Notation 2.7. For K compact Hausdorff and $(a_s)_{s:S}$ an S-family of points in K, that is, a map of sets $a : S\to K$ , we denote by $\int _{s:\mu }a_s$ the image of $\mu $ under the extension $\beta (S)\to K$ given by Proposition 2.4. We will think of $\int _{s:\mu }a_s$ as the limit of the family $(a_s)_{s:S}$ directed by $\mu $ . For example, $f_*(\mu ) = \int _{s:\mu }\delta _{f(s)}$ . More generally, given $(\nu _s)_{s:S}$ an S-family of ultrafilters on T and $\mu $ an ultrafilter on S, we can consider $\nu := \int _{s:\mu }\nu _s\in \beta (T)$ the limit of $(\nu _s)_{s:S}$ directed by $\mu $ : a subset is $\nu $ -large if and only if it is $\nu _s$ -large for a $\mu $ -large number of s.

1. i. Either the set of s satisfying $\varphi (s)$ or its complement is $\mu $ -large.

2. ii. By negating both sides of the equivalence and applying part (i), this amounts to the following:

   $$\begin{align*}(\forall s:\mu)(\exists a\in A_s)\neg \psi_s(a) \Longleftrightarrow (\exists (a_s)_{s:\mu}\in\textstyle\int_{s:\mu}A_s)(\forall s:\mu)\neg \psi_s(a_s),\end{align*}$$

   and this follows from Lemma 2.12.

Notation 2.20. We write instead of , this recovers the usual category of ultrafilters, first introduced by Blass in [Reference Blass3]. Explicitly, for $\mu $ and $\nu $ two ultrafilters on, respectively, S and T, a map of ultrafilters is given by a $\mu $ -family $(f(s))_{s:\mu }$ in T such that $\int _{s:\mu }f(s) = \nu $ . Note that isomorphisms of recover the notion of Definition 2.8.

Construction 2.24. We denote by $\star $ the unique ultrafilter on the singleton set, so all principal ultrafilters have the same type of ultraconvergence as $\star $ . This plays the role of a neutral for the sum operation: we have the following isomorphism of ultrafilters:

$$ \begin{align*} \mu\otimes\star \simeq \mu \hspace{30pt} \star\otimes\ \mu \simeq \mu \hspace{30pt} \sum_{r:\lambda}\sum_{s:\mu_r}\nu_s \simeq \sum_{s:\sum_{r:\lambda}\mu_r}\nu_s. \end{align*} $$

Construction 2.25. For a category $\mathscr {C}$ , we have a canonical curryfication isomorphism:

$$\begin{align*}\mathscr{C}^{\sum_{s:\mu}\nu_s}\stackrel{\sim}\longrightarrow \int_{s:\mu}\mathscr{C}^{\nu_s}\end{align*}$$

that we can write in a more suggestive way

$$\begin{align*}((s:\mu\otimes \nu_s)\to\mathscr{C}) \cong (s:\mu\to(\nu_s\to\mathscr{C})).\end{align*}$$

Proof. Suppose $a\in A$ open and $\mu $ converges to a, then by definition of the ultraconvergence A contains $\mu $ . Conversely, let $a\in A$ and consider the filter of neighborhoods of a, note that $T\setminus A$ is never in . So, if A were not in , we could have extended to an ultrafilter converging to a, contradicting the hypothesis. Hence A is in , showing that A is a neighborhood of a.

Proof. Suppose f continuous and $\mu \succcurlyeq a$ , then for any open U containing $f(a)$ , $f^{-1}(U)$ is an open containing a, so $\mu $ is in $f^{-1}(U)$ , and so $f_*(\mu )$ is in U as wanted. Conversely, take U an open of B; if $\mu \succcurlyeq a$ with $a\in f^{-1}(U)$ , then (by hypothesis) $f_*(\mu )\succcurlyeq f(a)$ with $f(a)\in U$ open, so $f_*(\mu )$ is in U and $\mu $ is in $f^{-1}(U)$ , this concludes that $f^{-1}(U)$ is open by Proposition 3.5.

Proof. Suppose f satisfies the hypothesis, and let U be an open of A, we use Proposition 3.5 to show that $f(U)$ is an open of B. Let $\mu \succcurlyeq f(a)$ with $a\in U$ , by hypothesis $\mu = f_*(\nu )$ for some $\nu \succcurlyeq a$ . As U is open, $\nu \in U$ , and so $\mu = f_*(\nu ) \in f(U)$ .

