2.1 Calibrations and families

We fix a category $\mathcal{B}$ with finite limits.

Since $\mathcal{B}$ has a terminal object, the terminal functor $1:\mathcal{B}^{\mathrm{op}}\to \mathsf{CAT}$ is a subfunctor of the universe $\mathbb{B}$ . The corresponding fibration is the identity of $\mathcal{B}$ . Equivalently, it corresponds to the subcodomain fibration $\mathcal{B}=\mathcal{B}^\simeq \subset{\mathcal{B}}^{\to }$ spanned by the isomorphisms.Footnote ²

A regular calibration defines a wide subcategory of $\mathcal{B}$ (since it contains all isomorphisms). In fact, regular calibrations are in bijection with wide subcategories of $\mathcal{B}$ whose maps are closed under base change in $\mathcal{B}$ .

Definition 4. (Family construction). We fix a $\mathcal{B}$ -category $\mathbb{C}$ , with associated fibration $\mathcal{C}\to \mathcal{B}$ . Let $\mathcal{U}\subset{\mathcal{B}}^{\to } \xrightarrow{cod} \mathcal{B}$ be a subfibration of the codomain fibration. The fibration $\mathsf{Fam}_{\mathcal{U}}(\mathcal{C})$ of $\mathcal{U}$ -indexed families of objects in $\mathcal{C}$ is defined as the functor $\phi$ in the diagram

(where the square is a fiber product). Following our typographic convention, we denote by $\mathsf{Fam}_{\mathbb{U}}(\mathbb{C})$ the associated functor $\mathcal{B}^{\mathrm{op}}\to \mathsf{CAT}$ (and call it the $\mathcal{B}$ -category of $\mathbb{U}$ -indexed families of objects in $\mathbb{C}$ ). Its value at an object $I$ in $\mathcal{B}$ is the category of pairs $(u:U\to I \in \mathcal{U},C\in \mathbb{C}(U))$ (with the obvious notion of morphism).

2.2 Sums and products

The family construction associated with the constant calibration is the identity: $\mathbb{C} = \mathsf{Fam}_1(\mathbb{C})$ . If $\mathbb{U}$ is a pointed calibration we get a canonical functor $\Delta _{\mathbb{U}}:\mathbb{C}=\mathsf{Fam}_1(\mathbb{C})\to \mathsf{Fam}_{\mathbb{U}}(\mathbb{C})$ .

The following result is proved for 1-categories in Streicher (2020). The statement extends to $\infty$ -categories as well. So do a range of expected results from Streicher (2020) and Moens (Reference Moens1982) about fibrations with internal sums. This has been developed by Buchholtz and Weinberger (Reference Buchholtz and Weinberger2023) and Weinberger (Reference Weinberger2024) in the type theory of synthetic $\infty$ -categories (Riehl and Shulman, Reference Riehl and Shulman2017; Riehl, Reference Riehl2023a) (internally to any $\infty$ -topos, see Shulman (2020); Riehl (Reference Riehl2023b); Weinberger (Reference Weinberger2022)). See also the work on internal $\infty$ -categories in an arbitrary $\infty$ -topos by Martini (Reference Martini2022), Martini and Wolf (Reference Martini and Wolf2023a, Reference Martini and Wolf2023b).

2.3 Smooth and proper maps

We fix a category $\mathcal{B}$ with finite limits and a $\mathcal{B}$ -category $\mathbb{C}:\mathcal{B}^{\mathrm{op}}\to \mathsf{CAT}$ .

Definition 9. (Smooth & proper maps). A map $u:X\to Y$ in $\mathcal{B}$ is called left (right) Beck–Chevalley for $\mathbb{C}$ if for any cartesian square

the maps $u^*$ and $\bar u ^*$ have left (right) adjoints

and the corresponding mate natural transformation is invertible

\begin{equation*} {\bar u_!} \bar v^* \xrightarrow \sim v^* {u_!} \qquad \qquad \big (\ v^*{u_*} \xrightarrow \sim {\bar u_*}\bar v^*\ \big )\,. \end{equation*}

A map $u:X\to Y$ in $\mathcal{B}$ is called smooth, or stably left Beck–Chevalley (proper, or stably right Beck–Chevalley) if every base change of $u$ is left (right) Beck–Chevalley. The relation of these operations with quantification in logic is recalled in Table 2.

