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?
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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 CampionCommented Apr 27, 2017 at 13:44
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$\begingroup$ @TimCampion fair enough, though I was hoping for a characterization separate from S, so that we could say that S classifies such arrows. $\endgroup$– ColinCommented Apr 27, 2017 at 14:19
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$\begingroup$ 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. $\endgroup$– Peter HeinigCommented Apr 27, 2017 at 17:55
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$\begingroup$ Roughly speaking, the suggestion was to think about dualities between monos and epis w.r.t. some known and natural functor. $\endgroup$– Peter HeinigCommented Apr 27, 2017 at 18:13
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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 CampionCommented Apr 28, 2017 at 19:10
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Yes, see my answer here. The main idea is to define Sierpinski objects as small spaces.
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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 $x$? $\endgroup$ Commented Apr 28, 2017 at 2:37
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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
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$\begingroup$ 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 $Y$). $\endgroup$– ColinCommented May 2, 2017 at 0:59