Proof. We first show that q is étale. Let $\mu $ be in $\beta (E/T)$ , by hypothesis we can find a $\mu $ -large subset $E_0$ on which p is injective, then $\beta (E_0)\subseteq \beta (E)$ is an open contained in $\beta (E/T)$ and p induces a homeomorphism $\beta (E_0)\to \beta (p(E_0))$ .

The fiber of q over the principal ultrafilter $\delta _x$ is given by the ultrafilters on E that are principal on an element of $E_x$ , so we can identify the fiber of q above $\delta _x$ with $E_x$ .

Let $T_0\subseteq T$ , we now look at the sections of q on $\beta (T_0)$ . The restriction of such a section to the principal ultrafilters gives a section $e : T_0\to E$ (by the description above) and as $\beta (E/T)$ is Hausdorff the section is determined by this e. Conversely, for an arbitrary section $e :T_0 \to E$ , the map sending $\mu \in \beta (T_0)$ to $e_*(\mu )$ induces a section of q on $\beta (T_0)$ (p is injective on $e(T_0)$ that is $e_*(\mu )$ -large). Hence, sections of q on the basic open $\beta (T_0)$ correspond to sections of p on $T_0 = \delta ^{-1}(\beta (T_0))$ , showing that q is indeed the pushforward of p along $\delta $ .

For the last claim, the fiber of q above an arbitrary ultrafilter $\mu \in \beta (T)$ is given by the filtered colimit of the set of sections of q on the $\beta (T_0)$ for $T_0$ a $\mu $ -large subset, and sections of q on $\beta (T_0)$ correspond to sections of p on $T_0$ that are elements of $\prod _{x:T_0}E_x$ .

Proof. This is a standard sheaf-theoretical fact; a proof can be found, for example, in [Reference Schnürer and Soergel26, Proposition 10.1].

Proof of Theorem 3.11.

The direct implication is straightforward. Suppose p is étale and $\mu \succcurlyeq p(e)$ . We take a homeomorphism $U\to p(U)$ from an open neighborhood U of e and denote by $\xi $ its inverse. Any $\nu \succcurlyeq e$ above $\mu $ is in the open U and so $\nu = \xi _*(p_*(\nu )) = \xi _*(\mu )$ showing the uniqueness of the lifting. And by continuity $\xi _*(\mu )$ converges indeed to $\xi (p(e)) = e$ , and $\xi _*(\mu )\in \beta (E/T)$ as p is injective on U.

The converse is more demanding. Suppose $p : E \to T$ satisfies conditions (i) and (ii) of Theorem 3.11 and fix some $e\in E$ , we want to construct a homeomorphism from an open neighborhood of e onto a neighborhood of $a := p(e)$ . The hypothesis (i) gives a set section $\sigma _a : \beta _a(T)\to \beta (E)$ , where $\beta _a(T)$ is the subspace of $\beta (T)$ of ultrafilters converging to a, mapping $\mu \succcurlyeq a$ to the unique $\sigma _a(\mu )$ over $\mu $ converging to e, and the hypothesis (ii) ensures that actually $\sigma _a$ factorizes through $\beta (E/T)$ .

We would like to glue together all these lifts $(\sigma _a(\mu ))$ to get a local section of p around a; the crucial ingredient is given by Lemma 3.18, we apply it to the sheaf $\beta (E/T)\stackrel {\acute{\text{e}}{\text{t}}}\to \beta (T)$ of Lemma 3.17 and the closed subset $\beta _a(T)$ . We get a continuous section $\zeta $ on an open V of $\beta (T)$ , such that any $\mu $ converging to a is in V and satisfies $\zeta (\mu )\succcurlyeq e$ .

To get a section on T we pull back (in $\mathsf{Set}$ ) along $\delta : T\to \beta (T)$ . We get the following diagram:

where $\delta ^* V$ is the subset of $x\in T$ such that $\delta _x\in V$ , and $\xi : \delta ^*V\to E$ is the unique map such that $\zeta (\delta _x) = \delta _{\xi (x)}$ .