The classes of $\mathbb{C}$ -smooth and $\mathbb{C}$ -proper maps are closed under base change and define subfibrations of the codomain fibration of $\mathcal{B}$ . Equivalently, they define regular calibrations $\Sigma (\mathbb{C}) \subset \mathbb{B}\supset \Pi (\mathbb{C})$ of the universe of $\mathbb{B}$ , called the smooth calibration and the proper calibration of $\mathbb{C}$ . The interest of the notions is in the following result.

The characterization of the smooth and proper maps can be quite difficult in practice. In the setting where the category $\mathbb{C}$ has a forgetful functor into the universe $\mathbb{B}$ (typically, when $\mathbb{C}$ classifies objects in $\mathbb{B}$ with an extra structure, notably when $\mathbb{C}$ is a calibration), stricter notions of smooth and proper maps can be defined, which are easier to characterize in practice.

Definition 12. (Strict smoothness/properness). Let $\mathbb{C}:\mathcal{B}^{\mathrm{op}} \to \mathsf{CAT}$ be equipped with a natural transformation $U:\mathbb{C}\to \mathbb{B}$ to the universe of $\mathbb{B}$ (typically the inclusion of a subuniverse). For $f:X\to Y$ a map in $\mathcal{B}$ , we have a commutative square

Recall from Proposition 8 that the functor $f^*:\mathcal{B}{_{/Y}}\to \mathcal{B}{_{/X}}$ has always a left adjoint $f_!$ given by composition with $f$ and that the right adjoint $f_*$ exists if and only if $f$ is an exponentiable map in $\mathcal{B}$ . We shall say that a $\mathbb{C}$ -smooth map is strictly smooth if the mate $f_!U_X \to U_Y\;f_!$ is invertible. We shall say that a $\mathbb{C}$ -proper map is strictly proper if it is exponentiable in $\mathcal{B}$ (i.e., if $f_*$ exists) and if the mate $U_Yf_* \to f_*U_X$ is invertible. Essentially, this says that a map $f$ is strictly smooth if $f_!$ can be computed by composition with $f$ in $\mathcal{B}$ , and that it is strictly proper if $f_*$ can be computed by exponentiation along $f$ in $\mathcal{B}$ .

A modality on the category $\mathcal{B}$ is a (unique) factorization system $(\mathcal{L},\mathcal{R})$ such that the factorization (or equivalently the left class $\mathcal{L}$ ) is stable under base change. In this situation, both classes $\mathcal{L}$ and $\mathcal{R}$ define calibrations $\mathbb{L}\subset \mathbb{B}\supset \mathbb{R}$ and $\mathbb{R}$ is even a reflective calibration (with the reflection given by the factorization) (Anel et al., Reference Anel, Biedermann, Finster and Joyal2020; Rijke et al., Reference Rijke, Shulman and Spitters2019).

Notice that Lemmas 14 and 15 together provide examples where smooth and strictly smooth maps do not coincide.

We consider now the case of a weak factorization system $(\mathcal{L},\mathcal{R})$ on $\mathcal{B}$ . The class $\mathcal{R}$ is still closed under base change and defines a regular calibration $\mathbb{R}\subset \mathbb{B}$ (but not the class $\mathcal{L}$ in general). The following lemma is a classical trick to recognize proper maps.

The name “acyclic” is motivated by the following result. The name “localic” is motivated by an application to topos theory (see third example of Section 3.3). In an $\infty$ -topos $\mathcal{E}$ , an acyclic class is a class of maps containing all isomorphisms, closed under composition and base change, and under colimits in the arrow category of $\mathcal{E}$ (Anel et al., Reference Anel, Biedermann, Finster and Joyal2022, Definition 3.2.8). If $(\mathcal{L},\mathcal{R})$ is a (unique) factorization system on $\mathcal{E}$ , the class $\mathcal{L}$ is acyclic if and only if $(\mathcal{L},\mathcal{R})$ is a modality.

The $\mathbb{C}$ -acyclic and $\mathbb{C}$ -localic maps sometimes form a factorization system on $\mathcal{B}$ . When this the case, it is always a modality since both classes are stable by base change by definition.

