4
$\begingroup$

In Top, the monos are the injective maps and the regular monos are the subspace inclusions. Is there a (similarly pithy) categorical description for the open subspace inclusions?

$\endgroup$
8
  • 4
    $\begingroup$ An open inclusion $U \hookrightarrow X$ is the same as a map $X \to S$, where $S$ is the Sierpinski space (the two-point space which is neither discrete nor codiscrete). The equivalence, analogous to a subobject classifier, comes by pulling back along the inclusion of the open point into $S$. Maybe not quite as pithy as one might like... One might be able to describe open maps via lifting properties though. $\endgroup$
    – Tim Campion
    Commented Apr 27, 2017 at 13:44
  • $\begingroup$ @TimCampion fair enough, though I was hoping for a characterization separate from S, so that we could say that S classifies such arrows. $\endgroup$
    – Colin
    Commented Apr 27, 2017 at 14:19
  • $\begingroup$ A necessary condition in terms of mapping properties is this. There is a well-known (contravariant) functor $F$ from Top to the category of complete Heyting algebras. Now if $j\colon U\to X$ is an open inclusion in Top, then $j$ is mapped by $F$ to the epi $F(j)$ $=$ $\mathrm{Ouv}(X)\twoheadrightarrow\{ O\in\mathrm{Ouv}(X)\colon O\leq U\}$. It might be fruitful to analyse whether this can be made into a characterization of open inclusions. $\endgroup$ Commented Apr 27, 2017 at 17:55
  • $\begingroup$ Roughly speaking, the suggestion was to think about dualities between monos and epis w.r.t. some known and natural functor. $\endgroup$ Commented Apr 27, 2017 at 18:13
  • 1
    $\begingroup$ Regarding my vague suggestion about lifting properties: note that open inclusions are not closed under products, so they do not form a right orthogonality class. But I think they are closed under colimits and cobase change, so they ought to form a left orthogonality class. However, maps $X \to Y$ which have the right lifting property with respect to every open inclusion seem to be very weird... $\endgroup$
    – Tim Campion
    Commented Apr 28, 2017 at 19:10

1 Answer 1

2
$\begingroup$

Yes, see my answer here. The main idea is to define Sierpinski objects as small spaces.

$\endgroup$
3
  • 1
    $\begingroup$ I'm confused by a few things in the answer you link to. 1) The codiscrete category with two elements meets your definition of a Sierpinski object (a connected object with two global points), but I don't see where you consider this. 2) I don't understand what you mean by "distinguishing two isomorphic Sierpinski spaces from each other". Do you mean you want to distinguish the two points $x,y: 1 \to S$ of the Sierpinski space $S$ from each other? 3.) You mention certain maps that do certain things to $y$; are you claiming there are no maps that do similar things to $x$? $\endgroup$
    – Tim Campion
    Commented Apr 28, 2017 at 2:37
  • 1
    $\begingroup$ 1) Corrected. 2) Well we have to say which point is closed and which point is open. Notice that my first definition of a Sierpinski is completely symmetric. So it cannot be complete. 3) For $Top$ this is the case. $\endgroup$ Commented Apr 28, 2017 at 21:34
  • $\begingroup$ I think the Sierpinski space can also be defined as the lax pushout of $Id_1$ along itself. (Considering Top as a poset-enriched category where $Hom(X,Y)$ is ordered pointwise using the specialization order on $Y$). $\endgroup$
    – Colin
    Commented May 2, 2017 at 0:59

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

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