Factorization systems in the category of toposes with geometric morphisms

Question

I have been thinking about geometric morphisms of topoi recently, and I am trying to organize some of what I know so far.

Right now my question is, what are some (weak, strong, etc.) factorization systems in the category of toposes with geometric morphisms?

I suspect that surjective maps and embeddings form such a system, and that monomorphisms and regular epimorphisms do as well.

What about connected morphisms and injective morphisms (where injective means that $f_*$ is faithful)?

Thanks!

Answer 1

This is a broad question, but I think much of what you are looking for is contained in section VII.4 of Sheaves in Geometry and Logic by Mac Lane and Moerdijk. That is the section Embeddings and Surjections in the chapter Geometric Morphisms, so you can guess what it is about ;)

Briefly, we have the following results (numbering refers to the aforementioned book).

Definition. A geometric morphism $f: \mathcal{F} \to \mathcal{E}$ is a(n)

• surjection if the inverse image functor $f^*$ is faithful;
• embedding if the direct image functor $f_*$ is full and faithfull.

The condition for the embedding is equivalent to saying that the counit of the adjunction is an isomorphism. An example of an embedding is the sheafification functor $a: \mathbf{Set}^{\mathcal{C}^\text{op}} \to \operatorname{Sh}(\mathcal{C}, J)$. Together with the inclusion functor $i: \operatorname{Sh}(\mathcal{C}, J) \to \mathbf{Set}^{\mathcal{C}^\text{op}}$, they form a geometric morphism $a \dashv i$ and the counit is an isomorphism (which is saying that taking the sheafification of a sheaf does nothing). In fact, geometric embeddings correspond to topologies on the domain of the embedding (corollary VII.4.7).

In fact, that is a corollary to the following factorisation theorem.

Theorem (VII.4.6). Let $f: \mathcal{F} \to \mathcal{E}$ be a geometric morphism. Then there exists a topology $j$ on $\mathcal{E}$, such that $f$ factors through the geometric embedding $i: \operatorname{Sh}_j(\mathcal{E}) \to \mathcal{E}$ by a geometric surjection $p$.

There is also a uniqueness statement. However, the category of toposes with geometric embeddings should really be considered as a 2-category. Which loosely means that we should considering commuting diagrams up to natural isomorphism.

Theorem (VII.4.8, weakened). Suppose $\mathcal{F} \xrightarrow{p} \mathcal{A} \xrightarrow{u} \mathcal{E}$ and $\mathcal{F} \xrightarrow{q} \mathcal{B} \xrightarrow{v} \mathcal{E}$ are two factorisations of $f: \mathcal{F} \to \mathcal{E}$, in the sense that the relevant triangles commute up to isomorphism. If both are a geometric surjection follows by a geometric embedding, then there is an equivalence $g: \mathcal{A} \to \mathcal{B}$ making everything commute (up to natural isomorphism).

I wrote "weakened", because the actual theorem (in the book) is a lot stronger. It allows us to compare any two factorisations of $f$ where $p$ is a surjection and $v$ is an embedding. So this $g$ exists in those cases, but it is not always an equivalence for arbitrary factorisations.