nLab (geometric surjection, embedding) factorization system

Idea

Every geometric morphism between toposes factors into a geometric surjection followed by a geometric embedding. This exhibits an image construction in the topos-theoretic sense, and gives rise to a factorization system in a 2-category for Topos.

Statement

Proposition

There is a factorization system on the 2-category Topos whose left class is the surjective geometric morphisms and whose right class is the geometric embeddings.

Moreover, the factorization of a given geometric morphism $f : \mathcal{E} \to \mathcal{F}$ is, up to equivalence, through the canonical surjection onto the topos of coalgebras $f^* f_* CoAlg(\mathcal{E})$ of the comonad $f^* f_* : \mathcal{E} \to \mathcal{E}$:

$$\array{ \mathcal{E} &&\stackrel{f}{\to}&& \mathcal{F} \\ & {}_{\mathllap{F}}\searrow && \nearrow \\ && f^* f_* CoAlg(\mathcal{E}) } \,E.$$

This appears for instance as (MacLaneMoerdijk, VII 4., theorem 6).

We use the following lemma

Lemma

Let $j$ be a Lawvere-Tierney topology on a topos $\mathcal{E}$ and write $i : Sh_j(\mathcal{E}) \to \mathcal{E}$ for the corresponding geometric embedding.

Then a geometric morphism $f : \mathcal{F} \to \mathcal{E}$ factors through $i$ precisely if

• the direct image $f_*$ takes values in $j$-sheaves;

or, equivalently

• the inverse image $f^*$ sends $j$-dense monomorphisms to isomorphisms.

This appears as (MacLaneMoerdijk, VII 4. prop. 2).

Proof of the lemma

We first show the first statement, that for $f$ to factor it is sufficient for $f_*$ to take values in $j$-sheaves: in that case, set

$$p_* := i^* f_*: \mathcal{F} \to Sh_j(\mathcal{E}) \,.$$

Since by assumption the unit map $x \to i_* i^* x$ is an isomorphism on the image of $f_*$ this indeed serves to factor $f_*$:

$$i_* p_* \simeq i_* i^* f_* \simeq f_* \,.$$

The left adjoint to $p_*$ is then

$$p^* \simeq f^* i_* \,,$$

because

$$\begin{aligned} \mathcal{F}(g^* E, F) & \simeq \mathcal{F}(f^* i_* E, F) \\ & \simeq \mathcal{E}(i_* E, f_* F) \\ & \simeq \mathcal{E}(i_* E, i_* i^* f_* F) \\ & \simeq Sh_j\mathcal{E} (E, i^* f_* F) \\ & \simeq Sh_j(E, p_* F) \end{aligned} \,,$$

where in the middle steps we used that $f_* F$ is a $j$-sheaf, by assumption, and that $i_*$ is full and faithful.

It is clear that $p^*$ is left exact, and so $(p^* \dashv p_*)$ is indeed a factorizing geometric morphism.

We now show that $f_*$ taking values in sheaves is equivalent to $f^*$ mapping dense monos to isos.

Let $u : U \hookrightarrow X$ be a $j$-dense monomorphism and $A \in \mathcal{E}$ any object. Consider the induced naturality square

$$\array{ \mathcal{E}(X, f_* A) &\stackrel{\simeq}{\to}& \mathcal{F}(f^* X, A) \\ {}^{\mathllap{\mathcal{E}(u, f_* A)}}\downarrow && \downarrow^{\mathrlap{\mathcal{F}(f^* u, A)}} \\ \mathcal{E}(U, f_* A) &\stackrel{\simeq}{\to}& \mathcal{F}(f^* U, A) }$$

of the adjunction natural isomorphism. If now $f_* A$ is a $j$-sheaf and $u$ a dense monomorphism, then by definition the left vertical morphism is also an isomorphism and so is the right one. By the Yoneda lemma this being an iso for all $A$ is equivalent to $f^* u$ being an iso. And conversely.

Proof of the proposition

Let $f : \mathcal{F} \to \mathcal{E}$ be any geometric morphism.

We first discuss the existence of the factorization, then its uniqueness.