As $\zeta (\delta _a) = \delta _e$ we get $\xi (a) = e$ , so $\xi $ seems to be a good candidate for our local homeomorphism. However, except in trivial cases, $\delta : T\to \beta (T)$ is not continuous, so there is no reason for $\delta ^* V$ to be open, nor for $\xi $ to be continuous. We thus restrict $\xi $ to W the subset of points of $\delta ^*V$ where $\xi $ is locally defined and sequentially continuous, that is,

$$\begin{align*}W:=\{w\in {\mathrm{dom}}(\xi) \ |\ (\forall \mu\succcurlyeq w)\ \mu\in{\mathrm{dom}}(\xi) \text{ and } \xi_*(\mu)\succcurlyeq\xi(w) \}.\end{align*}$$

This shows together that $\xi (W)$ is an open and $\xi (W) \stackrel {p}\to W$ is a homeomorphism with inverse $\xi $ . As $\xi (a) = e$ this exhibits the desired local homeomorphism and shows that p is étale.

Construction 4.1. Ultraproducts of sets behave like a tensor operation: we have associators and a unitor

(1)

where we used, implicitly, the curryfication isomorphism from Construction 2.25 to define the associator:

$$\begin{align*}\mathsf{Set}^{\sum_{s:\mu}\nu_s}\simeq\int_{s:\mu}\mathsf{Set}^{\nu_s} \stackrel{(\int_{\nu_s})}\longrightarrow\int_{s:\mu}\mathsf{Set}=\mathsf{Set}^{\mu}.\end{align*}$$

Note that the associator corresponds to a currification: for $(A_{s,t})_{(s,t):\sum _{s:\mu }\nu _s}$ an ultrafamily of sets, the associator gives a bijection $\int _{\sum _{s:\mu }\nu _s}A_{s,t} \stackrel {\sim }\to \int _{s:\mu }\int _{t:\nu _s}A_{s,t}$ . Moreover, these natural isomorphisms satisfy the following three usual coherences:

(2)

(3)

There are also reindexing operations that give the ultraproduct an additive flavor (in the sense that there are diagonal maps $A\to A^{\mu }$ ). For any map of ultrafilters $f : \lambda \to \kappa $ , there is a natural transformation $f^{\#} : \int _{\kappa }\Rightarrow \int _{\lambda }\circ f^*$

(4)

whose value on a $\kappa $ -family $(A_k)_{k:\kappa }$ is given by

$$\begin{align*}f^{\#}_{(A_k)}(a_k)_{k:\kappa}:=(a_{f(l)})_{l:\lambda}.\end{align*}$$

Note that by looking at $f : \mu \to \star $ we recover the diagonal inclusion $(f^{\#})_A : A\hookrightarrow A^{\mu }$ . Moreover, these $f^{\#}$ are strictly functorial in $f,$ that is,

(5) $$ \begin{align} \mathsf{id}_{\kappa}^{\#} = \mathsf{id}_{\int_{\kappa}} \hspace{10pt} \text{ and } \hspace{10pt} (f\circ g)^{\#} = g^{\#}\circ f^{\#} \end{align} $$

and the associators of (1) are stable under the reindexing action, that is,

(6)

commutes; this can also be written as the commutation of the following diagram:

Proof. We give some details for the proof that $\mathsf{Ult}^{\mathsf{L}}\hookrightarrow \mathsf{vUlt}$ is 2-fully faithful. Let $\mathscr {M}$ and $\mathscr {N}$ be two ultracategories and $F : \underline {\mathscr {M}} \to \underline {\mathscr {N}}$ a functor between v-ultracategories. The ultraarrow of $\underline {\mathscr {M}}$ corresponding to the identity is sent to an ultraarrow of $\underline {\mathscr {N}}$ that corresponds to some arrow $\sigma : F(\int _{s:\mu }a_s)\to \int _{s:\mu }F(a_s)$ in $\mathscr {N}$ . One can check that these $(\sigma )$ induce a left-ultrafunctor structure $F : \mathscr {M} \to \mathscr {N}$ .

Conversely, from a left-ultrafunctor $F : \mathscr {M}\to \mathscr {N}$ we can define a functor of v-ultracategories as follows: an ultraarrow corresponding to some $f: a \to \int _{s:\mu }b_s$ is sent to the ultraarrow corresponding to the composite of $F(f):F(a)\to F(\int _{s:\mu }b_s)$ with the comparison map $F(\int _{s:\mu }b_s)\to \int _{s:\mu }F(b_s)$ .

One can then check that these two constructions establish a bijection between $\mathsf{Ult}^{\mathsf{L}}(\mathscr {M},\mathscr {N})$ and $\mathsf{vUlt}(\underline {\mathscr {M}},\underline {\mathscr {N}})$ .

