nLab weakly open subtopos



Topos Theory

topos theory



Internal Logic

Topos morphisms

Extra stuff, structure, properties

Cohomology and homotopy

In higher category theory




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


In other words, the sheafification a ja_j commutes with pseudo-complementation.

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

Since the associated sheaf functor a ja_j is just the inverse image of the inclusion i:Sh j()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()i:Sh_j(\mathcal{E})\hookrightarrow\mathcal{E} is weakly open. \qed

Less straightforward is the following


A dense subtopos i:Sh j()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 ff such that f(U)f(U) is dense in some open set, for UU open (cf. Pták (1958)).


Last revised on October 12, 2022 at 10:03:48. See the history of this page for a list of all contributions to it.