Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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?

share|improve this question
    
1) and 2) have been answered by Zhen. I appreciate any examples for 3). –  Martin Brandenburg Mar 7 '13 at 9:47
add comment

1 Answer 1

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.

share|improve this answer
    
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
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.