3.1 Set theory and type theory

The examples are summarized in Table 1.

Table 1. Examples from set theory and type theory

Any 1-category $C$ represents a functor $\mathcal{B}^{\mathrm{op}}=\mathsf{Set}^{\mathrm{op}}\to \mathsf{CAT}$ , sending $I$ to $C^I$ . In this setting, all conditions on maps can be computed fiberwise. A set $I$ is smooth (proper) if the coproduct $\coprod _I:C^I\to C$ (product functor $\prod _I:C^I\to C$ ) exists. Notice that in this case, the Beck–Chevalley condition is always fulfilled since everything is determined fiberwise (Remark 11). A map is smooth (proper) if its fibers are smooth (proper) sets. If $C$ is the terminal category, every set is smooth and proper. If $C$ is the initial category, the smooth (proper) maps are the surjections. A set $I$ is acyclic if $C\to C^I$ is an equivalence. If $C$ is the terminal category, every set is acyclic (and every map is acyclic). If $C$ is the empty category, the acyclic sets are the nonempty sets (the acyclic maps are surjections, and the localic maps are the injections). If $C$ is otherwise, the only acyclic sets are the singletons; thus, the acyclic maps are the bijections, and every map is localic.

Similar considerations hold when $\mathcal{B}=\mathcal{S}$ is the $\infty$ -category of spaces and $C$ some $\infty$ -category. In particular, Remark 11 applies, and the Beck–Chevalley conditions are always fulfilled.

The second example is the one detailed in the introduction of Section 2. It is an instance of the previous example if and only if $\kappa$ is a regular cardinal (use $C=\mathsf{Set}^{\lt \kappa }$ ). The example of the fibration of subsets (or of injections) is the case $\kappa =2$ . Every map $u:I\to J$ is smooth and proper. The functors $u_!$ and $u_*$ are the two direct images of subsets, classically related to existential and universal quantifiers ( $u_!A = \{j\ |\ \exists a\in A, a\in u^{-1}(j)\}$ , $u_*A = \{j\ |\ \forall a\in A, a\in u^{-1}(j)\}$ ). More generally, the setting of sets and subsets could be replaced by a hyperdoctrine in the sense of Lawvere (Reference Lawvere1970, Reference Lawvere2006) and Seely (Reference Seely1983). We recall the correspondence between logical quantifiers and the adjoints to base change in Table 2.

Table 2. Quantifiers and direct images

In the case of the codomain fibration, we have seen in Proposition 8 that all maps are smooth and that proper maps are the exponentiable maps whose pullbacks are also exponentiable. All the projections $X\times Y\to X$ are proper if and only if the category $\mathcal{B}$ is cartesian closed, and all the maps are proper if and only if the category $\mathcal{B}$ is locally cartesian closed. In this setting, only the isomorphisms are acylic and every map is localic.

Another example related to type theory is that of a category $\mathcal{B}$ with a $\pi$ -clan structure in the sense of Joyal (Reference Joyal2020). The clan structure distinguishes a class of maps in $\mathcal{D}\subset \mathcal{B}$ closed under base change and containing all isomorphisms, or equivalently a regular calibration $\mathbb{D}\subset \mathbb{B}$ . This implies that strict smooth maps are exactly the maps in $\mathcal{D}$ . And the definition of the $\pi$ -clan structure says that every map in $\mathcal{D}$ is strictly proper (Joyal, Reference Joyal2020, Definition 2.4.1).

If $\mathcal{B}$ is a cartesian closed 1-category with a subobject classifier (e.g., 1-topos), we consider the example of a dominance, which is a regular calibration of monomorphisms classified by a subobject $\mathcal{O}\subset \Omega$ of the subobject classifier (Escardó, Reference Escardó2004; Hyland, Reference Hyland, Carboni, Pedicchio and Rosolini1991; Rosolini, 2020). Intuitively, the object $\mathcal{O}$ classifies the subobjects meant to be “open” in the sense of topology, and the exponential $\mathcal{O}^A\subset \Omega ^A$ is the lattice of open subobjects sitting within that of all subobjects. The smooth (proper) maps define a notion of open (compact) maps in the sense of topology. The strictly smooth maps are those classified by $\mathcal{O}$ (Lemma 14). The strictly proper maps are those maps for which the direct image can be computed within the posets of all subobjects. Acyclic and localic maps are considered in the literature, but (seemingly) only with the base change property along cartesian projections and not along arbitrary maps. They are called $\mathcal{O}$ -equable maps and $\mathcal{O}$ -replete maps in Hyland (Reference Hyland, Carboni, Pedicchio and Rosolini1991).

