nLab
open subtopos

Contents

Context

Topos Theory

topos theory

Background

Toposes

Internal Logic

Topos morphisms

Extra stuff, structure, properties

Cohomology and homotopy

In higher category theory

Theorems

Contents

Idea

The concept of an open subtopos generalizes the concept of an open subspace from topology to toposes.

Definition

Let UU be a subterminal object of a topos. Then o U(V)(UV)o_U(V)\coloneqq (U\Rightarrow V) defines a Lawvere-Tierney topology on \mathcal{E}, whose corresponding subtopos is called the open subtopos associated to UU.

The reflector into the topos of sheaves can be constructed explicitly as O U(X)=X UO_U(X) = X^U.

A topology that is of this form for some subterminal object UU is called open.

Example

In case =Sh(X)\mathcal{E}=Sh(X) is the topos of sheaves on a topological space XX, a subterminal object is just an open subset UU of XX and the open subtopos corresponding to it is equivalent to Sh(U)Sh(U).

As one would expect from the topological situation, for any topos \mathcal{E}, the empty subtopos (given by o(V)(V)=o(V) \coloneqq (\bot \Rightarrow V) = \top) and \mathcal{E} itself (given by o(V)(V)=Vo(V) \coloneqq (\top \Rightarrow V) = V) are open subtoposes of \mathcal{E}.

Remark

The subterminal object UU in \mathcal{E} is associated with a closed subtopos Sh c(U)()Sh_{c(U)}(\mathcal{E}) as well e.g. in the case of =Sh(X)\mathcal{E}=Sh(X) on a space XX this yields Sh(XU)Sh(X\setminus U).

Moreover, given a Lawvere-Tierney topology jj on a topos \mathcal{E} with corresponding subtopos Sh j()Sh_j(\mathcal{E})\hookrightarrow\mathcal{E}, we a get a canonical subterminal object ext(j)ext(j) associated to jj by taking the jj-closure of O1O\rightarrowtail 1. The corresponding closed and open subtoposes associated to ext(j)ext(j) provide a ‘closureSh c(ext(j))()Sh_{c(ext(j))}(\mathcal{E}), respectively, an ‘exteriorSh o(ext(j))()Sh_{o(ext(j))}(\mathcal{E}) for Sh j()Sh_j(\mathcal{E}) (cf. SGA4, p.461). More on this below.

Properties

Proposition

Let Sh j()Sh_j(\mathcal{E})\hookrightarrow\mathcal{E} be a subtopos with corresponding topology jj. The following are equivalent:

See Johnstone (1980, pp.219-220; 2002, pp.609-610).

Proposition

Let UU a subterminal object and Sh c(U)()Sh_{c(U)}(\mathcal{E}) and Sh o(U)()Sh_{o(U)}(\mathcal{E}) the corresponding closed, resp. open subtoposes. Then Sh c(U)()Sh_{c(U)}(\mathcal{E}) and Sh o(U)()Sh_{o(U)}(\mathcal{E}) are complements for each other in the lattice of subtoposes.

See Johnstone (2002, pp.212,215).

Remark

Whereas, general open morphisms are only bound to preserve first order logic, open inclusions preserve also higher order logic since their inverse images are logical.

That the inverse image is logical is a special case of the general fact that the pullback functor /X\mathcal{E}\to\mathcal{E}/X along X1X\to 1 is logical for arbitrary objects XX. In particular, Sh o(U)()/USh_{o(U)}(\mathcal{E})\cong\mathcal{E}/U.

Open localizations

Subtoposes of a topos \mathcal{E} correspond to localizations of \mathcal{E} i.e. replete, reflective subcategories whose reflector preserves finite limits. Just as this notion makes sense more generally for categories with finite limits, the notion of open localization makes sense more generally for locally presentable categories:

Given a locally α\alpha-presentable category 𝒞\mathcal{C} with subcategory of α\alpha-presentable objects 𝒫\mathcal{P}, the subobject Ω 𝒞\Omega_\mathcal{C} of the subobject classifier of Set 𝒫 opSet^{\mathcal{P}^{op}} given by Ω 𝒞(P):=\Omega_\mathcal{C}(P):= set of α\alpha-exact subpresheaves of 𝒫(_,P)\mathcal{P}(\text{_},P), classifies subobjects of 𝒞\mathcal{C}. Furthermore, localizations of 𝒞\mathcal{C} correspond to topologies j:Ω 𝒞Ω 𝒞j:\Omega_\mathcal{C}\to\Omega_\mathcal{C}.

At this level of generality, an open subtopos of a Grothendieck topos corresponds to the notion of an open localization of a locally presentable category that is studied in Borceux-Korotenski (1991). The main result in their paper is the following

Proposition

Let li:𝒟𝒞l\dashv i: \mathcal{D}\hookrightarrow\mathcal{C} be a localization of a locally presentable category 𝒞\mathcal{C}. The localization is called open if the following equivalent conditions are satisfied:

  • The corresponding closure operator admits a universal dense interior operator.

