Timeline for A nice subcategory of the category of measurable spaces
Current License: CC BY-SA 3.0
13 events
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| Apr 13, 2017 at 12:58 | history | edited | CommunityBot |
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| Feb 4, 2016 at 7:23 | comment | added | Zhen Lin | In situations like this we should a class-locally presentable pretopos. I'm not so sure whether we get a cartesian closed category, or even whether we get a subobject classifier. Restricting to a small "subsite" (not standard terminology) would give a Grothendieck topos but it seems like an arbitrary thing to do. | |
| Feb 4, 2016 at 2:49 | comment | added | Dmitri Pavlov | @ZhenLin: In fact, taking sheaves on the full subcategory of measurable locales consisting of two objects, the point and the real line, gives us an honest Grothendieck topos satisfying the desired properties. | |
| Feb 4, 2016 at 1:01 | comment | added | Dmitri Pavlov | @ZhenLin: You're right, we need some additional work to show that we have (or don't have) a Grothendieck topos, or something close to it. We do seem to have a problem with a set of small generators: taking cartesian powers of arbitrary high cardinality of the discrete two-point measurable locale will eventually give us measurable locales that cannot be probed by any fixed set of measurable locales. Perhaps we have here a class-locally presentable elementary topos (in the sense of class-locally presentable categories of Chorny and Rosický)? | |
| Feb 4, 2016 at 0:00 | comment | added | Zhen Lin | I suppose coaccessibility is what we want, though. Small presheaves on an coaccessible categories are the same as (covariant) accessible functors, if I recall correctly. On the other hand, restricting to small sheaves is not going to give you a Grothendieck topos... | |
| Feb 3, 2016 at 22:23 | history | edited | Dmitri Pavlov | CC BY-SA 3.0 |
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| Feb 3, 2016 at 22:17 | comment | added | Dmitri Pavlov | @ZhenLin: You are right, of course, measurable locales are coaccessible, not accessible. (In my mind I had the example of the category of Hilbert spaces, which is accessible and coaccessible—but it's not complete or cocomplete either.) | |
| Feb 3, 2016 at 22:06 | comment | added | Zhen Lin | Err, the opposite of a locally presentable category is accessible if and only if it is a preorder... | |
| Feb 3, 2016 at 20:54 | comment | added | Dmitri Pavlov | @ZhenLin: The category of measurable locales is accessible because it is the opposite of the category of commutative von Neumann algebras, and the latter is locally presentable. | |
| Feb 3, 2016 at 17:20 | comment | added | Zhen Lin | Is the category of measurable locales accessible? Or else what do you mean? | |
| Feb 3, 2016 at 12:42 | comment | added | Dmitri Pavlov | @ToddTrimble: I guess the only way to find out is to ask the OP… | |
| Feb 3, 2016 at 12:34 | comment | added | Todd Trimble | I think I read the question a little differently. It seems to me OP wants a subcategory of measurable spaces and measurable maps to form a topos (where "subcategory" is in the set-theoretic sense: subclass of objects, and homs are subsets -- which is of course not a notion that is invariant under equivalence). Here you've instead expanded the category (which might be the more sensible thing to consider, but still). | |
| Feb 3, 2016 at 12:29 | history | answered | Dmitri Pavlov | CC BY-SA 3.0 |