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dense subtopos

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Idea

The concept of dense subtopos generalizes the concept of a dense subspace from topology to toposes. A subtopos is dense if it contains the initial object \emptyset of the ambient topos.

Definition

Let i: ji:\mathcal{E}_j\hookrightarrow \mathcal{E} be a subtopos with corresponding Lawvere-Tierney topology jj. j\mathcal{E}_j is called dense if the following equivalent conditions hold:

  • j=j\circ\bot=\bot (with :1Ω\bot: 1\to\Omega the classifying map of 1\emptyset\rightarrowtail 1).

  • 1\emptyset\rightarrowtail 1 is jj-closed.

  • \emptyset is a jj-sheaf.

  • the direct image satisfies i *()i_\ast (\emptyset)\simeq\emptyset.

  • the inverse image satisfies: from i *(Z)i^\ast (Z)\simeq\emptyset follows ZZ\simeq\emptyset.

A topology jj satisfying these conditions is also called dense.

Remark

jj\circ\bot classifies the jj-closure ¯\bar{\emptyset} of \emptyset whence j=j\circ\bot = \bot iff ¯=\bar{\emptyset}=\emptyset i.e. 1\emptyset\rightarrowtail 1 is jj-closed. Since 11 is a jj-sheaf for any topology jj and subobjects of jj-sheaves in general are jj-closed precisely when they are jj-sheaves, this is equivalent to \emptyset being a jj-sheaf. Another way to say this is that \emptyset is preserved by i *i_\ast.

The equivalence between the last two formulations follows from the adjunction i *i *i^\ast\dashv i_\ast and the strictness of \emptyset in a topos: id i *()id_{i_\ast(\emptyset)} corresponds under the adjunction to a map i *(i *())i^\ast( i_\ast (\emptyset))\to \emptyset showing that i *(i *())i^\ast (i_\ast(\emptyset))\simeq\emptyset in general. Conversely, id i *(Z)id_{i^\ast (Z)} corresponds to a map Zi *(i *(Z))Z\to i_\ast(i^\ast(Z)) showing that ZZ\simeq\emptyset provided i *(Z)i^\ast(Z)\simeq\emptyset and i *()i_\ast(\emptyset)\simeq\emptyset.

In SGA4 (p.430) another equivalent formulation is on offer, namely it suffices to check the last condition on subterminal objects ZZ (because i *()i_\ast(\emptyset) is a subterminal in general since i *i_\ast as a right adjoint preserves monos hence subterminals). An even more comprehensive list can be found in (Caramello 2012, p.9).

The last two conditions make sense not only for embeddings: general geometric morphisms fulfilling them are called dominant. So another way to express that i: ji:\mathcal{E}_j\hookrightarrow\mathcal{E} is a dense subtopos is to say that the inclusion ii is dominant.

Notice that there is also a certain Grothendieck topology on small categories 𝒞\mathcal{C} called the dense topology whose corresponding Lawvere-Tierney topology on Set 𝒞 opSet^{\mathcal{C}^{op}} is dense in the above sense, and coincides with the double-negation topology ¬¬\neg\neg on Set 𝒞 opSet^{\mathcal{C}^{op}}.

Properties

Some basic observations

Of course, the composition kjk\circ j of dense inclusions j,kj,k is again dense. Conversely, we have

Proposition

Given the following commutative diagram

i j i k k \array{ \mathcal{E}_i &\overset{j}{\to}& \mathcal{E} \\ &{i}{\searrow}& \uparrow k \\ & & \mathcal{E}_k }

where i,j,ki,j,k are subtopos inclusions. When jj is dense, then ii and kk are dense as well.

Proof: Suppose =i *(Z)\emptyset =i^\ast (Z). Since kk is an inclusion we have that the counit ϵ:k *k *id k\epsilon:k^\ast k_\ast\to id_{\mathcal{E}_k} is a natural isomorphism whence Z=k *k *(Z)Z= k^\ast k_\ast (Z) and therefore

=i *(Z)=i *(k *k *(Z))=i *k *(k *(Z))=j *(k *(Z)).\emptyset =i^\ast(Z)=i^\ast(k^\ast k_\ast (Z))=i^\ast k^\ast (k_\ast (Z))=j^\ast(k_\ast(Z))\quad .

From which k *(Z)=k_\ast(Z)=\emptyset since jj is dense by assumption.

Then Z=k *k *(Z)=k *()=Z=k^\ast k_\ast (Z)=k^\ast (\emptyset)=\emptyset since k *k^\ast preserves colimits, in other words, we have shown that ii is dense.

But from k *(Z)=k_\ast (Z)=\emptyset and Z=Z=\emptyset follows that kk is dense as well. \qed

Given two dense topologies j 1j_1, j 2j_2 on a topos \mathcal{E}, their join j 1j 2j_1\vee j_2 is again dense.

This follows from the general fact that Sh j 1j 2()Sh_{j_1\vee j_2}(\mathcal{E}) corresponds to the meet, i.e. the intersection of the corresponding subtoposes Sh j 1()Sh j 2()Sh_{j_1}(\mathcal{E})\cap Sh_{j_2}(\mathcal{E}) in the lattice of subtoposes, and this obviously contains \emptyset_\mathcal{E} for j 1j_1, j 2j_2 dense.

