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Let $X,Y$ be topological spaces. Define a partial order on $\hom(Y,X)$ as follows: $f \leq g$ if $f^{-1}(U) \subseteq g^{-1}(U)$ for all open subsets $U \subseteq X$. Equivalently, $f(y)$ is a specialization of $g(y)$ for all $y \in Y$. In particular this endows $\mathsf{Top}$ with the structure of a $2$-category. I have several questions:

1) Is this partial order well-known and studied somewhere in the literature?

2) What are some properties of this partial order? What about the case that $X,Y$ are sober?

3) What are interesting and nontrivial examples of maps $f,g$ with $f \leq g$, but $f \neq g$? I am mostly interested in scheme morphisms.

EDIT: I would like to add:

4) If $X,Y$ are actually ringed spaces, then we can define $f \leq g$ if $f^{-1}(U) \subseteq g^{-1}(U)$ for all $U \subseteq X$ and $f^\# = \mathrm{res} \circ g^\#$. What about this $2$-categorical structure on the category of ringed spaces, is this already known?

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1) and 2) have been answered by Zhen. I appreciate any examples for 3). – Martin Brandenburg Mar 7 '13 at 9:47

The short answer is yes: for locales (hence sober spaces), anyway. The 2-category (or rather, $\textbf{Poset}$-enriched category) $\mathfrak{Loc}$ has hom-sets with precisely this ordering. Some of its properties are mentioned in passing in [Johnstone, Sketches of an elephant, Chapter C1], but to some extent the 2-dimensional properties of $\mathfrak{Loc}$ are subsumed by the 2-dimensional properties of the 2-category $\mathfrak{BTop}_{/ \textbf{Set}}$ of Grothendieck toposes and geometric morphisms. Indeed:

Theorem. The pseudofunctor $\textbf{Sh}(-) : \mathfrak{Loc} \to \mathfrak{BTop}_{/ \textbf{Set}}$ is 2-full embedding, in the sense that the induced functors on hom-categories $\mathfrak{Loc}(X, Y) \to \mathfrak{BTop}_{/ \textbf{Set}}(\textbf{Sh}(X), \textbf{Sh}(Y))$ is (half of) an equivalence of categories.

Here is a non-trivial property of the hom-poset $\mathfrak{Loc}(X, Y)$:

Proposition. $\mathfrak{Loc}(X, Y)$ has joins for directed subsets, i.e. $\mathfrak{Loc}(X, Y)$ is dcpo and moreover $\mathfrak{Loc}(X, Y)$ is an accessible category.

The last one generalises to $\mathfrak{BTop}_{/ \textbf{Set}}$: for any two Grothendieck toposes $\mathcal{E}$ and $\mathcal{F}$, the category of geometric morphisms $\mathcal{E} \to \mathcal{F}$ forms an accessible category.

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Thanks. Actually I have proven a generalization of the Theorem, and that's why I was asking. It is Proposition C.1.4.5 in Johnstone's book. – Martin Brandenburg Mar 7 '13 at 7:55

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