\gamma,0}:\left(\mathcal{P}\times^{G^{\Gamma}}G\right)$
$\displaystyle\to\left(\mathcal{P}\times^{G^{\Gamma}}G\right)\times^{G,\gamma}G$
(A.2.36) $\displaystyle(p,g)$
$\displaystyle\mapsto((p,1),\alpha_{\gamma}(g)),$

and which sends a morphism

(A.2.37)
${\mathcal{P}}$${\mathcal{P}^{\prime}}$${T}$${T^{\prime}.}$$\scriptstyle{b}$$\scriptstyle{a}$

to the induced diagram

(A.2.38)
${\mathcal{P}\times^{G^{\Gamma}}G}$${\mathcal{P}^{\prime}\times^{G^{\Gamma}}G}$${T}$${T^{\prime},}$$\scriptstyle{b_{G}}$$\scriptstyle{a}$

where $b_{G}(p,g)=(b(p),g)$, which is well defined by
$G^{\Gamma}$-equivariance of $b$. Similarly there is a natural morphism

(A.2.39)
$\displaystyle\left[X^{\Gamma}/G^{\Gamma}\right]\to\left[X/G\right]^{h\Gamma}$

of categories fibered in groupoids over $S$ which sends
$(T,\mathcal{P},\omega)$ to
$(T,\mathcal{P}\times^{G^{\Gamma}}G,\\{\tau_{\gamma,0}\\}_{\gamma\in\Gamma},\omega_{G})$,
where

(A.2.40) $\displaystyle\omega_{G}:\mathcal{P}\times^{G^{\Gamma}}G\to X$

is defined by $\omega_{G}(p,g)=A(g,\omega(p))$. By construction, we see that
the following diagram of categories fibered in groupoids over $S$ is strictly
commutative

(A.2.41)
${\left[X^{\Gamma}/G^{\Gamma}\right]}$${\left[X/G\right]^{h\Gamma}}$${(\mathbb{B}G^{\Gamma})}$${(\mathbb{B}G)^{h\Gamma}.}$

We let $\underline{H}^{1}(\Gamma,G)$ be the sheafification of the presheaf
$T\mapsto H^{1}(\Gamma,G(T))$ on $S$. We have the following fundamental
results.

######  Proposition A.2.7.

The morphism $(\mathbb{B}G^{\Gamma})\to(\mathbb{B}G)^{h\Gamma}$ is an
isomorphism if $\underline{H}^{1}(\Gamma,G)$ is trivial.

######  Proposition A.2.8.

The diagram in equation (A.2.41) is $2$-Cartesian.

######  Corollary A.2.9.

If $\underline{H}^{1}(\Gamma,G)$ is trivial, then the natural map

(A.2.42)
$\displaystyle\left[X^{\Gamma}/G^{\Gamma}\right]\to\left[X/G\right]^{h\Gamma}$

is an equivalence.

We start by proving a lemma.

######  Lemma A.2.10.

The natural map $\mathbb{B}G^{\Gamma}\to\left(\mathbb{B}G\right)^{h\Gamma}$ is
fully faithful.

###### Proof.

Since this is a morphism of categories fibered in groupoids over $S$, it
suffices to prove that the map on fibers is fully faithful. Let $T\in S$, let
$\mathcal{P},\mathcal{P}^{\prime}$ be $G^{\Gamma}$-torsors over $T$ and let
$f:(T,\mathcal{P}\times^{G^{\Gamma}}G,\\{\tau_{\gamma,0}\\}_{\gamma\in\Gamma})\to(T,\mathcal{P}^{\prime}\times^{G^{\Gamma}}G,\\{\tau^{\prime}_{\gamma,0}\\}_{\gamma\in\Gamma})$
be an isomorphism in $\left(\mathbb{B}G\right)^{h\Gamma}(T)$.

We need to show that $f$ is induced from a unique isomorphism
$\mathcal{P}\to\mathcal{P}^{\prime}$. well defined up to left multiplication
by $G^{\Gamma}(T)$. If we can show that $f_{2}(p,g)=g$, then $f$ is induced
from a unique isomorphism $\mathcal{P}\to\mathcal{P}^{\prime}$; namely, the
unique $G^{\Gamma}$-equivariant morphism sending $p$ to $f_{1}(p,1)$. It
follows from the definitions of morphisms in
$\left(\mathbb{B}G\right)^{h\Gamma}$ that for $\gamma\in\Gamma$ the following
diagram must commute

(A.2.43)
${\left(\mathcal{P}\times^{G^{\Gamma}}G\right)}$${\left(\mathcal{P}\times^{G^{\Gamma}}G\right)\times^{G,\gamma}G}$${\left(\mathcal{P}^{\prime}\times^{G^{\Gamma}}G\right)}$${\left(\mathcal{P}^{\prime}\times^{G^{\Gamma}}G\right)\times^{G,\gamma}G.}$$\scriptstyle{\tau_{\gamma,0}}$$\scriptstyle{f}$$\scriptstyle{(f,1)}$$\scriptstyle{\tau^{\prime}_{\gamma,0}}$