To construct the factorization, we shall give a Lawvere-Tierney topology on $\mathcal{E}$ and factor $f$ through the geometric embedding of the corresponding sheaf topos.

Take the closure operator $\overline{(-)} : Sub(-)_{\mathcal{E}} \to Sub(-)_{\mathcal{E}}$ to be given by sending $U \hookrightarrow X$ to the pullback

$$\array{ \overline{U} &\to& f_* f^* U \\ \downarrow && \downarrow \\ X &\to& f_* f^* X } \,,$$

where the bottom morphism is the $(f^* \dashv f_*)$-unit. One checks that this is indeed a closure operator by the fact that $f^*$ preserves both pullbacks and pushouts.

Notice that this implies that for two subobjects $U_1, U_2 \hookrightarrow X$ we have

(1)$$(U_1 \subset \overline{U_2}) \;\;\; \Leftrightarrow \;\;\; (f^* U_1 \subset f^* U_2)$$

Write $j$ for the corresponding Lawvere-Tierney topology and

$$i : Sh_j(\mathcal{E}) \to \mathcal{E}$$

for the corresponding geometric embedding.

By lemma we get a factorization through $I$ if $f^*$ sends $j$-dense monomorphisms to isomorphisms. But if $U \hookrightarrow X$ is dense so that $X \subset \overline{U}$ then, by (1), $f^* X \subset f^* U$ and hence $f^* X = f^* U$.

Write

$$f : \mathcal{F} \stackrel{p}{\to} Sh_j(\mathcal{E}) \stackrel{i}{\to} \mathcal{E}$$

for the factorization thus established. It remains to show that $p$ here is a geometric surjection. By one of the equivalent characterizations discussed there, this is the case if $p^*$ induces an injective homomorphism of subobject lattices.

So suppose that for subobjects $U_1, U_2 \subset X$ we have $p^* U_1 \simeq p^* U_2$. Observe that then also $f^* i_* U_1 \simeq f^* i_* U_2$, because

$$\begin{aligned} f^* i_* U_1 & \simeq p^* i^* i_* U_1 \\ & \simeq p^* U_1 \\ & \simeq p^* U_2 \\ & \simeq p^* i^* i_* U_2 \\ & \simeq f^* i:* U_2 \end{aligned}$$

by the fact that $i_*$ is full and faithful. With (1) it follows that also

$$i_* U_1 \simeq \overline{i_* U_2}$$

and hence

$$\cdots \simeq i_* U_2$$

by the very fact that $i_*$ includes $j$-sheaves in general, hence $j$-closed subobjects in particular. Finally since $i_*$ if a full and faithful functor this means that

$$U_1 \simeq U_2 \,.$$

So $p^*$ is indeed injective on subobjects and so $p$ is a geometric surjection.

This establishes the existence of a surjection/embedding factorization. Next we discss that this is unique.

So consider two factorizations

$$\array{ && \mathcal{A} \\ & {}^{\mathllap{p_1}}\nearrow &\Downarrow^\simeq& \searrow^{\mathrlap{i_1}} \\ \mathcal{F} &&\stackrel{f}{\to}&& \mathcal{E} \\ & {}_{\mathllap{p_2}}\searrow &\downarrow^{\simeq}& \nearrow_{\mathrlap{i_2}} \\ && \mathcal{B} }$$

into a geometric surjection followed by a geometric embedding.

We will now argue that $i_1$ factors – essentially uniquely – through $i_2$ in a way that makes

$$\array{ && \mathcal{A} \\ & {}^{\mathllap{p_1}}\nearrow && \searrow^{\mathrlap{i_1}} \\ \mathcal{F} &&\downarrow^g&& \mathcal{E} \\ & {}_{\mathllap{p_2}}\searrow && \nearrow_{\mathrlap{i_2}} \\ && \mathcal{B} }$$

commute up to natural isomorphisms. By the same argument for the upside-down situation we find an essentially unique middle vertical morphism $h : \mathcal{B} \to \mathcal{A}$ the other way round. Then essential uniqueness of these factorizations implies that $g \circ h \simeq Id$ and $h \circ g \simeq Id$. This means that the original two factorizations are equivalent.

