The concept of weakly open subtopos is weakening of the concept of a dense subtopos in topos theory.
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 $\mathcal{E}$, $a_j(\neg A)\cong \neg^j a_j(A)$.
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.
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
An open subtopos $i:Sh_j(\mathcal{E})\hookrightarrow\mathcal{E}$ is weakly open. $\qed$
Less straightforward is the following
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).
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)).
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)
Last revised on October 12, 2022 at 10:03:48. See the history of this page for a list of all contributions to it.