nLab closed subtopos

Idea

The concept of a closed subtopos generalizes the concept of a closed subspace from topology to toposes.

Definition

Let $U$ be a subterminal object of a topos $\mathcal{E}$. Then $c_U(V) \coloneqq U\cup V$ defines a Lawvere-Tierney topology on $\mathcal{E}$, whose corresponding subtopos is called the closed subtopos associated to $U$.

The reflector into the sheaves can be constructed explicitly as the join with $U$, i.e. $C_U(X)$ is the pushout of $X$ and $U$ under $X\times U$.

A topology $j$ that is of this form for some subterminal object $U$ is called closed.

Example

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

Remark

The subterminal object $U$ in $\mathcal{E}$ is associated with an open subtopos $Sh_{o(U)}(\mathcal{E})$ as well e.g. in the case of $\mathcal{E}=Sh(X)$ on a space $X$ this yields $Sh(U)$.

Moreover, given a Lawvere-Tierney topology $j$ on a topos $\mathcal{E}$ with corresponding subtopos $Sh_j(\mathcal{E})\hookrightarrow\mathcal{E}$, we a get a canonical subterminal object $ext(j)$ associated to $j$ by taking the $j$-closure of $O\rightarrowtail 1$. The corresponding closed and open subtoposes associated to $ext(j)$ provide a ‘closure’ $Sh_j(\mathcal{E})\hookrightarrow Sh_{c(ext(j))}(\mathcal{E})$, called l’adhérence in (SGA4, p.461), respectively, an ‘exterior’ $Sh_{o(ext(j))}(\mathcal{E})$ for $Sh_j(\mathcal{E})$.

Properties

Proposition

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

For the proof see (Johnstone (2002), pp.212,215).

Closed subtoposes that are Boolean are parametrized by automorphisms of the subobject classifier as the next proposition documents:

Proposition

Let $\mathcal{E}$ be a topos. Then automorphisms of $\Omega$ correspond bijectively to closed Boolean subtoposes. The group operation on $Aut(\Omega)$ corresponds to symmetric difference of subtoposes.

This result appears in Johnstone (1979).

The following result is a part of the so called (dense,closed)-factorization.

Proposition

Let $i:Sh_{c(U)}(\mathcal{E})\hookrightarrow\mathcal{E}$ be dominant and a closed inclusion at the same time. Then $i$ is an isomorphism.

Proof: Recall that $X\in\mathcal{E}$ are in the closed subtopos precisely when they satisfy $X\times U\cong U$ with $U$ the subterminal object associated to $i$. But $i$ is dominant, or what comes to the same for inclusions: dense, hence $\emptyset$ is in $Sh_{c(U)}(\mathcal{E})$ and therefore $\emptyset\times U\cong U$ . From this follows $U\cong\emptyset$, which in turn implies that all $X\in\mathcal{E}$ are in $Sh_{c(U)}(\mathcal{E})$ . $\qed$

This says that the only closed subtopos of a topos $\mathcal{E}$ that is dense, is $\mathcal{E}$ itself.

It follows e.g. that Boolean toposes have no non-trivial dense subtoposes since all their subtoposes are closed (and open, as well). Of course, this follows also from the fact that a Boolean topos coincides with its double negation subtopos and the latter is the smallest dense subtopos!

Related pages

• (dense,closed)-factorization
• open subtopos
• locally closed subtopos
• dense subtopos
• Artin gluing
• closed subspace

References

• M. Artin, A. Grothendieck, J. L. Verdier, Théorie des Topos et Cohomologie Etale des Schémas (SGA4), LNM 269 Springer Heidelberg 1972. (Exposé IV 9.3.4-9.4., pp.456ff)

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

• Peter Johnstone, Automorphisms of $\Omega$ , Algebra Universalis 9 (1979) pp.1-7.

• Peter Johnstone, Sketches of an Elephant vol. I, Oxford UP 2002. (A4.5., pp.204-220)