When $\mathcal{B}$ is a 1-topos, any Grothendieck topology defines a dominance $\Omega _j\subset \Omega$ where $\Omega _j$ is the classifier of closed monos. In this case, all maps are smooth, the closed monos are the strictly smooth maps. The acyclic maps are the maps inverted in the localization by the topology, and the localic maps are the relative sheaves. A map $A\to 1$ is proper if and only if $A$ it is quasi-compact (every covering family has a covering finite subfamily). General proper maps can be described by a relative version of the same condition.

The $\infty$ -category $\mathcal{S}$ of $\infty$ -groupoids is a higher categorical model for dependent type theory with univalence: precisely, the small subuniverses correspond to univalent maps in $\mathcal{S}$ . It seems difficult to describe the smooth, proper, acyclic, and localic maps for an arbitrary subuniverse, but it is easier with strict smooth and proper maps. We shall only consider the subuniverse of $\kappa$ -small spaces. When $\kappa$ is regular, the strict smooth maps are the maps with $\kappa$ -small fibers by Lemma 14. When $\kappa$ is inaccessible, the strict proper maps are the maps with $\kappa$ -small fibers by Lemma 16. In particular, if $\kappa$ is the inaccessible cardinal bounding the size of small objects, all maps are smooth and proper. In this case, the acyclic maps are reduced to equivalences and all maps are localic.

In the example of an $\infty$ -topos $\mathcal{E}$ , we denote the universe by $\mathbb{E}$ .Footnote ³ By definition of an $\infty$ -topos, the functor $\mathbb{E}:\mathcal{E}^{\mathrm{op}}\to \mathsf{CAT}$ sends colimits of $\mathcal{E}$ to limits in $\mathsf{CAT}$ , and the interesting subuniverses are those satisfying a similar condition (corresponding to local classes of maps (Lurie, Reference Lurie2009, Proposition 6.1.3.7)). For $n\geq -2$ , we denote by $\mathcal{C}_n$ and $\mathcal{T}_n$ the classes of $n$ -connected and $n$ -truncated maps. For example, the class $\mathcal{C}_{-1}$ and $\mathcal{T}_{-1}$ are the classes of surjections and monomorphisms. Every map can be factored uniquely into an $n$ -connected map followed by an $n$ -truncated maps, and this factorization is stable under base change (it is a modality). The class $\mathcal{T}_n$ contains all isomorphisms, is closed under composition, and is local. It defines a regular calibration $\mathbb{T}_n\subset \mathbb{E}$ , which is also a reflective subuniverse (where the reflection is given by the factorization). Every map is smooth for $\mathbb{T}$ , and the strict smooth maps are those in $\mathbb{T}$ . Every map is strictly proper (hence proper) for $\mathbb{T}$ because dependent products preserve the truncation level of objects. The $\mathbb{T}_n$ -acyclic maps and $\mathbb{T}_n$ -localic maps can be characterized as the $(n+1)$ -connected maps and $(n+1)$ -truncated maps (see below).

More generally, a modality on $\mathcal{E}$ is a (unique) factorization system $(\mathcal{L},\mathcal{R})$ on $\mathcal{E}$ such that both classes are stable under base change. Then both classes $\mathcal{L}$ and $\mathcal{R}$ are local by Anel et al. (Reference Anel, Biedermann, Finster and Joyal2020, Proposition 3.6.5) and define subuniverses $\mathbb{L},\mathbb{R}\subset \mathbb{E}:\mathcal{E}^{\mathrm{op}}\to \mathsf{CAT}$ . We shall consider the case $\mathbb{C}=\mathbb{R}$ . Since the class $\mathcal{R}$ is closed under composition and base change, Lemma 14 shows that the strict smooth maps are all maps in $\mathcal{R}$ . But the subuniverse actually admits sums indexed by all maps in $\mathcal{E}$ . Given a map $u:X\to Y$ in $\mathcal{E}$ and a map $r:\bar X\to X$ in $\mathcal{R}$ , the $(\mathcal{L},\mathcal{R})$ -factorization of the composite map $\bar X\to X\to Y$ gives maps $\bar X\to \bar Y$ in $\mathcal{L}$ and $r':\bar Y\to Y$ in $\mathcal{R}$ . The image of $r$ by $u_!$ is the map $\bar Y\to Y$ . The Beck–Chevalley condition holds because the factorization is stable under base change. A more conceptual way to see this is to say that $\mathbb{R}$ is in fact a reflective subuniverse (where the reflection is given by the factorization) and therefore complete under all sums existing in $\mathbb{E}$ .

