nLab
proper geometric morphism

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 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.)

Definition

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

Definition

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;

Remark

An n-coherent (∞,1)-topos is compact of height nn 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

    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)

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 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).

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

then

  • The slice topos H /X\mathbf{H}_{/X} is a compact topos, def. 1.
Remark

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. 6 below.

Proof

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).

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.

(MV, Examples III.1.1)

Proposition

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. 3.

Proof

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

Proposition

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)

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)