Using the definition $\tau_{\gamma,0}(p,g)=(p,1,\gamma(g))$ this gives us the
equality

(A.2.44)
$\displaystyle((f_{1}(p,g),1),\gamma(f_{2}(p,g))=((f_{1}(p,g),1),\gamma(g)).$

Thus we see that $f_{2}(p,g)=g$ and thus $f$ is induced from the unique
$G^{\Gamma}$-equivariant map $\mathcal{P}\to\mathcal{P}^{\prime}$ sending
$p\mapsto f_{2}(p,1)$. ∎

#### A.2.11.

Consider the sheaf $\underline{Z}^{1}(\Gamma,G)$ whose $T$-points are given by
the set of cocycles $\underline{Z}^{1}(\Gamma,G(T))$ with values in $G(T)$. In
other words, these are the functions $\sigma:\Gamma\to G(T)$ satisfying

(A.2.45)
$\displaystyle\sigma(\gamma\cdot\gamma^{\prime})=\sigma(\gamma)\cdot\alpha_{\gamma}(\sigma(\gamma^{\prime})).$

There is an action of $G$ on $\underline{Z}^{1}(\Gamma,G)$, which takes an
element $g\in G(T)$ and a cocycle $\sigma\in Z^{1}(\Gamma,G(T))$ and sends it
to the cocycle

(A.2.46)
$\displaystyle\sigma^{\prime}(\gamma)=g\cdot\sigma(\gamma)\cdot\alpha_{\gamma}(g^{-1}).$

The quotient set for this action is per definition the first (nonabelian)
cohomology group of $\Gamma$ with coefficients in $G(T)$, in formulas,

(A.2.47) $\displaystyle Z^{1}(\Gamma,G(T)/G(T)=H^{1}(\Gamma,G(T)).$

Given a cocycle $\sigma:\Gamma\to G(T)$ and $\gamma\in\Gamma$ we get an
isomorphism of trivial $G_{T}$-torsors

(A.2.48) $\displaystyle\epsilon_{\gamma,\sigma}:G_{T}$ $\displaystyle\to
G_{T}\times^{G,\gamma}G$ (A.2.49) $\displaystyle g$
$\displaystyle\mapsto(\alpha_{\gamma^{-1}}(g),\sigma(\gamma)).$

To check that $(T,G_{T},\\{\epsilon_{\gamma,\sigma}\\}_{\gamma\in\Gamma}))$
defines an object of $(\mathbb{B}G)^{h\Gamma}$ we need to check that the
following diagram commutes

(A.2.50)
${G_{T}}$${G_{T}\times^{G,\gamma_{2}}G}$${G_{T}\times^{G,\gamma_{2}\gamma_{1}}G}$${\left(G_{T}\times^{G,\gamma_{1}}G\right)\times^{G,\gamma_{2}}G.}$$\scriptstyle{\epsilon_{\gamma_{2}\gamma_{1},\sigma}}$$\scriptstyle{\epsilon_{\gamma_{2}}}$$\scriptstyle{\mathbb{B}(\gamma_{2})(\epsilon_{\gamma_{1},\sigma})}$$\scriptstyle{\eta_{\mathbb{B},\gamma_{1},\gamma_{2}}}$

This comes down to the equality

(A.2.51)
$\displaystyle(\alpha_{\gamma_{2}^{-1}\gamma_{1}}(g),\sigma(\gamma_{1}\gamma_{2})),1)=((\alpha_{\gamma_{2}^{-1}}\alpha_{\gamma_{1}^{-1}}(g),\sigma(\gamma_{1})),\sigma(\gamma_{1}))$

which follows from the cocycle condition

(A.2.52)
$\displaystyle\sigma(\gamma\cdot\gamma^{\prime})=\sigma(\gamma)\cdot\alpha_{\gamma}(\sigma(\gamma^{\prime})).$

If we now consider $\underline{Z}^{1}(\Gamma,G)$ as a fibered category over
$S$, the construction above defines a natural map
$\underline{Z}^{1}(\Gamma,G)\to(\mathbb{B}G)^{h\Gamma}$ which takes
$(T,\sigma\in Z^{1}(\Gamma,G(T))$ to
$(T,G_{T},\\{\epsilon_{\gamma,\sigma}\\}_{\gamma\in\Gamma})$.

######  Lemma A.2.12.

This map induces an isomorphism

(A.2.53)
$\displaystyle\left[\underline{Z}^{1}(\Gamma,G)/G\right]\to(\mathbb{B}G)^{h\Gamma}.$