Proof. The proof relies on the characterization of étale maps of Theorem 3.11. From an ultrasheaf $A\in \mathfrak {sh}(\mathfrak {pt}(T))$ we get a sheaf over the space T by taking $E := \bigsqcup _{a:T}A_a$ with the topology given by: $B\subseteq E$ is open, if

$$\begin{align*}\text{for all } a\preccurlyeq \mu \text{ , } A([\![ \mathit{a}\preccurlyeq\mu]\!]) : A_a \longrightarrow \int_{x:\mu}A_x \text{ restricts to } B_a \longrightarrow \int_{x:\mu}B_x,\end{align*}$$

where $B_x := B\cap A_x$ . The projection map $E\to T$ satisfies the hypotheses of Theorem 3.11 and so is étale.

One can then check that this construction induces a functor $\mathfrak {sh}(\mathfrak {pt}(T))\to \mathsf{sh}(T)$ and that this functor is a pseudoinverse to $\mathsf{ev} : \mathsf{sh}(T)\to \mathfrak {sh}(\mathfrak {pt}(T))$ .

Proof. Let be a small category and an -diagram in $\mathfrak {sh}(X)$ .

For any $(x_s)_{s:\mu }$ ultrafamily of objects of X, there is a comparison map

allowing us to define an ultrasheaf structure on $({\mathrm {colim}}((A^i)_x))_{x:\mathsf{Ob}(X)}$ that we will denote by $A\in \mathfrak {sh}(X)$ . There is a natural cocone of A above the $(A^{i})$ and this cocone is universal as maps of ultrasheaves are given objectwise.

For limits, the comparison map is in the wrong direction

but this comparison map is bijective when is finite, allowing us to conclude as above.

Proof. The proof is essentially the same as [Reference Lurie20, Proposition 5.4.6]. Let $\alpha : A \to B$ be a map of ultrasheaves over X and an -diagram in $\mathfrak {sh}(X)/B$ for a small category. We need to show that the canonical map

is an isomorphism of sheaves as it is a bijection on each object of X. This follows from the fact that small colimits and pullbacks are computed objectwise by Lemma 6.5 and that small colimits and pullbacks commute in $\mathsf{Set}$ .

Let now $R\subseteq A\times A$ be an equivalence relation in $\mathfrak {sh}(X)$ . We need to show that the canonical map

$$\begin{align*}R\longrightarrow A \times_{A/R} A\end{align*}$$

is an isomorphism, where $A/R := {\mathrm {coeq}}(R\rightrightarrows A)$ is the quotient of A by R. Again, this follows from the fact that the quotient $A/R$ and the pullback $A \times _{A/R} A$ are computed objectwise by Lemma 6.5 and that equivalence relations are effective in $\mathsf{Set}$ .

Proof. Let $F : X\to Y$ be a functor between bounded v-ultracategories. Precomposition by F gives a functor of categories $\mathfrak {sh}(Y)\to \mathfrak {sh}(X)$ and this functor is lex and cocontinuous by Lemma 6.5. Hence, as X and Y are bounded, it induces a geometric morphism $\mathfrak {sh}(F) : \mathfrak {sh}(X)\to \mathfrak {sh}(Y)$ . The rest of the 2-functorial structure is straightforward.

Construction 7.2. The 2-categories $\mathsf{TopSp}$ and $\mathsf{vUlt}$ have strict 2-pullbacks that can be constructed in a similar fashion to strict 2-pullbacks of categories.

1. i. For two continuous maps and of topological spaces, the 2-pullback of f and g is the subspace of given by the pairs $(x,y)$ with $f(x)$ and $g(y)$ equivalent for the specialization order (i.e., $f(x)$ and $g(y)$ are contained in the same opens). Note that this is slightly different from the usual 1-pullback of topological spaces, but that this difference disappears if is T0-separated.

2. ii. For two functors of v-ultracategories and , the 2-pullback is given by the v-ultracategory whose objects are triplets $(\theta ,x,y)$ of an object , an object , and an isomorphism ; an ultraarrow from $(\theta ,x,y)$ to $(\theta _s,x_s,y_s)_{s:\mu }$ is given by a pair $(f,g)$ of an ultraarrow and an ultraarrow such that $G(g)\circ \theta = (\theta _s)_{s:\mu }\circ F(f)$ .

These two constructions are strict 2-pullbacks, and the strict 2-functor $\mathfrak {pt} : \mathsf{TopSp}\to \mathsf{vUlt}$ preserves these strict 2-pullbacks.

