nLab proper geometric morphism



Topos Theory

topos theory



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In higher category theory


Compact objects



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 Γ:Set\Gamma : \mathcal{E} \to Set preserves directed joins of subterminal objects.

A geometric morphism f: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 Γ\Gamma commutes even with all filtered colimits.

A geometric morphism f: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


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

Accordingly a geometric morphism is κ\kappa-proper of height nn if it exhibits a κ\kappa-compact of height nn (,1)(\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 nn in the sense of def. , this is (Lurie XIII, prop. 2.3.9).


Stability and closure properties

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

  2. If in the diagram

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

    pp is a surjective geometric morphism and ff is proper then so is gg.

  3. If hh is proper and gg is a geometric embedding then pp is proper.

  4. Any hyperconnected geometric morphism is proper.

  5. f:FGf:F\to G is proper iff its localic reflection Sh G(X)GSh_G(X)\to G is, i.e. iff XX is a compact internal locale in GG.

  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)


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 ff of toposes is said to satisfy the stable (weak) Beck-Chevalley condition if any pullback of ff satisfies the (weak) Beck-Chevalley condition ((weak)BCC).


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).



Compact toposes


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


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

Beware that XX 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.


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

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

Γ X([EX])=H(X,E)× H(X,X){id X}. \Gamma_X([E \to X]) = \mathbf{H}(X, E) \times_{\mathbf{H}(X,X)} \{id_X\} \,.

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

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

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


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)


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

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

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


The direct image Γ X\Gamma_X of the global section geometric morphism

(()×XΓ X):H /XΓ X()×XHH(*,)ΔSet ((-) \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 H /X\mathbf{H}_{/X} is the identity morphism id X:XXid_X : X \to X. So the terminal geometric morphism takes any [EX][E \to X] in H /X\mathbf{H}_{/X} to the set of sections, given by the pullback of the hom set along the inclusion of the identity

Γ X([EX])=H(X,E)× H(X,X){id}. \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 H /X\mathbf{H}_{/X} are computed in H\mathbf{H}. So if [EX]lim i[E iX][E \to X] \simeq \underset{\longrightarrow_i}{\lim}{[E_i \to X]} is a filtered colimit in H /X\mathbf{H}_{/X}, then Elim iE iE \simeq \underset{\longrightarrow_i}{\lim}{E_i } is a filtered colimit in H\mathbf{H}.

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

Γ X([EX]) =H(X,lim iE i)× H(X,X){id} (lim iH(X,E i))× H(X,X){id} lim i(H(X,E i)× H(X,X){id}) lim iΓ X([E iX]), \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 Γ X():H /XSet\Gamma_X(-) : \mathbf{H}_{/X} \to Set commutes over filtered colimits if XX is a compact object.

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

lim i([X×F iX])[X×(lim iF i)X]. \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

Γ X([X×F iX])H(X,F i). \Gamma_X([X \times F_i \to X]) \simeq \mathbf{H}(X,F_i) \,.

So it follows that XX is a compact object:

H(X,lim iF i) Γ X([X×(lim iF i)X]) Γ X(lim i[X×F iX]) lim iΓ X([X×F iX]) lim iH(X,F i). \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


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

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

out of the slice topos is a proper geometric morphism. And precisely if XX 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).


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 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

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

Last revised on November 15, 2023 at 09:45:52. See the history of this page for a list of all contributions to it.