nLab proper geometric morphism

Idea

The notion of compact topos is the generalization from topology to topos theory of the notion of compact topological space.

More generally, over a general base topos, the notion of proper geometric morphism is the generalization to morphisms between toposes of the notion of proper map between topological spaces.

Definition

Definition

A sheaf topos $\mathcal{E}$ is called a compact topos if the direct image of the global section geometric morphism $\Gamma : \mathcal{E} \to Set$ preserves directed joins of subterminal objects.

A geometric morphism $f : \mathcal{F} \to \mathcal{E}$ is called proper if it exhibits $\mathcal{F}$ as a compact topos over $\mathcal{E}$. (The stack semantics of $\mathcal{E}$ can be used to formalize this.)

Definition

A topos is called strongly compact if $\Gamma$ commutes even with all filtered colimits.

A geometric morphism $f : \mathcal{F} \to \mathcal{E}$ is called tidy if it exhibits $\mathcal{F}$ as a strongly compact topos over $\mathcal{E}$.

(MV, p. 53)

This are the first stages of a notion that in (∞,1)-topos theory continue as follows

Definition

Let $\kappa$ be a regular cardinal and $-1 \leq n \leq \infty$. Then an (∞,1)-topos is $\kappa$-compact of height $n$ if the global section geometric morphism preserves $\kappa$-filtered (∞,1)-colimits of n-truncated objects.

Accordingly a geometric morphism is $\kappa$-proper of height $n$ if it exhibits a $\kappa$-compact of height $n$ $(\infty,1)$-topos over a base (∞,1)-topos.

In this terminology

• a topos compact of height (-1) is the same as a compact topos;

• a topos compact of height 0 is the same as a strongly compact topos;

Remark

An n-coherent (∞,1)-topos is compact of height $n$ in the sense of def. , this is (Lurie XIII, prop. 2.3.9).

Properties

Stability and closure properties

Proposition

1. Any equivalence is proper and the class of proper maps is closed under composition.

2. If in the diagram

   $$\array{ A&\xrightarrow{p}&B \\ \downarrow^f&\swarrow^g \\ C }$$

   $p$ is a surjective geometric morphism and $f$ is proper then so is $g$.

3. If $h$ is proper and $g$ is a geometric embedding then $p$ is proper.

4. Any hyperconnected geometric morphism is proper.

5. $f:F\to G$ is proper iff its localic reflection $Sh_G(X)\to G$ is, i.e. iff $X$ is a compact internal locale in $G$.

6. If in a pullback square the bottom morphism is open and surjective and the left morphism is proper then so is the right.

(VM, I.1, I.2)

Proposition

The pullback of a proper geometric morphism is again proper.

The pullback of a tidy geometric morphism is again tidy.

(VM, theorem 5.8)

Properness and Beck-Chevalley conditions

A geometric morphism $f$ of toposes is said to satisfy the stable (weak) Beck-Chevalley condition if any pullback of $f$ satisfies the (weak) Beck-Chevalley condition ((weak)BCC).

Proposition

A map satisfies the stable weak BCC iff it is proper.

(MV, Corollary I.5.9)

Compact sites

We discuss classes of sites such that their sheaf topos is a compact topos, def. (VM, I.5).

(…)

Strongly compact sites

We discuss classes of sites such that their sheaf topos is a strongly compact topos, def. (VM, III.4).

(…)

Examples

Compact toposes

Proposition

Let $\mathbf{H}$ be a topos and $X \in \mathbf{H}$ an object. If

• $X$ is a “compact topological space-object” in that:

  for every set of morphisms $\{U_i \to X\}_{i \in I}$ such that $\coprod_{i \in I} U_i \to X$ is an effective epimorphism, there is a finite subset $J \subset I$ such that $\coprod_{i \in J} U_i \to X$ is still an effective epimorphism;

then

• The slice topos $\mathbf{H}_{/X}$ is a compact topos, def. .

Remark

Beware that $X$ being a “compact topological space-object” is different from it being a compact object (the difference being that between compactness of height (-1) and height 0). For the latter case see prop. below.