In other words, the intersection of two dense subtoposes is still dense!

Somewhat surprisingly, this still holds if one takes the intersection of all dense subtoposes, as the next section details.

Relation to double-negation topology

For any topos \mathcal{E}, its double negation topology gives the smallest dense subtopos. This agrees with the situation for locales but contrasts with the situation for topological spaces where, in general, smallest dense subspaces do not exist.

Proposition

Sh ¬¬()Sh_{\not\not}(\mathcal{E}) \hookrightarrow \mathcal{E} is the smallest dense subtopos.

(Johnstone, below Corollary 4.5.20)

In fact, dense topologies are characterized by their relation to ¬¬\neg\neg:

Proposition

Let \mathcal{E} be a topos. A topology jj satisfies j¬¬j\le\neg\neg , i.e. jj is dense, iff ( j) ¬¬= ¬¬(\mathcal{E}_j)_{\not\not}=\mathcal{E}_{\not\not}.

(Blass-Scedrov 1983, p.19, Caramello 2012, p.9, see also at double negation).

From this and the fact that \mathcal{E} is trivially dense, follows:

Proposition

A topos \mathcal{E} is Boolean iff \mathcal{E} has exactly one dense subtopos, namely ¬¬=\mathcal{E}_{\neg\neg}=\mathcal{E}.

Notice that, though these results prevent a topos from having more than one dense Boolean subtopos, nothing prevents a topos from having more than one Boolean subtopos e.g. the Sierpinski topos Set Set^{\to} has two non trivial ones that complement each other in the lattice of subtoposes. This example, incidentally, also shows that in the above proposition just ( j) ¬¬ ¬¬(\mathcal{E}_j)_{\neg\neg}\cong\mathcal{E}_{\neg\neg} wouldn’t do.

The (dense,closed)-factorization

A geometric embedding of elementary toposes

Sh j() Sh_j(\mathcal{E}) \hookrightarrow \mathcal{E}

factors as

Sh j()Sh c(ext(j))() Sh_j(\mathcal{E}) \hookrightarrow Sh_{c(ext(j))}(\mathcal{E}) \hookrightarrow \mathcal{E}

where ext(j)ext(j) (the “exterior” of jj) denotes the jj-closure of 1\emptyset \rightarrowtail 1 and

j¯c(ext(j)) \bar j \coloneqq c(ext(j))

the closed topology corresponding to the subterminal object ext(j)ext(j).

Here the first inclusion exhibits a dense subtopos and the second a closed subtopos.

This is the so called (dense,closed)-factorization and implies e.g. that proper dense subtoposes aren’t closed.

Dense inclusions participate also in the description of skeletal inclusions as the closure of open inclusions under composition with dense inclusions.

Some parallels to topology

The above terminology suggests to view a dense subtopos as one with an empty exterior.

This analogy to topology is pursued further in (SGA4, p.462) where a dense subtopos is characterized as a subtopos Sh j()Sh_j(\mathcal{E}) whose ‘exterior’ Ext(Sh j())Ext(Sh_j(\mathcal{E})) (i.e. the open subtopos that corresponds to the subterminal object ext(j)ext(j)) is trivial and whose ‘closure’ Cl(Sh j()):=Sh c(ext(j))()Cl(Sh_j(\mathcal{E})):=Sh_{c(ext(j))}(\mathcal{E}) (i.e. the closed subtopos corresponding to ext(j)ext(j)) coincides with the ‘whole space’ \mathcal{E}.

Let’s have a look at some of the details:

Due to the construction of open subtoposes we know that the objects of Ext(Sh j())Ext(Sh_j(\mathcal{E})) have the form X ext(j)X^{ext(j)} for some XX\in\mathcal{E}. Hence the exterior is trivial, i.e. X ext(j)=1X^{ext(j)}=1 for all XX\in\mathcal{E}, precisely when ¯=ext(j)\bar\emptyset =ext(j)\simeq \emptyset which means that Sh j()Sh_j(\mathcal{E}) is dense. By construction Cl(Sh j())Cl(Sh_j(\mathcal{E})) is the complement of Ext(Sh j())Ext(Sh_j(\mathcal{E})) in the lattice of subtoposes hence Cl(Sh j())=Cl(Sh_j(\mathcal{E}))=\mathcal{E} in case the latter is trivial. This follows also directly from the description of objects in Cl(Sh j())Cl(Sh_j(\mathcal{E})) as those objects XX\in\mathcal{E} with X×ext(j)ext(j)X\times ext(j)\cong ext(j).

E.g. let Sh k()Sh_k(\mathcal{E}) be a subtopos that has a trivial intersection with a non-trivial open subtopos Sh o()Sh_o(\mathcal{E}). Then Sh k()Sh_k(\mathcal{E}) is contained in the (closed) complement of Sh o()Sh_o(\mathcal{E}) hence Cl(Sh k())Cl(Sh_k(\mathcal{E}))\neq \mathcal{E} and we see that Sh k()Sh_k(\mathcal{E}) cannot be dense: we have recuperated the familiar fact from point-set topology that a dense subset intersects all non-trivial open sets non-trivially.