Construction 7.11. Recall that our objective is to prove Proposition 7.1. We know from Proposition 6.4 that the result holds for topological spaces, our strategy will thus consist of representing the topos by a topological groupoid to reduce to the zero-dimensional case.

Suppose that $\mathscr {E}$ is represented by $(T_{\bullet })$ , that is, there is a universal descent-cocone $(\pi ,\theta )$ above $(\mathsf{sh}(T_{\bullet }))$ :

Going through the functor $\mathfrak {pt}$ we get a descent-cocone $(\mathfrak {pt}(\pi ),\mathfrak {pt}(\theta ))$ over $(\mathfrak {pt}(T_{\bullet }))$ :

Question 7.12. The cocone $(\pi ,\theta )$ is universal by hypothesis, is the descent-cocone $(\mathfrak {pt}(\pi ),\mathfrak {pt}(\theta ))$ also universal? that is, is $\mathfrak {pt}(\mathscr {E};T_0)$ represented by the $\mathsf{vUlt}$ -groupoid $(\mathfrak {pt}(T_{\bullet }))$ ?

Proof. We denote by $(X_1,\theta )$ the 2-pullback of $\pi $ along itself so that $(\pi ,\theta )$ forms a descent-cocone above $(X_{\bullet })$ and we want to show that this descent-cocone is universal. Let us fix another v-ultracategory; we want to show that the functor is an equivalence of categories.

The faithfulness comes from the surjectivity on objects of $\pi $ : let and $\alpha ,\beta :F\Rightarrow G$ such that $\alpha $ and $\beta $ have the same image by $(\pi ,\theta )^*$ , so $\alpha \pi = \beta \pi $ , and we conclude that $\alpha = \beta $ using that $\pi $ is surjective on objects.

The proofs of fullness and essential surjectivity are a bit more involved. Using the hypotheses, we can fix the following liftings: for each object there is $\mathsf{h}(a)\in X_0$ such that $\pi (\mathsf{h}(a)) = a$ and for each ultraarrow in and x such that $\pi (x) = a$ , there is such that $\pi (\mathsf{h}(f;x)) = f$ .

We first show the fullness. Let and $\alpha : F\circ \pi \Rightarrow G\circ \pi $ be a morphism of descent-cocones from $(\pi ,\theta )^*(F)$ to $(\pi ,\theta )^*(G)$ , that is,

$$\begin{align*}(\alpha \cdot t) \circ (F \cdot \theta) = (G \cdot \theta)\circ(\alpha \cdot s).\end{align*}$$

A candidate for a preimage of $\alpha $ is given by $\gamma : F\Rightarrow G$ defined by $\gamma _a := \alpha _{\mathsf{h}(a)}$ ; we have to check the naturality of $\gamma $ . Let be an ultraarrow in , the following diagram shows the naturality of $\gamma $ at f:

The upper square is a naturality square of $\alpha $ , and the lower square is the equation $(\alpha \cdot t) \circ (F \cdot \theta ) = (G \cdot \theta )\circ (\alpha \cdot s)$ applied to $(\mathsf{id}_{b_s},\mathsf{k}(f;\mathsf{h}(a))_s,\mathsf{h}(b_s))\in X_1$ .

We now show the essential surjectivity. Let $(H,\eta )$ be a descent-cocone over $(X_{\bullet })$ of apex , we want to construct a preimage . We define as follows:

Only remains to show that F defines a functor (it is not obvious as the lifts are chosen arbitrarily) and that it is a preimage of $(H,\eta )$ . There are a lot of coherences to check, but everything flows fluently.

• • First, notice that for any with two lifts , , the following square commutes:

  (*)

  

  It is the naturality square of $\eta $ for the ultraarrow of $X_1$ given by $(g,g')$ .

• • $F(\mathsf{id}_a) = \mathsf{id}_{F(a)}$ : For a an object of , the following square

  

  (where the left equality is $\eta $ by the unit condition) commutes by (*).

• • $F((g_s)\circ f) = (F(g_s))\circ F(f)$ : For and ultraarrows in , we need to show the commutation of the following diagram:

  

  The right hand side commutes by the cocycle condition, the left equality is $\eta $ by the unit condition and the middle and upper squares commute by (*).