Proof

The terminal object of $\mathbf{H}_{/X}$ is the identity $id_X : X \to X$ in $\mathbf{H}$. A subterminal object of $\mathbf{H}_{/X}$ is a monomorphism $U \hookrightarrow X$ in $\mathbf{H}$.

The global section geometric morphism $\Gamma_X : \mathbf{H}_{/X} \to Set$ sends an object $[E \to X]$ to its set of sections

$$\Gamma_X([E \to X]) = \mathbf{H}(X, E) \times_{\mathbf{H}(X,X)} \{id_X\} \,.$$

Therefore it sends all subterminal object in $\mathbf{H}_{/X}$ to the empty set except the terminal object $X$ itself, which is sent to the singleton set.

So let $X$ now be a compact-topological-space-object and $U_\bullet : I \to \mathbf{H}_{/X}$ is directed system of subterminals.

If their union $\vee_i U_i$ does not cover $X$, then $\Gamma_X(\vee_i U_i) = \emptyset$. But then also none of the $U_i$ can be $X$ itself, and hence also $\Gamma_X(U_i) = \emptyset$ for all $i \in I$ and so $\vee_i \Gamma_X(U_i) = \emptyset$. On the other hand, if $\vee_i U_i = X$ then the $\{U_i \to X\}_{i \in I}$ form a cover, hence then by assumption there is a finite subset $\{U_i \to X\}_{i \in J}$ which still covers. By the assumption that the system $U_\bullet$ is a directed set it also contains the union $X = \vee_{i \in J} U_i$. Therefore $\vee_{i \in I} \Gamma_X(U_i) = \Gamma_X(X) = *$ is the singleton, as is $\Gamma_X(\vee_{i \in I} U_i) = \Gamma_X(X)$. So $\Gamma_X$ preserves directed unions of subterminals and hence $\mathbf{H}_{/X}$ is a compact topos.

Strongly compact toposes

The following propositions say in summary that

1. the petit topos over a compact topological space that is also Hausdorff is strongly compact.

2. the gros topos over a compact object is strongly compact.

See also (VM, III.1).

Proposition

Examples of strongly compact toposes $\mathcal{E}$, def. , include the following.

1. Every coherent topos is strongly compact.

2. The sheaf topos over a compact Hausdorff topological space is strongly compact.

(MV, Examples III.1.1)

Proposition

Let $\mathbf{H}$ be a topos over Set and $X \in \mathbf{H}$ an object. Then the following are equivalent

1. $X$ is a compact object (in the sense that the hom functor $\mathbf{H}(X,-)$ preserves filtered colimits)

2. the slice topos $\mathbf{H}_{/X}$ is strongly compact, def. .

Proof

The direct image $\Gamma_X$ of the global section geometric morphism

$$((-) \times X \dashv \Gamma_X) : \mathbf{H}_{/X} \stackrel{\overset{(-) \times X}{\leftarrow}}{\underset{\mathbf{\Gamma}_X}{\to}} \mathbf{H} \stackrel{\overset{\Delta}{\leftarrow}}{\underset{\mathbf{H}(*,-)}{\to}} Set$$

is given by the hom functor out of the terminal object. The terminal object in $\mathbf{H}_{/X}$ is the identity morphism $id_X : X \to X$. So the terminal geometric morphism takes any $[E \to X]$ in $\mathbf{H}_{/X}$ to the set of sections, given by the pullback of the hom set along the inclusion of the identity

$$\Gamma_X([E \to X]) = \mathbf{H}(X,E) \times_{\mathbf{H}(X,X)} \{id\} \,.$$

By the discussion at overcategory – limits and colimits we have that colimits in $\mathbf{H}_{/X}$ are computed in $\mathbf{H}$. So if $[E \to X] \simeq \underset{\longrightarrow_i}{\lim}{[E_i \to X]}$ is a filtered colimit in $\mathbf{H}_{/X}$, then $E \simeq \underset{\longrightarrow_i}{\lim}{E_i }$ is a filtered colimit in $\mathbf{H}$.

