Timeline for Categorical Description of Open Subspaces
Current License: CC BY-SA 3.0
13 events
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| May 23, 2017 at 14:34 | comment | added | Peter Heinig | @ColinZwanziger: good to hear. While this is not what you are asking for (dictionary between attributes of maps and attributes of morphisms), let me mention a known categorical description of open subsets: open subsets are precisely the coalgebras of the interior-endofunctor. Let $\mathsf{C}$ denote the small posetal category with objects precisely the subsets of $X$ and let $\mathrm{int}$ denote the endofunctor sending any subset to its interior w.r.t. the topology of $X$. Then the open subsets bijectively correspond to the coalgebras of $\mathrm{int}$. This is almost tautological, though. | |
| May 20, 2017 at 20:12 | comment | added | Colin | Actually, @PeterHeinig, can use your suggestion, since there is a notion of "local homeomorphism" of complete heyting algebras (locales) which doesn't reference open sets. My one concern is that the functor from $Top$ to complete heyting algebras is defined with reference to open sets. | |
| May 20, 2017 at 19:51 | comment | added | Colin | The subspace inclusion $A \subset X$ is open iff it is a local homeomorphism. But unclear how to purge the notion of "open set" from the definition of "local homeomorphism". | |
| Apr 28, 2017 at 19:10 | comment | added | Tim Campion | 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... | |
| Apr 27, 2017 at 22:40 | answer | added | Martin Brandenburg | timeline score: 2 | |
| Apr 27, 2017 at 18:13 | comment | added | Peter Heinig | Roughly speaking, the suggestion was to think about dualities between monos and epis w.r.t. some known and natural functor. | |
| Apr 27, 2017 at 17:55 | comment | added | Peter Heinig | 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. | |
| Apr 27, 2017 at 16:37 | history | edited | Colin | CC BY-SA 3.0 |
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| Apr 27, 2017 at 15:51 | history | edited | Colin | CC BY-SA 3.0 |
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| Apr 27, 2017 at 14:19 | comment | added | Colin | @TimCampion fair enough, though I was hoping for a characterization separate from S, so that we could say that S classifies such arrows. | |
| Apr 27, 2017 at 13:44 | comment | added | Tim Campion | 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. | |
| Apr 27, 2017 at 12:29 | history | edited | Colin | CC BY-SA 3.0 |
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| Apr 27, 2017 at 12:19 | history | asked | Colin | CC BY-SA 3.0 |