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(geometric surjection, embedding) factorization system

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

f F f *f *CoAlg()E. \array{ \mathcal{E} &&\stackrel{f}{\to}&& \mathcal{E} \\ & {}_{\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 jj be a Lawvere-Tierney topology on a topos \mathcal{E} and write i:Sh j()i : Sh_j(\mathcal{E}) \to \mathcal{E} for the corresponding geometric embedding.

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

or, equivalently

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

Proof of the lemma

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

p *:=i *f *:Sh j(). p_* := i^* f_*: \mathcal{F} \to Sh_j(\mathcal{E}) \,.

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

i *p *i *i *f *f *. i_* p_* \simeq i_* i^* f_* \simeq f_* \,.

The left adjoint to p *p_* is then

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

because

(g *E,F) (f *i *E,F) (i *E,f *F) (i *E,i *i *f *F) Sh j(E,i *f *F) Sh j(E,p *F), \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 *Ff_* F is a jj-sheaf, by assumption, and that i *i_* is full and faithful.

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

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

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

(X,f *A) (f *X,A) (u,f *A) (f *u,A) (U,f *A) (f *U,A) \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 *Af_* A is a jj-sheaf and uu 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 AA is equivalent to f *uf^* u being an iso. And conversely.

Proof of the proposition

Let f: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 ff through the geometric embedding of the corresponding sheaf topos.

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

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

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

Notice that this implies that for two subobjects U 1,U 2XU_1, U_2 \hookrightarrow X we have

(1)(U 1U 2¯)(f *U 1f *U 2) (U_1 \subset \overline{U_2}) \;\;\; \Leftrightarrow \;\;\; (f^* U_1 \subset f^* U_2)

Write jj for the corresponding Lawvere-Tierney topology and

i:Sh j() i : Sh_j(\mathcal{E}) \to \mathcal{E}

for the corresponding geometric embedding.

By lemma 1 we get a factorization through II if f *f^* sends jj-dense monomorphisms to isomorphisms. But if UXU \hookrightarrow X is dense so that XU¯X \subset \overline{U} then, by (1), f *Xf *Uf^* X \subset f^* U and hence f *X=f *Uf^* X = f^* U.

Write

f:pSh j()i 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 pp here is a geometric surjection. By one of the equivalent characterizations discussed there, this is the case if p *p^* induces an injective homomorphism of subobject lattices.

So suppose that for subobjects U 1,U 2XU_1, U_2 \subset X we have p *U 1p *U 2p^* U_1 \simeq p^* U_2. Observe that then also f *i *U 1f *i *U 2f^* i_* U_1 \simeq f^* i_* U_2, because

f *i *U 1 p *i *i *U 1 p *U 1 p *U 2 p *i *i *U 2 f *i:*U 2 \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 *i_* is full and faithful. With (1) it follows that also

i *U 1i *U 2¯ i_* U_1 \simeq \overline{i_* U_2}

and hence

i *U 2 \cdots \simeq i_* U_2

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

U 1U 2. U_1 \simeq U_2 \,.

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

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

So consider two factorizations

𝒜 p 1 i 1 f p 2 i 2 \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 1i_1 factors – essentially uniquely – through i 2i_2 in a way that makes

𝒜 p 1 i 1 g p 2 i 2 \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:𝒜h : \mathcal{B} \to \mathcal{A} the other way round. Then essential uniqueness of these factorizations implies that ghIdg \circ h \simeq Id and hgIdh \circ g \simeq Id. This means that the original two factorizations are equivalent.

To find gg and hh, use again that every geometric embedding (by the discussion there) is, up to equivalence, an inclusion of jj-sheaves for some jj. Find such a jj the bottom morphism and then use again lemma 1 that i 1i_1 factors through i 2i_2 – essentially uniquely – precisely if i 1 *i_1^* sends dense monomorphisms to isomorphisms.

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

p 2 *i 2 *U p 1 *i 1 *U p 2 *i 2 *X p 1 *i 1 *X. \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 *(UX)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 1p_1 is a geometric surjection we have (by the discussion there) that p 1 *p_1^* is conservative, and hence also i 1 *Ui 1 *Xi_1^* U \to i_1^* X is an isomorphism.

Hence i 1i_1 factors via some gg through i 2i_2 and the proof is completed by the above argument.

Examples

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

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

    is a geometric surjection/geometric embedding factorization.

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

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

    is a geometric surjection/embedding factorization.

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

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

    is a geometric surjection/embedding factorization.

See (MacLaneMoerdijk, p. 377).

References

Section VII. 4. of

Revised on October 11, 2011 22:36:31 by Mike Shulman (71.136.248.27)