If now $X \in \mathbf{H}$ is a compact object, then this commutes over this colimit and hence

$$\begin{aligned} \Gamma_X([E \to X]) &= \mathbf{H}(X,\underset{\longrightarrow_i}{\lim} E_i) \times_{\mathbf{H}(X,X)} \{id\} \\ & \simeq (\underset{\longrightarrow_i}{\lim}\mathbf{H}(X, E_i)) \times_{\mathbf{H}(X,X)} \{id\} \\ &\simeq \underset{\longrightarrow_i}{\lim} (\mathbf{H}(X, E_i) \times_{\mathbf{H}(X,X)} \{id\}) \\ & \simeq \underset{\longrightarrow_i}{\lim} \Gamma_X([E_i \to X]) \end{aligned} \,,$$

where in the second but last step we used that in the topos Set colimits are preserved by pullback.

This shows that $\Gamma_X(-) : \mathbf{H}_{/X} \to Set$ commutes over filtered colimits if $X$ is a compact object.

Conversely, assume that $\Gamma_X(-)$ commutes over all filtered colimits. For every (filtered) diagram $F_\bullet : I \to \mathbf{H}$ there is the corresponding filtered diagram $X \times F_\bullet : I \to \mathbf{H}_{/X}$, where $[X \times F_i \to X]$ is the projection. As before, the product with $X$ preserves forming colimits

$$\underset{\longrightarrow_i}{\lim} ([X \times F_i \to X]) \simeq [X \times (\underset{\longrightarrow_i}{\lim} F_i) \to X] \,.$$

Moreover, sections of a trivial bundle are maps into the fiber

$$\Gamma_X([X \times F_i \to X]) \simeq \mathbf{H}(X,F_i) \,.$$

So it follows that $X$ is a compact object:

$$\begin{aligned} \mathbf{H}(X, \underset{\longrightarrow_i}{\lim} F_i) & \simeq \Gamma_X( [X \times (\underset{\longrightarrow_i}{\lim} F_i) \to X]) \\ & \simeq \Gamma_X(\underset{\longrightarrow_i}{\lim} [X \times F_i \to X]) \\ & \simeq \underset{\longrightarrow_i}{\lim} \Gamma_X( [X \times F_i \to X]) \\ & \simeq \underset{\longrightarrow_i}{\lim} \mathbf{H}(X,F_i) \end{aligned} \,.$$

Finite objects

Proposition

An object $X \in \mathcal{T}$ in a topos $\mathcal{T}$ is a Kuratowski finite object precisely if the étale geometric morphism

$$\mathcal{T}_{/X} \to \mathcal{T}$$

out of the slice topos is a proper geometric morphism. And precisely if $X$ is even decidable is this a tidy geometric morphism.

(Moerdijk-Vermeulen, examples III 1.4)

Geometric stacks

A typical condition on a geometric stack to qualify as an orbifold/Deligne-Mumford stack is that its diagonal be proper. This is equivalent to the corresponding map of toposes being a proper geometric morphism (e.g. Carchedi 12, section 2, Lurie Spectral, section 3).

Related concepts

• separated geometric morphism, Hausdorff topos

References

The theory of proper geometric morphisms is largly due to

• Ieke Moerdijk, Jacob Vermeulen, Relative compactness conditions for toposes (pdf)

• Ieke Moerdijk, Jacob Vermeulen, Proper maps of toposes , Memoirs of the American Mathematical Society, no. 705 (2000)

based on the localic case discussed in

• Jacob Vermeulen, Proper maps of locales, J. Pure Applied Alg. 92 (1994)

A textbook account is in section C3.2 of

• Peter Johnstone, Sketches of an Elephant

Discussion with relation to properness of geometric stacks includes

• David Carchedi, section 2 of Étale Stacks as Prolongations (arXiv:1212.2282)

Discussion of higher compactness conditions in (∞,1)-topos theory is in section 3 of

• Jacob Lurie, Spectral Schemes (pdf)

and in section 2.3 of

• Jacob Lurie, Rational and p-adic Homotopy Theory

and for the special case of spectral Deligne-Mumford stacks in section 1.4 of

• Jacob Lurie, Quasi-Coherent Sheaves and Tannaka Duality Theorems

More on proper geometric morphisms between $(\infty,1)$-toposes:

• Louis Martini, Sebastian Wolf, Proper morphisms of ∞-topoi [arXiv:2311.08051].