Another easy result in this vein is

Proposition

Let i:Sh j()i:Sh_j(\mathcal{E})\hookrightarrow\mathcal{E} be a dense subtopos that is connected in the sense that 11 is indecomposable: if WZ=1W\coprod Z=1 then W=W=\emptyset or Z=Z=\emptyset. Then \mathcal{E} is connected as well.

Proof: Let XY=1X\coprod Y = 1 be a decomposition of 11 in \mathcal{E}. Since i *i^\ast is a left exact left adjoint, it preserves coproducts and the terminal object and i *(X)i *(Y)i^\ast(X)\coprod i^\ast (Y) is therefore a decomposition of 11 in Sh j()Sh_j(\mathcal{E}) hence trivial by assumption. Let’s say i *(X)i^\ast(X)\simeq\emptyset but Sh j()Sh_j(\mathcal{E}) is dense and therefore we can conclude XX\simeq\emptyset hence XY=1X\coprod Y = 1 is trivial as well. \qed

Example I: two-valued toposes

Presheaf toposes Set M opSet^{M^{op}} of actions of a monoid MM are classical examples of toposes whose truth value objects Ω\Omega have exactly two global points 1Ω1\to\Omega without the toposes being necessarily Boolean. In fact they are Boolean precisely when MM is a group.

As the next proposition shows, they are also instances of toposes in which only the degenerate subtopos fails to be dense:

Proposition

The non-degenerate subtoposes Sh j()Sh_j(\mathcal{E}) of a two-valued topos \mathcal{E} are precisely the dense subtoposes of \mathcal{E}.

Proof: Truth values 1Ω1\to\Omega correspond precisely to subobjects of 11. Hence the jj-closure ext(j)ext(j) of 1\emptyset\rightarrowtail 1 is either \emptyset or 11. In the first case, Sh j()Sh_j(\mathcal{E}) is dense, in the second, from X×1=1X\times 1=1 for XCl(Sh j())X\in Cl(Sh_j(\mathcal{E})) follows triviality. \qed

Combining this with the above shows that two-valued and Boolean toposes are opposite extremes when it comes to dense subtoposes and the following observation (cf. Caramello (2009); prop. 10.1) follows immediately:

Proposition

A topos \mathcal{E} that is two-valued and Boolean has no non-trivial subtoposes. \qed

In other words, two-valued Boolean toposes are atoms in the lattice of subtoposes. Notice that this applies e.g. to well-pointed toposes.

Example II: persistent localizations

Recall that a persistent localization is given by a Lawvere-Tierney topology jj with the property that every jj-separated object is a jj-sheaf. But separated objects are closed under taking subobjects and therefore in the case of persistent jj, subobjects of jj-sheaves are themselves jj-sheaves.

In particular, this applies to 1\emptyset\rightarrowtail 1, since 11 is always a sheaf. Whence \emptyset is a jj-sheaf and we see that persistent localizations are dense. This includes e.g. ‘quintessential localizations’ aka quality types.

This observation is due to Johnstone (1996).

By the above proposition it follows immediately that every persistent localization of a Boolean topos is trivial.

Relation to Aufhebung

Notice that, since the localization LL corresponding to a subtopos is a left exact functor, all subtoposes necessarily contain the terminal object *\ast of the ambient topos. Moreover, the idempotent comonad and idempotent monad constant on the initial object and terminal object, respectively, are adjoint to each other (forming an adjoint modality). Denoting by “\vee” the inclusion of modal objects, then the general situation for any subtopos localized on by LL is depicted by

L *. \array{ && L \\ && \vee \\ \emptyset &\dashv& \ast } \,.

In view of this, the subtopos being dense says that not only *\ast, but this whole adjoint modality that it participates in sits inside the subtopos. Lawvere had proposed to call this situation resolution or (a special minimal version of it) Aufhebung of the unity of opposites expressed by *\emptyset \dashv \ast (“becoming”).

In other words, for an essential subtopos being dense is equivalent to resolve *\emptyset \dashv \ast in the Hegelian calculus of levels!

Example: essential subtoposes of the topos of globular sets

Kennett-Riehl-Roy-Zaks (2011) show that in the gros topos Set 𝒢 opSet^{\mathcal{G}^{op}} of reflexive globular sets essential subtoposes correspond to dimensional truncations (plus the level Set 𝒢 op{Set^{\mathcal{G}^{op}}} ‘at infinity’). Then level n+1n+1 is the Aufhebung of nn starting from *\emptyset\dashv\ast at level 00. In general, the Aufhebung l¯\bar{l} of a level ll resolves all the levels that ll resolves. Therefore in Set 𝒢 opSet^{\mathcal{G}^{op}} all essential subtoposes (above 0) resolve *\emptyset\dashv\ast and hence are dense!

References

Last revised on May 20, 2018 at 08:40:47. See the history of this page for a list of all contributions to it.