To find $g$ and $h$, use again that every geometric embedding (by the discussion there) is, up to equivalence, an inclusion of $j$-sheaves for some $j$. Find such a $j$ the bottom morphism and then use again lemma that $i_1$ factors through $i_2$ – essentially uniquely – precisely if $i_1^*$ sends dense monomorphisms to isomorphisms.

To see that it does, let $IU \to X$ be a dense mono and consider the naturality square

$$\array{ p_2^* i_2^* U &\stackrel{\simeq}{\to}& p_1^* i_1^* U \\ \downarrow && \downarrow \\ p_2^* i_2^* X &\stackrel{\simeq}{\to}& p_1^* i_1^* X } \,.$$

Since $i_2^*(U \to X)$ is an iso by definition, the left vertical morphism is, and thus so is the right vertical morphism. But since $p_1$ is a geometric surjection we have (by the discussion there) that $p_1^*$ is conservative, and hence also $i_1^* U \to i_1^* X$ is an isomorphism.

Hence $i_1$ factors via some $g$ through $i_2$ and the proof is completed by the above argument.

Examples

• For $f : X \to Y$ a continuous function between topological spaces and $X \to im(f) \to Y$ its ordinary image factorization through an embedding, the corresponding composite of geometric morphisms of sheaf toposes

  $$Sh(X) \to Sh(im(f)) \to Sh(Y)$$

  is a geometric surjection/geometric embedding factorization.

• For $\mathcal{E}$ any topos, $f : X \to Y$ any morphism in $\mathcal{E}$, and $X \to im(f) \to Y$ its image factorization, the corresponding composite of base change geometric morphisms

  $$\mathcal{E}/X \to \mathcal{E}/im(f) \to \mathcal{E}/Y$$

  is a geometric surjection/embedding factorization.

• For $f : C \to D$ any functor between categories, write $C \to im(f) \to D$ for its essential image factorization. Then the induced composite geometric morphism of presheaf toposes

  $$[C^{op}, Set] \stackrel{}{\to} [im(f)^{op}, Set] \to [D^{op}, Set]$$

  is a geometric surjection/embedding factorization.

See (MacLaneMoerdijk, p. 377).

Properties

As idempotent approximation

A geometric morphism $f:\mathcal{F}\to\mathcal{E}$ induces via the adjunction $f^\ast\vdash f_\ast$ a monad on $\mathcal{E}$. Due to a general result by S. Fakir this induces an associated idempotent monad on $\mathcal{E}$ and this idempotent approximation coincides with the monad induced by $i^\ast\vdash i_\ast$ given by the inclusion $i$ from the factorization $f=i\circ q$.

For references and further details on the idempotent approximation see at idempotent monad.

A logical description

Let $T$ be a geometric theory over a signature $\Sigma$ and $f:\mathcal{E}\to Set[T]$ a geometric morphism to its classifying topos. Then by the general properties of a classifying topos, $f$ corresponds to a certain $T$-model $M$ in $\mathcal{E}$.

Notice that every geometric morphism $f$ between Grothendieck toposes is of this form for some geometric theory $T$ and hence corresponds to some model $M$ ! This model permits to attach a geometric theory to $f$ as well:

The theory of M $Th(M)$ consists of all geometric sequents $\sigma$ over $\Sigma$ such that $M\models \sigma$.

Then the following holds (Caramello 2009, p.57):

Proposition

The topos occurring in the middle of the surjection-embedding factorization of $f$ is precisely the classifying topos for $Th(M)$: $\mathcal{E}\twoheadrightarrow Set[Th(M)]\hookrightarrow Set[T]$.

Related entries

• open subtopos

• (dense,closed)-factorization

References

• Peter Johnstone, Topos Theory , Academic Press 1977 (Dover reprint 2014). (section 4.1, pp.103-107)

• Peter Johnstone, Sketches of an Elephant vol. I, Oxford UP 2002. (section A4.2, pp.172ff)

• Saunders MacLane, Ieke Moerdijk, Sheaves in Geometry and Logic , Springer Heidelberg 1994. (section VII.4)

• O. Caramello, Lattices of theories , arXiv:0905.0299v1 (2009). (pdf)