This shows the functoriality of . To show that F is a preimage of $(H,\eta )$ we construct a natural isomorphism $\Phi : H \stackrel {\sim }\Rightarrow F\circ \pi $ by

$$\begin{align*}\Phi_x : H(x) \stackrel{\eta}\longrightarrow H(\mathsf{h}(\pi(x))) = F(\pi(x)), \text{ for } x \text{ an object of } X_0.\end{align*}$$

• • $\Phi $ is natural: For any in $X_0$ , we need to show the following commutation:

  

  The lower square is the definition of F, and the upper square commutes by (*).

• • $\Phi $ induces an isomorphism between $(F\pi ,F\theta )$ and $(H,\eta )$ : For any $(v,x,y)\in X_1$ (i.e., $x,y\in X_0$ and $v:\pi (x)\to \pi (y)$ isomorphism), we need to show the following commutation:

  

  The lower square is again the definition of F, and using the cocycle and the unit condition of $\eta $ , the commutation of the upper square amounts to the one of

  

  where we denote $x':= \mathsf{h}(\pi (x))$ , $y':= \mathsf{k}(v;\mathsf{h}(\pi (x))),$ and $f := \mathsf{h}(v;\mathsf{h}(\pi (x)))$ . And this square is the naturality square of $\eta $ for the arrow $(v,x',y')\to (\mathsf{id}_{y'},y',y')$ given by $(f,\mathsf{id}_{y'})$ .

Construction 8.1. We want to construct a topological groupoid $(T_{\bullet })$ representing $\mathscr {E}$ . The points of $T_0$ will induce points of $\mathscr {E}$ , so (as $T_0$ is small) we have to choose a small set of points of $\mathscr {E}$ , and we want this set of points to be separating to have some hope to be able to reconstruct $\mathscr {E}$ .

We look now at two points a and b of $T_0$ that are sent onto two points p and q of X. If $a\preccurlyeq b$ in $T_0$ the ultraconvergence induces a map of points $p\to q$ , but in general, there are plenty of such maps, we thus need a way to add some information on a and b to be able to trace out the map $p\to q$ to which $a\preccurlyeq b$ will be sent. The idea is to add an indexing: a point a will be a pair $(p,\alpha )$ where $\alpha $ is an indexing of the model pointed by p, so that if $a = (p,\alpha )\preccurlyeq b=(q,\beta )$ , the two indexings $\alpha $ and $\beta $ will suffice to determine to which map $p\to q$ the ultraconvergence $a\preccurlyeq b$ is sent.

The data of such $\mathscr {E}$ , X, $M,$ and $\kappa $ will be fixed until the end of this section.

1. i. Let p be a point in X. By assumption on $\kappa $ , $\kappa \geq |M_p|$ , and as $\kappa $ is infinite $|\kappa \sqcup \kappa | = \kappa $ , hence

   $$\begin{align*}\kappa \simeq\kappa\sqcup\kappa\supset\kappa\twoheadrightarrow M_p\end{align*}$$

   gives an ample indexing of p.

2. ii. We choose a support set S of $\mu $ such that $\prod _{s:S}M_{q_s}\to \int _{s:\mu }M_{q_s}$ is surjective. So that we can lift $f_M\circ \alpha : {\mathrm {dom}}(\alpha )\to \int _{s:\mu }M_{q_s}$ to a map into the product $\prod _{s:S}M_{q_s}$ , getting a family $(\alpha ^{\prime }_s: {\mathrm {dom}}(\alpha ) \to M_{q_s})_{s:S}$ . As $\alpha $ is ample, $|\kappa \setminus {\mathrm {dom}}(\alpha )| = \kappa $ and we extend each $\alpha ^{\prime }_s : {\mathrm {dom}}(\alpha )\to M_{q_s}$ to an ample indexing $\beta _s : \kappa \rightharpoonup \mathrel {\mspace {-15mu}}\rightharpoonup M_{q_s}$ :

   

Construction 8.4. The points of $T_0^{\mathsf{amp}}$ are amply indexed models $(p,\alpha )$ . The topology is the logical topology, that is, the topology is given by the following basic opens:

$$ \begin{align*} U_{\underline{i},A} := \{(p,\alpha) : \alpha(\underline{i})\in A_p\} \text{, with } \underline{i} \in \kappa^n \text{ and } A\subseteq M^n. \end{align*} $$

From a logical point of view, A is seen as a formula with n parameters so that $U_{\underline {i},A}$ is the set of amply indexed models satisfying this formula, the parameters being specified by the indexing.

