Timeline for Topologizing the category of measure spaces
Current License: CC BY-SA 3.0
23 events
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| Dec 27, 2012 at 15:19 | comment | added | Dmitri Pavlov | Thus you can take (equivalence classes of) measurable subsets of Z as the space of objects and the appropriate subset of Ob×Ob×End(Z) (namely, triples (A,B,f) such that f is supported on A and maps A into B isometrically) as the space of morphisms. Both objects and morphisms have several natural topologies. For objects you can take the norm topology or the ultraweak topology and End(Z) can be equipped with the topology given in my answer or with one of the other well-known topologies, like Haagerup's u-topology. | |
| Dec 27, 2012 at 15:11 | comment | added | Dmitri Pavlov | @Tom Leinster: If you equip your measurable spaces with actual measures (which is a bad idea from the categorical viewpoint, because you lose pretty much every nice categorical property, like completeness and cocompleteness), then you can certainly get a nontrivial internal category. Assuming a restriction on cardinality, all measurable spaces equipped with a measure embed isometrically into some fixed (big) measurable space Z equipped with a measure (e.g., Z can be taken to be pt^{⊔m} ⊔ R^{⊔n} for some appropriate cardinal numbers m and n). | |
| Dec 26, 2012 at 23:46 | comment | added | Tom Leinster | Dmitri, it seems to me that there are interesting non-discrete possibilities for the topology on the objects. For example, take a skeleton of the category of measure spaces with finite underlying set. The set of objects is in natural bijection with the disjoint union of the topological simplices $\Delta^n$ ($n \geq 0$). This set carries a natural, non-discrete topology, namely the disjoint union of the Euclidean topologies. | |
| Dec 26, 2012 at 19:17 | comment | added | Dmitri Pavlov | @Tom Leinster: Even if we put some restrictions on cardinality, there is still not much we can do: isomorphism classes of measurable spaces are classified by a pair of cardinal numbers, hence there is no reasonable way to organize objects of the category under consideration into something nondiscrete. So the answer to the modified question is yes, but the only interesting part in the internal category structure is the structure on homs, in other words, the enrichment. | |
| Dec 26, 2012 at 15:37 | comment | added | Tom Leinster | Dmitri, you're technically right of course (and I mentioned this in my first comment). But the question about internalization (not enrichment!) is still a good one if you interpret it with a bit of generosity: e.g. take the category of measure spaces whose underlying sets have cardinality less than some fixed cardinal. | |
| Dec 26, 2012 at 9:26 | vote | accept | TWalker | ||
| Dec 26, 2012 at 9:18 | comment | added | Dmitri Pavlov | Clearly, the category of measurable spaces cannot be interpreted as an internal category in spaces, because measurable spaces (or even their isomorphism classes) form a proper class, whereas a space can only have a set of points. I presume that the question actually means enriched categories. | |
| Dec 26, 2012 at 8:26 | answer | added | Dmitri Pavlov | timeline score: 4 | |
| Dec 26, 2012 at 2:51 | comment | added | Tom LaGatta | Also, as @Gerald Edgar and @Michael Greinecker point out in an answer to a recent question of mine, there is a nice functor $M$ on the category of measurable spaces which sends each space $X$ to its space of measures $M(X)$, and each measurable function $f : X \to Y$ to the map $M(f)$ defined by $\mu \mapsto \mu \circ f^{-1}$. i.e., $M(f)$ pushes forward a measure through the function $f$. Take a look: mathoverflow.net/questions/117118/… | |
| Dec 26, 2012 at 2:48 | comment | added | Tom LaGatta | @Walker: you should check out Dmitri Pavlov's posts on this site dating back over the past three(!) years, as well as the nLab entry on measurable spaces. In short, the philosophy is that the category of measurable spaces is nicer to work with than that of measured spaces. * mathoverflow.net/questions/49426/… * mathoverflow.net/questions/20740/… * ncatlab.org/nlab/show/measurable+space | |
| Dec 26, 2012 at 1:39 | comment | added | Theo Buehler | @Ronnie Brown: It is possible to develop Mackey's theory of virtual groups along lines closely parallel to the approach to orbifolds via localizations of suitable categories of Lie groupoids. First steps in this direction were taken in the seventies mainly by Arlan Ramsay and more recently by Anantharaman-Delaroche and Renault in their book on amenable groupoids. None of the available references are very "algebraic" or "categorical", though. | |
| Dec 26, 2012 at 0:58 | history | reopened |
Tom Leinster Kevin Walker algori Todd Trimble Dan Petersen |
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| Dec 25, 2012 at 22:29 | comment | added | Todd Trimble | I'm not convinced it's a good question (yet), but having entered a vote to close this 8 hours ago, I just vote to reopen. | |
| Dec 25, 2012 at 21:13 | history | edited | TWalker | CC BY-SA 3.0 |
added 71 characters in body; edited tags; edited title
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| Dec 25, 2012 at 20:54 | comment | added | TWalker | @Tom ,excellent ,thank you for the help,I will do | |
| Dec 25, 2012 at 19:41 | comment | added | Tom Leinster | Ronnie, you may be right; all the same, the question does seem reasonable to me. For example, in my own work I've used the fact that the category of measures spaces with finite underlying set can naturally be regarded as an internal category in Top. This question removes the restriction of finiteness, which makes it harder. | |
| Dec 25, 2012 at 18:54 | comment | added | Ronnie Brown | My impression is that this question is somehow the wrong way round, and should be more of the form: is there a category of measured spaces? I mention this because G.W. Mackey did work on measured groupoids and I have always assumed these were groupoids internal to the appropriate category! See G.W. Mackey, ‘Ergodic theory and virtual groups’, Math. Ann. 166 (1966) 187-207. He asked the question: if a transitive action of a group $G$ corresponds to a subgroup of $G$ (or conjugacy class thereof) what then corresponds to an erodic action, which is "almost transitive"! | |
| Dec 25, 2012 at 17:55 | comment | added | Tom Leinster | Walker, I'm afraid I don't know a reference. The best thing to do is to edit your answer to make it more clear. If you're having trouble with the English, try this: "Is there a natural way to interpret the category of measure spaces as an internal category in the category of topological spaces? Any references would be helpful." Also, you could usefully change the title to "Topologizing the category of measure spaces". If you do this, I will vote to re-open. | |
| Dec 25, 2012 at 16:29 | comment | added | TWalker | @Tom ,thank you; can I have some paper that tell about these ways if any. Walker | |
| Dec 25, 2012 at 15:11 | comment | added | Tom Leinster | It seems to me that this could be a sensible question, if only it were more clear. I guess it's asking whether there's a natural way to interpret the category of measure spaces as an internal category in Top. Presumably the answer is yes (disregarding niggles over set-theoretic size); perhaps there are several ways to do it. | |
| Dec 25, 2012 at 13:32 | history | closed |
Qiaochu Yuan Fernando Muro user9072 Kevin Walker Todd Trimble |
off topic | |
| Dec 25, 2012 at 11:03 | comment | added | Qiaochu Yuan | mathoverflow.net/howtoask | |
| Dec 25, 2012 at 10:51 | history | asked | TWalker | CC BY-SA 3.0 |