In this example, it seems difficult to describe the proper maps without further assumptions on $\mathbb{R}$ . However, the acyclic maps are exactly the maps in the décalage of the class $\mathcal{L}$ (Anel et al., Reference Anel, Biedermann, Finster and Joyal2024, 2.2.7) (they are also the “fiberwise $\mathbb{R}$ -equivalences” of Anel et al. (Reference Anel, Biedermann, Finster and Joyal2022, Def. 3.3.1)). In particular, they form an acyclic class in $\mathcal{E}$ (Anel et al., Reference Anel, Biedermann, Finster and Joyal2022, Theorem 3.3.9). In this context, acyclic and localic maps define a modality on $\mathcal{E}$ , which is detailled in Anel et al. (Reference Anel, Biedermann, Finster and Joyal2023, Theorem 2.3.32).

Finally, any left-exact localization of $\infty$ -topoi $q^*:\mathcal{E}\to \mathcal{E}[\mathcal{W}^{-1}]$ provides a modality $(\mathcal{W},\mathcal{F})$ on $\mathcal{E}$ , where $\mathcal{W}$ is the class if maps inverted by $q^*$ and $\mathcal{F}$ are the class of “relative sheaves” (Rijke et al., Reference Rijke, Shulman and Spitters2019; Anel et al., Reference Anel, Biedermann, Finster and Joyal2022). Such a modality is left-exact in the sense that the factorization of a map preserves finite limits (in the arrow category). In fact, there is a bijection between left-exact localizations and left-exact modalities (Anel et al., Reference Anel, Biedermann, Finster and Joyal2022). Let $\mathbb{W}\subset \mathbb{E}\supset \mathbb{F}$ be the corresponding subuniverses. If $\mathbb{E}':\mathcal{E}[\mathcal{W}^{-1}]^{\mathrm{op}}\to \mathsf{CAT}$ is the universe of the $\infty$ -topos $\mathcal{E}[\mathcal{W}^{-1}]$ , one can show that $\mathbb{F}=\mathbb{E}'\circ q^*:\mathcal{E}^{\mathrm{op}}\to \mathsf{CAT}$ . Since left-exact modalities are fixed by décalage (Anel et al., Reference Anel, Biedermann, Finster and Joyal2024, Lemma 2.4.6 (1)), the acyclic (localic) maps coincides with the class $\mathcal{W}$ ( $\mathcal{F}$ ).

3.2 Category theory

We now consider examples from category theory. The study of fibrations of categories was the motivation of Grothendieck for introducing his notion of smooth and proper functors. The examples are summarized in Table 3.

Table 3. Examples from category theory

The first example is that of the universe of $\mathcal{B}=\mathsf{Cat}$ . It will be the base of all the other examples. We denote by $\mathsf{\mathbb{C} at}$ the universe of $\mathsf{Cat}$ (which can be thought of either as the codomain fibration, or as the slice functor $C\mapsto \mathsf{Cat}{_{/C}}$ ). The category $\mathsf{Cat}$ is cartesian closed but not locally cartesian closed. The exponentiable functors are the Conduché fibrations (see Ayala and Francis (Reference Ayala and Francis2020, Lemma 1.11) for a higher categorical account). The acyclic maps are the equivalences of categories and every functor is localic.

The following examples will study various kinds of “fibrations” conditions on functors. This will lead to consider functors $\mathbb{C}:\mathsf{Cat}^{\mathrm{op}} \to \mathsf{CAT}$ with a natural forgetful morphism $\mathbb{C}\to \mathsf{\mathbb{C} at}$ into the universe of $\mathsf{Cat}$ (or equivalently a morphism between the corresponding fibrations over $\mathsf{Cat}$ ).

