nLab weakly open subtopos

Idea

The concept of weakly open subtopos is weakening of the concept of a dense subtopos in topos theory.

Definition

Let $\mathcal{E}$ be a topos and $Sh_j(\mathcal{E})$ a subtopos of $\mathcal{E}$ with $j$ the corresponding Lawvere-Tierney topology, $a_j$ the corresponding associated sheaf functor and $\neg$ and $\neg^j$ the pseudo-complementation operators on the subobject lattices in $\mathcal{E}$ and $Sh_j(\mathcal{E})$, respectively. The subtopos $Sh_j(\mathcal{E})$ is called weakly open if for all subobjects $A\rightarrowtail E$ in $\mathcal{E}$, $a_j(\neg A)\cong \neg^j a_j(A)$.

Remark

In other words, the sheafification $a_j$ commutes with pseudo-complementation.

If $i:Sh_j(\mathcal{E})\hookrightarrow\mathcal{E}$ is weakly open then the topology $j$ and the inclusion $i$ are called weakly open as well.

Since the associated sheaf functor $a_j$ is just the inverse image of the inclusion $i: Sh_j(\mathcal{E})\hookrightarrow\mathcal{E}$ the definition can be generalized to general geometric morphisms.

Properties

As already the name suggests, the concept of a weakly open subtopos can equally be viewed as a weakening of the concept of an open subtopos. Indeed, since the associated sheaf functor of an open inclusion is logical we have

Proposition

An open subtopos $i:Sh_j(\mathcal{E})\hookrightarrow\mathcal{E}$ is weakly open. $\qed$

Less straightforward is the following

Proposition

A dense subtopos $i:Sh_j(\mathcal{E})\hookrightarrow\mathcal{E}$ is weakly open.

This occurs as prop.6.3 in Caramello (2012a). See also Johnstone (1982).

Remark

The concept and terminology goes back to Johnstone (1982).

The equivalent concept for frames is called ‘nearly open’ in Banaschewski-Pultr (1996). The maps they call ‘weakly open’ are called ‘skeletal’ by Johnstone.

The concept of map in point-set topology that corresponds to weakly open is that of a continuous map $f$ such that $f(U)$ is dense in some open set, for $U$ open (cf. Pták (1958)).

Related entries

• dense subtopos

• open subtopos

• closed subtopos

• locally closed subtopos

• skeletal geometric morphism

• double negation

• De Morganization

References

• B. Banaschewski, A. Pultr, Variants of openness, Appl. Cat. Struc. 2 (1994) 1-21 [doi:10.1007/BF00873038]

• B. Banaschewski, A. Pultr, Booleanization, Cah. Top. Géom. Diff. Cat. XXXVII 1 (1996) 41-60 [numdam:CTGDC_1996__37_1_41_0]

• Olivia Caramello, Universal models and definability , Math. Proc. Cam. Phil. Soc. (2012) pp.279-302. (arXiv:0906.3061)

• Olivia Caramello, Topologies for intermediate logics , arXiv:1205.2547 (2012). (abstract)

• Peter Johnstone, Factorization theorems for geometric morphisms II , pp.216-233 in LNM 915 Springer Heidelberg 1982.

• Peter Johnstone, Sketches of an Elephant I, Oxford UP 2002. (pp.209f)

• V. Pták, Completeness and the open mapping theorem , Bull. Soc. Math. France 86 (1958) pp.41-74. (pdf)