Moreover, there is a geometric morphism $\pi : \mathsf{sh}(T_0^{\mathsf{amp}})\to \mathscr {E}$ , for $E\in \mathscr {E}$ the sheaf $\pi ^*(E)\to T_0^{\mathsf{amp}}$ is defined by $\pi ^*(E) := \bigsqcup _{(p,\alpha ):T_0^{\mathsf{amp}}}E_p$ with the topology given by the following basic opens:

$$ \begin{align*} W_{\underline{i},A,h} := \bigsqcup_{(p,\alpha):U_{\underline{i},A}}\{h(\alpha(\underline{i}))\} \text{, with } \underline{i} \in \kappa^n , A\subseteq M^n \text{ and } h : A\to E. \end{align*} $$

One can check easily that these are bases for topologies and that $\pi $ is a geometric morphism, we refer the reader to [Reference Butz and Moerdijk4] for more details.

Proof. First notice that there can be at most one ultramorphism of amply indexed models from $(p,\alpha )$ to $(q_s,\beta _s)$ : let f be a witness of such an ultramorphism, then as $\alpha $ is surjective the value of $f_M$ is determined, and so by the generating property of M the ultraarrow f is uniquely determined.

Let f be a witness of an ultramorphism of amply indexed models from $(p,\alpha )$ to $(q_s,\beta _s)$ , we show that $(p,\alpha )\preccurlyeq (q_s,\beta _s)$ in $T_0^{\mathsf{amp}}$ . Let $U_{\underline {i},A}$ be a basic open containing $(p,\alpha )$ , that is, $\alpha (\underline {i})\in A_p$ , then by functoriality $(\beta _s(i)) = f_M(\alpha (i)) \in \int _{s:\mu }A_{q_s}$ and so $(q_s,\beta _s)\in U_{\underline {i},A}$ for $\mu $ -all s.

Conversely, suppose that $(p,\alpha )\preccurlyeq (q_s,\beta _s)$ , this induces an ultraarrow in $\mathfrak {pt}(\mathsf{sh}(T_0^{\mathsf{amp}}))$ and by $\pi $ an ultraarrow . We will show that this f witnesses an ultramorphism from $(p,\alpha )$ to $(q_s,\beta _s)$ . We look at the action of the ultraconvergence on the sheaf $\pi ^*(M)\to T_0^{\mathsf{amp}}$ ; for any $\alpha (i)\in M_p$ , the open $W_{i,M,\mathsf{id}_M}$ witnesses the local homeomorphism property, and so $f_M(\alpha (i)) = (\beta _s(i))$ as wanted.

Proof. Lemma 8.3 together with Lemma 8.5 shows that $\mathfrak {pt}(\pi )$ satisfies the hypotheses of Proposition 7.13.

Construction 8.7. The points of $T_1^{\mathsf{amp}}$ are triplets $(\theta ,(p,\alpha ),(q,\beta ))$ where $(p,\alpha )$ and $(q,\beta )$ are amply indexed models, and $\theta $ is an isomorphism between p and q in the category of points of $\mathscr {E}$ (note that no conditions on the indexings are required on $\theta $ ). The topology is the logical topology, that is, the topology given by the following basic opens:

$$ \begin{align*} V_{\underline{i},A,\underline{j},B} := \{(\theta,(p,\alpha),(q,\beta)) : \alpha(\underline{i})\in A_p,\beta(\underline{j})\in B_q,\theta(\alpha(\underline{i})) = \beta(\underline{j})\}, \\\text{with } \underline{i},\underline{j} \in \kappa^n \text{ and } A,B\subseteq M^n. \end{align*} $$

There is a natural topological groupoidal structure $(T^{\mathsf{amp}}_{\bullet })$ . Moreover, noticing that for any $E\in \mathscr {E}$ the sheaf $\pi ^*(E)\to T_0^{\mathsf{amp}}$ is naturally an equivariant sheaf over $(T^{\mathsf{amp}}_{\bullet })$ , we can define a natural isomorphism

and we obtain a descent-cocone $(\pi ,\theta )$ over $(\mathsf{sh}(T^{\mathsf{amp}}_{\bullet }))$ .

Proof. We describe $\mathfrak {pt}(T^{\mathsf{amp}}_1)$ and show that we get the 2-pullback of $\mathfrak {pt}(\pi )$ along itself given by Construction 7.2. It is clear at the level of objects: objects of $\mathfrak {pt}(T^{\mathsf{amp}}_1)$ are given by $(\theta ,a,b)$ with a and b objects of $\mathfrak {pt}(T^{\mathsf{amp}}_0)$ and $\theta $ an isomorphism between $\mathfrak {pt}(\pi )(a)$ and $\mathfrak {pt}(\pi )(b)$ in $\mathfrak {pt}(\mathscr {E};X)$ .