Let us start with the the functor $\mathsf{LFib}:\mathsf{Cat}^{\mathrm{op}}\to \mathsf{CAT}$ sending a category $C$ to its category $\mathsf{LFib}(C) = \left [C,\mathcal{S}\right ]$ of (small) left fibrations. This functor is (up to size issues) representable by the category $\mathcal{S}$ . The Grothendieck construction provides a natural transformation $\mathsf{LFib}(C) \to \mathsf{Cat}{_{/C}}$ , and we can talk about strict notions. Recall that the left fibrations are the right class of a (unique) factorization system on $\mathsf{Cat}$ , whereas the left class is that of initial functors. The characterization of smooth and proper maps is open. The previous factorization system is not stable under base change, and we cannot apply Lemma 15. But the strict smooth maps are exactly the left fibrations themselves. The class of proper functors contains right fibrations And we can use Lemma 16 to characterize the strict proper maps as the exponentiable functors $u:C\to D$ such that for any base change $\bar u\to u$ , the pullback along $\bar u$ preserves the class of initial functors. This is exactly the notion of proper functor of Lurie (Reference Lurie2009, dual of Definition 4.1.2.9) and the notion of smooth functor of Grothendieck in Pursuing Stacks and also in Joyal (Reference Joyal2008, 21.1) and Cisinski (Reference Cisinski2019, dual of Definition 4.4.1).Footnote ⁴ In particular, right fibrations are strict proper functors for $\mathsf{LFib}$ . The previous characterization of strict proper functors shows that they are also left-final functors in the sense of Ayala–Francis since they satisfy the condition of Ayala and Francis (Reference Ayala and Francis2020, Lemma 4.1.1 (a)). However, the condition to be left-final seems strictly weaker. In the dual situation of right fibrations, the notion of smooth and proper are reversed.

Finally, we can look at the example of (small) cocartesian (and cartesian) fibrations. The functor of interest is $C\mapsto \mathsf{CcFib}(C) = \left [C, \mathsf{Cat}\right ]$ . Up to size issues, it is represented by $\mathsf{Cat}$ itself. The class of cocartesian fibrations is almost the right class of a weak factorization system on $\mathsf{Cat}$ (in 1-categories, it is the right class of an algebraic factorization system, Bourke and Garner (Reference Bourke and Garner2014)). The corresponding left class is the class of fully faithful left adjoint functor. Any functor $f:C\to D$ admits a factorization $C\to f\downarrow D\to D$ into a fully faithful left adjoint functor followed by a cocartesian fibration (where $f\downarrow D$ if the fiber product of $C\to D \xleftarrow{dom}{D}^{\to }$ ). Again, the smooth and proper maps are difficult to find. The strict smooth maps are the cocartesian fibrations themselves. The previous factorization is enough to apply Lemma 16 to characterize the strictly proper maps as the exponentiable functors $u:C\to D$ such that $u^*$ preserves fully faithful left adjoint functors. This class includes that of cartesian fibrations. In the dual situation of cartesian fibrations, the notion of smooth and proper are reversed.

In all the “fibration” examples, the acyclic maps are the equivalences and every functor is localic. To see this, recall that a functor $u:C\to D$ induces an equivalence $u_!:\mathcal{S}^C\rightleftarrows \mathcal{S}^D:u^*$ if and only if it is a Morita equivalence. The acyclic maps are Morita equivalences, which are stable under base change. This implies that they must be (essentially) surjective functors. Since $u_!$ is an equivalence, $u$ is fully faithful and is therefore an actual equivalence.

3.3 Topology and geometry

We now consider examples where the base category $\mathcal{B}$ is a category of topological or geometrical objects. We are going to see a tight connection between direct images functors and classical topological conditions. The examples are summarized in Table 4.

Table 4. Examples from topology and geometry

In the first example, $\mathcal{B}=\mathsf{Locale}$ is the category of localesFootnote ⁵ with the codomain fibration. Every map is smooth, and the proper maps are the exponentiable ones. The exponentiable locales are the locally compact ones (i.e., the locales such that ${\mathrm{Op}\left (X\right )}$ is a continuous lattice) (Johnstone, Reference Johnstone1982, Theorem 4.11). Acyclic maps are the isomorphisms, and every map is localic.

