Topos notions coming from topology and uniqueness of generalizations

Question

Let's note that the bifunctor $Sh \circ Op : Top \rightarrow Topos$, taking Topos as Grothendieck topoi + geometric morphisms, does not have any adjoints as discussed here on MO before. Yet we call geometric morphisms open by analogy to an open map. Similar are the cases for closed geometric morphisms, $T_0$ geometric embeddings etc. Considering the lack of adjoints, is there a way to make this process of generalization precise? Maybe a relative adjoint to $Sh \circ Op$, that would cover this process for large class of geometric morphism properties?

Answer 1

If the absence of adjoints is what worries you, you can consider this to be a two-step process - and I would argue that in practice this is the case in the vast majority of cases:

One first generalize from topological space to locales, and then from local to toposes.

While it is indeed true that the sheaf functor from topological spaces to toposes has no adjoints, it factors into two functor

$$ \text{Top} \overset{\mathcal{O}}\to \text{Loc} \overset{Sh}\to \text{Topos} $$

that both have adjoints: The functor $\mathcal{O}$ has a left adjoint $Pt$ taking a locale to the topological space of its points, the functor $Sh$ has a right adjoint taking taking a topos to (the locale corresponding to) its frame of subterminal objects.

And as I say most notion from topology that have been generalize to topos have generally first been generalized to locale - or at least the people doing the generalization had some thought about the cases of locales.

Now I would also mention that these adjoints alone are not the explanation of why these extention make sense. Sometimes yes, but generalizing is an art and some other time the most correct generalization is not even a generalization in any practical sense.

To give some examples:

• the Hausdorff property for locales (stated as $X \to X \times X$ is closed) and for topological space are not equivalent (that is for a topological space X, X being Hausdorf as a space and as a locale is not the same thing).

• General Localic groups behave quite differently from topological groups (they are more like locally compact or polish group) and not every topological group give rise to a localic group through these adjunctions.

• The notion of properness for locales has different generalization for toposes ("proper" and "Tidy") that both deserve to be called good generalization of the notion of properness to toposes depending on your point of view. (I would say Proper is a generalization and Tidy is a categorification).