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.
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.)
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
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;
An n-coherent (∞,1)-topos is compact of height $n$ in the sense of def. 3, this is (Lurie XIII, prop. 2.3.9).
Any equivalence is proper and the class of proper maps is closed under composition.
If in the diagram
$p$ is a surjective geometric morphism and $f$ is proper then so is $g$.
If $h$ is proper and $g$ is a geometric embedding then $p$ is proper.
Any hyperconnected geometric morphism is proper.
$f:F\to G$ is proper iff its localic reflection $Sh_G(X)\to G$ is, i.e. iff $X$ is a compact internal locale in $G$.
If in a pullback square the bottom morphism is open and surjective and the left morphism is proper then so is the right.
The pullback of a proper geometric morphism is again proper.
The pullback of a tidy geometric morphism is again tidy.
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).
A map satisfies the stable weak BCC iff it is proper.
We discuss classes of sites such that their sheaf topos is a compact topos, def. 1 (VM, I.5).
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We discuss classes of sites such that their sheaf topos is a strongly compact topos, def. 1 (VM, III.4).
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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
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.
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
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.
The following propositions say in summary that
the petit topos over a compact topological space that is also Hausdorff is strongly compact.
the gros topos over a compact object is strongly compact.
See also (VM, III.1).
Examples of strongly compact toposes $\mathcal{E}$, def. 3, include the following.
Every coherent topos is strongly compact.
The sheaf topos over a compact Hausdorff topological space is strongly compact.
Let $\mathbf{H}$ be a topos over Set and $X \in \mathbf{H}$ an object. Then the following are equivalent
$X$ is a compact object (in the sense that the hom functor $\mathbf{H}(X,-)$ preserves filtered colimits)
the slice topos $\mathbf{H}_{/X}$ is strongly compact, def. 3.
The direct image $\Gamma_X$ of the global section geometric morphism
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
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
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
Moreover, sections of a trivial bundle are maps into the fiber
So it follows that $X$ is a compact object:
An object $X \in \mathcal{T}$ in a topos $\mathcal{T}$ is a Kuratowski finite object precisely if the étale geometric morphism
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)
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).
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
A textbook account is in section C3.2 of
Discussion with relation to properness of geometric stacks includes
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