The next example is that of the functor $\mathsf{Locale}^{\mathrm{op}}\to \mathsf{CAT}$ sending a locale $X$ to its frame $\mathrm{Op}\left (X\right )$ of open domains. This functor is represented by the Sierpiński space. Let us say that a map is an open immersion if it is isomorphism to the inclusion of an open. The fibration corresponding to $\mathrm{Op}\left (-\right )$ is the subfibration of the codomain fibration of $\mathsf{Locale}$ spanned by open immersions. This shows that the notion of strict smooth and proper maps makes sense.

The smooth morphisms are almost by definition the open morphisms of topology (this can be shown directly, or deduced from the similar result for topoi (Johnstone, Reference Johnstone2002, Lemma C.3.1.10), see also Escardó (Reference Escardó2004)). By Lemma 14, the strict smooth maps are the open immersions that are also open morphisms, but that is the case of every open immersion. The proper morphisms are the proper morphisms of topology (i.e., the universally closed morphisms, this can be seen directly by considering left adjoints on the posets ${\mathrm{Op}\left (X\right )}^{\mathrm{op}}$ , or deduced from topos theory (Johnstone, Reference Johnstone2002, Lemma C.3.2.81), see also Escardó (Reference Escardó2004, Reference Escardó2020)). The characterization of strict proper maps is open. An open immersion is proper if and only if it is isomorphic to the inclusion of a clopen. The computation of acyclic and localic maps is straighforward.

In the third example, $\mathcal{B}$ is the category of 1-topoi and geometric morphisms (Johnstone, Reference Johnstone2002); Anel and Joyal, Reference Anel, Joyal, Anel and Catren2021)). If we look at the codomain fibration, the only nontrivial class of maps is that of exponentiable maps (Johnstone and Joyal, Reference Johnstone and Joyal1982).

Another important fibration is the one of open sub-1-topoi (or open immersions), sending a 1-topos $\mathsf X$ to the poset of subterminal objects in its category of sheaves $\mathrm{Sh}({\mathsf X})$ (its dual 1-logos in the sense of Anel and Joyal (Reference Anel, Joyal, Anel and Catren2021)). This fibration is represented by the Sierpiński topos (dual to the 1-logos ${\mathsf{Set}}^{\to }$ ). The identification of smooth maps as open morphisms of 1-topoi is done in Johnstone (Reference Johnstone2002, Theorem C.3.1.28), and that of proper maps as proper morphisms of 1-topoi is done in Moerdijk and Vermeulen (Reference Moerdijk and Vermeulen2000, Corollary I.5.9) and in Johnstone (Reference Johnstone2002, Theorem C.3.1.28). Both references use a “weak” version of Beck–Chevalley conditions (where the mate transformation is only monic) for arbitrary sheaves. But restricted to subterminal objects, this condition recovers the one from Definition 9, and this can be shown to be equivalent to the weak condition (see Johnstone, Reference Johnstone2002, Proposition A.4.1.17, and Moerdijk and Vermeulen, Reference Moerdijk and Vermeulen2000, Proposition 3.2).

Open immersions can be composed and define a regular calibration on $\mathsf{Topos}$ . Then, Lemma 14 shows that the strict smooth maps are exactly the open immersion of 1-topoi. The characterization of strict proper maps is open. An open immersion of 1-topoi is proper if it is also the inclusion of a closed sub-1-topoi (corresponding to a decidable subterminal object). Interestingly, in this example, the notion of acyclic and localic maps recover the hyperconnected and localic morphisms of topoi (this is in fact the motivation for the name localic). The proof is straightforward for hyperconnected maps, and the fact that the right orthogonal to hyperconnected morphisms are the localic morphisms is Johnstone (Reference Johnstone2002, Lemma A.4.6.4).

