# nLab proper geometric morphism

topos theory

## Theorems

#### Compact objects

objects $d \in C$ such that $C(d,-)$ commutes with certain colimits

# Contents

## 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 $\mathcal{E}$.

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. 3, 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.

###### Proposition

The pullback of a proper geometric morphism is again proper.

The pullback of a tidy geometric morphism is again tidy.

### 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.

### Compact sites

We discuss classes of sites such that their sheaf topos is a compact topos, def. 1 (VM, I.5).

(…)

### Strongly compact sites

We discuss classes of sites such that their sheaf topos is a strongly compact topos, def. 1 (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. 1.
###### 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. 6 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.

###### Proposition

Examples of strongly compact toposes $\mathcal{E}$, def. 3, include the following.

1. Every coherent topos is strongly compact.

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

###### 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. 3.

###### 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.

## 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

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

and in section 2.3 of

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

Revised on February 3, 2014 13:07:15 by Urs Schreiber (89.204.130.154)