###### Proof.

This comes down to showing that the sheaf $\underline{Z}^{1}(\Gamma,G)$
represents the functor on $S$ sending $T$ to the set of $\Gamma$-equivariant
structures on the trivial $G$-torsor over $T$. We have seen above how to go
from a cocycle $\sigma\in\underline{Z}^{1}(\Gamma,G)(T)$ to a
$\Gamma$-equivariant structure on the trivial $G$-torsor over $T$. Conversely,
let
$(\mathcal{P}^{0}\times^{G^{\sigma}}G,\\{\tau_{\gamma}\\}_{\gamma\in\Gamma})$,
be a $\Gamma$-equivariant structure on the trivial $G$-torsor
$\mathcal{P}^{0}\times^{G^{\sigma}}G$ where $\mathcal{P}^{0}$ is the trivial
$G^{\Gamma}$-torsor. Then we get a cocycle by considering the collection of
elements $\sigma(\gamma)\in G(T)$ defined to be the element of $G(T)$
corresponding to the following automorphism of
$\mathcal{P}^{0}\times^{G^{\sigma}}G$ over $T$

(A.2.54)
${\mathcal{P}^{0}\times^{G^{\sigma}}G}$${\left(\mathcal{P}^{0}\times^{G^{\sigma}}G\right)\times^{G,\gamma}G}$${\mathcal{P}^{0}\times^{G^{\sigma}}G.}$$\scriptstyle{\tau_{\gamma}}$$\scriptstyle{\tau_{\gamma,0}^{-1}}$

Here $\tau_{\gamma,0}$ is the isomorphism

(A.2.55)
$\displaystyle\tau_{\gamma,0}:\left(\mathcal{P}^{0}\times^{G^{\Gamma}}G\right)$
$\displaystyle\to\left(\mathcal{P}^{0}\times^{G^{\Gamma}}G\right)\times^{G,\gamma}G$
(A.2.56) $\displaystyle(p,g)$ $\displaystyle\mapsto((p,1),\gamma(g))$

from the beginning of Section A.2.6. It is a straightforward yet laborious
check that this defines an inverse to the construction above, and the lemma is
proved. ∎

###### Proof of Proposition A.2.7.

The functor is fully faithful by Lemma A.2.10. Essential surjectivity can be
checked locally in the Grothendieck topology on $S$ since both source and
target are stacks in groupoids over $S$, see Lemma A.2.5. Thus given an object
of $(\mathbb{B}G)^{h\Gamma}$ over an object $T$ of $S$ we may make a base
change to assume that the underlying $G$-torsor is trivial. Then the
$\Gamma$-equivariant structure corresponds to a cocycle $\sigma:\Gamma\to
G(T)$, and the vanishing of $\underline{H}^{1}(\Gamma,G)$ tells us that we may
make another base change to assume that this cocycle is trivial.

We now observe that the cocycle being trivial tells us precisely that the
$\Gamma$-equivariant structure on the trivial $G$-torsor over $T^{\prime}$ is
isomorphic to the _trivial_ $\Gamma$-equivariant structure on the trivial
$G$-torsor. In other words, that the corresponding object of
$(\mathbb{B}G)^{h\Gamma}(T)$ is in the essential image of
$(\mathbb{B}G^{h\Gamma})(T)$. ∎

###### Proof of Proposition A.2.8.

The fully faithfulness of the map follows from the fully faithfulness of
$\mathbb{B}G^{\Gamma}\to(\mathbb{B}G)^{h\Gamma}$. Thus it suffices to prove
essential surjectivity. The proposition then comes down to the following
claim: Given an object
$(T,\mathcal{P},\\{\tau_{\gamma}\\}_{\gamma\in\Gamma},\omega)$ of
$\left[X/G\right]^{h\Gamma}$ such that there is a $G^{\Gamma}$-torsor
$\mathcal{Q}\to T$ and a $G^{\Gamma}$-equivariant morphism
$f:\mathcal{Q}\to\mathcal{P}$ inducing an isomorphism

(A.2.57) $\displaystyle\mathcal{P}\times^{G^{\Gamma}}G\to\mathcal{Q}$

under which $\\{\tau_{\gamma}\\}_{\gamma\in\Gamma}$ corresponds to
$\\{\tau_{\gamma,0}\\}_{\gamma\in\Gamma}$, the composition of $f$ with
$\omega$ factors through $X^{\Gamma}$. Let us write $\omega^{\prime}$ for the
map $\mathcal{Q}\times^{G^{\Gamma}}G\to X$ induced by $f$. In this situation
the diagrams

(A.2.58)
${\mathcal{Q}\times^{G^{\Gamma}}G}$${\left(\mathcal{Q}\times^{G^{\Gamma}}\right)\times^{G,\gamma}G}$${X}$${X}$$\scriptstyle{\omega^{\prime}}$$\scriptstyle{\tau_{\gamma},0}$$\scriptstyle{\omega^{\prime}_{\gamma}}$$\scriptstyle{\operatorname{Id}_{X}}$

commute for all $\gamma\in\Gamma$, which means that for $q\in\mathcal{Q}$ we
have

(A.2.59) $\displaystyle\omega^{\prime}(q,1)$
$\displaystyle=\omega^{\prime}_{\gamma}((q,1),1)=\beta_{\gamma}(\omega^{\prime}(q,1))$

for all $\gamma\in\Gamma$. We deduce that $\omega^{\prime}$ maps
$\mathcal{Q}\subset\mathcal{Q}\times^{G^{\Gamma}}G$ to $X^{\Gamma}$, which
concludes our proof of essential surjectivity. ∎

## References