The next example is the case of the fibration of étale maps over the category of 1-topoi, sending a 1-topos $\mathsf X$ to its category of sheaves of sets (its dual 1-logos) $\mathrm{Sh}({\mathsf X})$ . This fibration is representable by the “object classifier” or the “topos line” (which we shall denote $\mathsf A$ , its 1-logos of sheaves is $\left [{\mathsf{finset}},\mathsf{Set}\right ]$ ) and the existence of sums and products for the étale fibration can be interpreted as sums and products structures on the 1-topos $\mathsf A$ . In this case, the smooth maps are the locally connected morphisms of 1-topoi (Johnstone, Reference Johnstone2002, Corollary C.3.3.16), and the proper maps are the tidy morphisms of 1-topoi (see Moerdijk and Vermeulen, Reference Moerdijk and Vermeulen2000, Corollary III.4.9, or Johnstone, Reference Johnstone2002, Corollary C.3.4.11). The strict smooth maps are all the étale morphisms. The characterization of strict proper maps is open. In this setting, the acyclic maps are the equivalences, and every map is localic.

Interestingly, not every étale map is proper: the intersection of the two classes is the class of finite maps. This means that the “topos line” $\mathsf A$ does not have arbitrary products indexed by itself, but only finite products. This might be surprising since categories of sheaves are always locally cartesian closed, but the dependent products are not preserved by geometric morphisms, only finite products are.

The generalization of the previous results to $\infty$ -topoi is quite rich! For the case of the codomain fibration, exponentiable $\infty$ -topoi have been characterized in Lurie (Reference Lurie2017, Theorem 21.1.6.12) and Anel and Lejay (Reference Anel and Lejay2019). The case of the fibration of étale morphisms has been studied recently by Martini and Wolf. If $\mathcal{B}=\mathsf{Topos}_\infty$ is the category of $\infty$ -topoi and $\mathbb{C}({\mathsf X}) ={\mathrm{Sh}_\infty \!\left ({\mathsf X} \right )}$ is the category of higher sheaves (the $\infty$ -logos dual to $\mathsf X$ ), then the smooth maps are the locally contractible morphisms of $\infty$ -topoi Martini (Reference Martini and Wolf2023), and the proper maps are the proper morphisms of $\infty$ -topoi (Lurie, Reference Lurie2009; Martini and Wolf Reference Martini and Wolf2023a, Reference Martini and Wolf2023b) The acyclic maps are equivalences, and every map is localic.

It is interesting to restrict the fibration of higher sheaves (aka higher etale morphisms) to $n$ -truncated sheaves only (the case $n=-1$ recovers the fibration of open immersions). In this case, it is easy to adapt the results of Martini and Wolf to characterize the smooth maps as some locally $n$ -connected morphisms of $\infty$ -topoi,Footnote ⁶ and the proper maps are the $n$ -proper morphisms of $\infty$ -topoi.Footnote ⁷ The strict smooth maps are the $n$ -truncated étale morphisms of topoi. The corresponding acyclic and localic maps define notions of hyper- $(n+1)$ -connected morphisms and $(n+1)$ -localic morphisms, and every geometric morphism should factor into a hyper- $(n+1)$ -connected morphism followed by a $(n+1)$ -localic morphism. Moreover, an $\infty$ -topos $\mathsf X$ should be $n$ -localic in the sense of Lurie (Reference Lurie2009, Definition 6.4.5.8) if and only if the morphism ${\mathsf X}\to 1$ is $n$ -localic in the previous sense.

Étale maps define a regular calibration of the universe of $\mathsf{Topos}_\infty$ , and the notions of strictly smooth and strictly proper maps can be defined. By Lemma 14, the strict smooth maps are exactly the etale maps. The strict proper maps are exponentiable maps $u$ whose direct image $u_*$ preserves etale maps. Their characterization is open.

The last example of Table 4 is in algebraic geometry, where the base category is the category $\mathsf{Sch}$ of (noetherian) schemes, and the functor $\mathsf{Sch}^{\mathrm{op}} \to \mathsf{CAT}$ is the one sending a scheme $X$ to its category of torsion étale sheaves (Freitag and Kiehl, Reference Freitag and Kiehl1988). The characterization of smooth and proper maps in general is open but Freitag and Kiehl (Reference Freitag and Kiehl1988, Theorems 6.1 and 7.3) show that smooth morphisms of schemes are smooth maps (if the torsion is prime to the characteristic) and that proper morphisms of schemes are proper maps. This setting is the one that inspired Grothendieck to name his notions of smooth and proper functors (Maltsiniotis, Reference Maltsiniotis2005). The characterization of acyclic and localic maps is an open question.