  • The associated topology j 𝒟:Ω 𝒞Ω 𝒞j_\mathcal{D}:\Omega_\mathcal{C}\to\Omega_\mathcal{C} has a left adjoint.

  • The localization is essential with klk\dashv l and, additionally, the first of the following diagrams1 being a pullback implies the second being a pullback, too:

D d D f p.b. g lC lc lCkD kd kD f¯ p.b. g¯ C c C.\array{ D&\overset{d}{\to}&D'\\ f\downarrow&p.b.&\downarrow g\\ lC&\underset{lc}{\to}&lC' } \qquad\Rightarrow \qquad \array{ kD&\overset{kd}{\to}&kD'\\ \bar{f}\downarrow&p.b.&\downarrow \bar{g}\\ C&\underset{c}{\to}&C' }\quad .

In case 𝒞\mathcal{C} is a topos, i:𝒟𝒞i:\mathcal{D}\hookrightarrow\mathcal{C} is an open subtopos.

Open localizations are special cases of essential localizations, and are in general better behaved than the latter. For example, the meet of two essential localizations in the lattice of essential localizations does not coincide with their meet in the lattice of localizations. Compare this with the following

Proposition

Let 𝒞\mathcal{C} a locally presentable category. The meet of two open localizations in the lattice of localizations is an open localization.

cf. Borceux-Korotenski (1991, p.235).

Proposition

Let 𝒞\mathcal{C} a locally presentable category where unions are universal. The supremum of a family of open localizations in the lattice of all localizations is again an open localization.

cf. Borceux-Korotenski (1991, p.236).

This applies e.g. to Grothendieck toposes since they are locally presentable and colimits are universal.

Proposition

Let 𝒞\mathcal{C} a locally presentable category where unions are universal. The open localizations in 𝒞\mathcal{C} constitute a locale.

cf. Borceux-Korotenski (1991, p.237).

The following proposition closes the circle and recovers the primordial example of sheaf subtoposes Sh(U)Sh(U) on open subsets UU as a special case:

Proposition

Let 𝒞\mathcal{C} a locally presentable category in which colimits are universal. Then the locale of open localizations of 𝒞\mathcal{C} is isomorphic to the locale of subterminal objects of 𝒞\mathcal{C}.

cf. Borceux-Korotenski (1991, p.238).

Open subtoposes associated to a subtopos

Let Sh j()Sh_j(\mathcal{E})\hookrightarrow\mathcal{E} be a subtopos of a (Grothendieck) topos with corresponding topology jj. From prop. it follows that the supremum of the family of all open subtoposes contained in Sh j()Sh_j(\mathcal{E}) is open again and, since it coincides with the supremum in the lattice of all localizations, is contained in Sh j()Sh_j(\mathcal{E}). Clearly, it is the biggest open subtopos contained in Sh j()Sh_j(\mathcal{E}) and therefore called the interior of Sh j()Sh_j(\mathcal{E}), denoted by Sh o(int(j))()Sh_{o(int(j))}(\mathcal{E}) and the corresponding subterminal object by int(j)int(j).

Whereas the other open subtopos Sh o(ext(j))()Sh_{o(ext(j))}(\mathcal{E}) connected with Sh j()Sh_j(\mathcal{E}) corresponding to ext(j)ext(j) is the biggest open subtopos disjoint from Sh j()Sh_j(\mathcal{E}) i.e. its exterior. Then the sum Sh o(ext(j))()Sh o(int(j))()Sh_{o(ext(j))}(\mathcal{E}) \vee Sh_{o(int(j))}(\mathcal{E}) is open again and corresponds to the subterminal object ext(j)int(j)ext(j)\vee int(j). Its closed complement Sh c(ext(j)int(j))()Sh_{c(ext(j)\vee int(j))}(\mathcal{E}) is called the boundary of Sh j()Sh_j(\mathcal{E}) in (SGA 4, p. 461).

For some further details see at dense subtopos.

References

  • M. Artin, A. Grothendieck, J. L. Verdier, Théorie des Topos et Cohomologie Etale des Schémas (SGA4), LNM 269 Springer Heidelberg 1972. (Exposé IV 9.2, 9.3.4-9.4., pp.451ff)

  • F. Borceux, M. Korostenski, Open Localizations , JPAA 74 (1991) pp.229-238.

  • C. Getz, M. Korostenski, Open Localizations and Factorization Systems , Quest. Math. 17 no.2 (1994) pp.225-230.

  • Peter Johnstone, Topos Theory , Academic Press New York 1977. (Dover reprint 2014, pp.93-95)

  • Peter Johnstone, Open maps of toposes , Manuscripta Math. 31 (1980) pp.217-247. (gdz)

  • Peter Johnstone, Sketches of an Elephant vols. I,II, Oxford UP 2002. (A4.5., pp.204-220; C3.1.5-7, pp.609f)


  1. Here ff corresponds to f¯\bar{f}, resp. gg to g¯\bar{g}, under the adjunction klk\dashv l.

Last revised on November 10, 2017 at 04:29:41. See the history of this page for a list of all contributions to it.