We fix $v := (\theta ,(p,\alpha ),(q,\beta ))$ and $(v_s := (\theta _s,(p_s,\alpha _s),(q_s,\beta _s)))_{s:\mu }$ points of $T^{\mathsf{amp}}_1$ . An ultraarrow from v to $(v_s)_{s:\mu }$ in the 2-pullback amounts, by Lemma 8.5, to ultraconvergences $(p,\alpha )\preccurlyeq (p_s,\alpha _s)$ (resp. $(q,\beta )\preccurlyeq (q_s,\beta _s)$ ) witnessed by some (resp. ) such that $(\theta _s)\circ f = g \circ \theta $ ; hence there cannot be more than one ultraarrow from v to $(v_s)_{s:\mu }$ in the 2-pullback.

Suppose that there is such an ultraarrow, we show that $v\preccurlyeq (v_s)_{s:\mu }$ in $T^{\mathsf{amp}}_1$ . Let $V_{\underline {i},A,\underline {j},B}$ be a basic open containing v, so $\alpha (\underline {i})\in A_p$ and $\beta (\underline {j})\in B_q$ . Using Lemma 8.5 we get $\alpha _s(\underline {i})\in A_{p_s}$ and $\beta _s(\underline {j})\in B_{q_s}$ for $\mu $ -all s, and $f_M(\alpha (\underline {i})) = (\alpha _s(\underline {i}))$ , $g_M(\beta (\underline {j})) = (\beta _s(\underline {j}))$ . The following sequence of equalities

$$\begin{align*}(\theta_s(\alpha_s(\underline{i}))) = (\theta_s(f_M(\alpha(\underline{i})))) = g_M(\theta(\alpha(\underline{i}))) = g_M(\beta(\underline{j})) = (\beta_s(\underline{j}))\end{align*}$$

shows that $(v_s)_{s:\mu }$ is in $V_{\underline {i},A,\underline {j},B}$ .

Conversely, suppose that $v\preccurlyeq (v_s)_{s:\mu }$ in $T^{\mathsf{amp}}_1$ , we have $(p,\alpha )\preccurlyeq (p_s,\alpha _s)$ (resp. $(q,\beta )\preccurlyeq (q_s,\beta _s)$ ) witnessed by some (resp. ). It only remains to show that $(\theta _s)\circ f = g \circ \theta $ , it is enough to show the equality on M; let $\alpha (i)\in M_p$ and j such that $\beta (j) = \theta _M(\alpha (i))$ , then $v\in V_{i,M,j,M}$ and so $(v_s)\in V_{i,M,j,M}$ , this yields

$$\begin{align*}\theta_s(f_M(\alpha(i))) = \theta_s(\alpha_s(i)) = \beta_s(j) = g_M(\beta(j)) = g_M(\theta(\alpha(i))),\end{align*}$$

showing $(\theta _s)\circ f = g \circ \theta $ on M as wanted.

Proof. Follows from Corollary 8.6 and Lemma 8.8.

Proof. We apply the theorem above. Let $\alpha : \kappa \rightharpoonup \mathrel {\mspace {-15mu}}\rightharpoonup M_p$ be an ample indexing, then for any partial surjection $f : \kappa \rightharpoonup \mathrel {\mspace {-15mu}}\rightharpoonup \kappa $ the composite $\alpha ' := \alpha \circ f : \kappa \rightharpoonup \mathrel {\mspace {-15mu}}\rightharpoonup M_p$ is also an ample indexing; indeed f induces a partial surjection $\kappa \setminus {\mathrm {dom}}(\alpha ')\rightharpoonup \mathrel {\mspace {-15mu}}\rightharpoonup \kappa \setminus {\mathrm {dom}}(\alpha )$ showing that $|\kappa \setminus {\mathrm {dom}}(\alpha ')|\geq |\kappa \setminus {\mathrm {dom}}(\alpha )|$ .

Proof. The following diagram commutes:

The left arrow is an equivalence by Corollary 8.11, the right one by Corollary 8.9, and the bottom one by the zero-dimensional case (see Remark 7.9). Thus the top arrow is also an equivalence.

Proof. For any small separating set X of points of $\mathscr {E}$ we can choose M and $\kappa $ as in Construction 8.1, and then apply Corollary 8.12.