Timeline for A nice subcategory of the category of measurable spaces
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23 events
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| Feb 5, 2016 at 17:36 | comment | added | Taylor Friesen | Unless I'm badly misunderstanding something, though, the quotient of the unit circle by an irrational rotation exists and is not a point in the classical theory: The points are orbits of the rotation and the measurable sets are sets of orbits whose union is Lebesgue measurable. It's not a space which is amenable to study by its $L^1$ functions, but in the category of sets equipped with $\sigma$-algebras, it's not isomorphic to a point. | |
| Feb 5, 2016 at 17:00 | comment | added | Dmitri Pavlov | By the way, another problem with measurable locales not being a Grothendieck topos is that they don't admit a set of generators. Indeed, consider the cartesian power W_κ of arbitrary cardinality κ of the two-point discrete locale. A map W_κ→W_λ exists only if λ≤κ, so in order to “see” all measurable locales one needs a proper class of objects of the form W_κ. (Any measurable locale can be presented as a coproduct of W_κ, which gives an exhaustive classification of measurable locales.) | |
| Feb 4, 2016 at 16:36 | comment | added | Dmitri Pavlov | Generally speaking, one cannot expect to have good quotients in the context of 1-toposes anyway, and passage to ∞-toposes is usually necessary. Indeed, the stacky quotient of a circle by an irrational rotation is a well-known (elementary) example in noncommutative geometry, and it is studied there by means of its groupoid convolution algebra. | |
| Feb 4, 2016 at 16:32 | comment | added | Dmitri Pavlov | Considering that virtually all nontrivial theorems in measure theory require a σ-ideal of null sets either explicitly or implicitly, it would be really surprising if one could do without it in this context. Also, it seems to me that the quotient of a circle by an irrational rotation should be a point. Indeed, functions on such a quotient are functions on a circle invariant under irrational rotations, and the only such functions are constant functions. In fact, the real line can be replaced by any target, and the Yoneda lemma tells us that the quotient is isomorphic to the point. | |
| Feb 3, 2016 at 20:55 | comment | added | Dmitri Pavlov | I would simply change “elementary topos” to “pretopos” (which will fix the only mistake), then convert to comments. | |
| Feb 3, 2016 at 18:59 | comment | added | Todd Trimble | Simon, let's roll back so that people can see what the comments were about. (It will also help others discern some of the issues involved.) You could also convert to CW so that any downvotes carry no penalty. Or, I could roll back and convert everything to comments as Dmitri suggested, which will have the effect of deleting this as an answer | |
| Feb 3, 2016 at 17:21 | comment | added | Dmitri Pavlov | Instead of deleting this post with the comments, you can flag it and ask the moderators to convert it to comments. In fact, this is what I will do right now. | |
| Feb 3, 2016 at 17:20 | comment | added | Dmitri Pavlov | So I guess we could say that measurable locales form a closed symmetric monoidal Boolean pretopos. | |
| Feb 3, 2016 at 17:08 | comment | added | Taylor Friesen | In the meantime, here's the intuition: There are various functors which take some kind of space $X$ to a space of measures on $X$. (The latter space has some linear or linear-like structure, depending on the type of measure.) If we have anything which looks like "the uniform measure on the unit interval", Vitali's theorem suggests that the source category can't be a topos with choice. My question is: Can we do the next best thing, and have the source category be a topos, while maintaining the ability to study at least the more common measur(abl)e spaces coming from other fields of mathematics? | |
| Feb 3, 2016 at 17:06 | comment | added | Taylor Friesen | I was looking for a subcategory of measurable spaces, although as multiple people have pointed out I didn't put enough restrictions on the morphisms involved. I also meant "measurable space" to mean a set equipped with a $\sigma$-algebra, not one additionally equipped with an ideal of null sets. I'm trying to rewrite my question to better capture what I'm looking for, but it's possible that I won't manage to make it rigorous and I'll need to make this into a soft question. | |
| Feb 3, 2016 at 17:01 | comment | added | Simon Henry | Indeed, I just found it. Kornell's paper also prove that there is no co-exponential in the categories of commutative VN algebras ( corrolary 6.7). So this shows I was wrong. But I have the impression that his argument say something one the initial question. | |
| Feb 3, 2016 at 16:54 | comment | added | Dmitri Pavlov | The main theorem of Kornell's paper is to show that the opposite of VNA is closed monoidal for the spatial tensor product, and his paper appeared earlier. | |
| Feb 3, 2016 at 16:42 | comment | added | Simon Henry | ... I knew I was missing something obvious ! This being said, a quick google search gave me this note math.ru.nl/~landsman/Uijlen.pdf (I think a master thesis of a student of K.Landsman) With inside a proof that the opposite of the category of VN algebras is monoidal closed for the spatial tensor product... This might be a start... | |
| Feb 3, 2016 at 16:24 | comment | added | Dmitri Pavlov | In fact, Theorem 5.8 in Kornell's “Quantum collections” shows that the opposite category of (noncommutative) von Neumann algebras is not cartesian closed. | |
| Feb 3, 2016 at 16:17 | comment | added | Dmitri Pavlov | How do you show that this category is cartesian closed? (Recall that an elementary topos is a finitely complete cartesian closed category with a subobject classifier.) I remember reading in one of Andre Kornell's papers that it isn't, which is why I'm confused now. | |
| S Feb 3, 2016 at 13:52 | comment | added | Simon Henry | Disclaimer: I have the impression that there is something very wrong with my answer... I'm hoping someone will tell me if it is wrong.) Let $C$ be the opposite category of the category of commutative von Neumann algebras, equivalently the category of measurable boolean locale Dimitri was talking about in his answer. Then: 1) C has all finite limits (fiber products are given by tensor product of Von Neumann algebras) 2) Monomorphisms are maps $A \twoheadrightarrow pA$ for $p$ a symmetric idempotent, or open inclusion in terms of locales. (continued) | |
| S Feb 3, 2016 at 13:52 | comment | added | Simon Henry | This second point implies that there is a sub-object classifier given by the algebra $\mathbb{C} \oplus \mathbb{C}$, or two points locales, and hence that this category is itself a boolean elementary topos (which seems extremely weird, hence the disclaimer in the beginning) in fact it also have arbitrary co-products and a generator (the standard measure space) hence it is a Grothendieck topos (which I found even weirder). Finally this category can be described as a category of nice measure spaces, with measurable map modulo equality almost everywhere. | |
| Feb 3, 2016 at 12:43 | comment | added | Dmitri Pavlov | Do you want your nice category to be a subcategory of measurable spaces, or do you want it to contain measurable spaces as a full subcategory? | |
| Feb 3, 2016 at 12:29 | answer | added | Dmitri Pavlov | timeline score: 3 | |
| Feb 3, 2016 at 5:54 | comment | added | Todd Trimble | We are told so little about the morphisms that it seems hard to deduce anything about any such topos. The only function that can equalize two distinct translations of $\mathbb{R}$ is the one with empty domain, so the initial object could only be the empty measurable space. But I'm honestly having trouble seeing that the terminal could only be the one-point space $1$ (e.g., we don't know that there are any nice maps to $1$, except the identity on $1$). How do you see the topos would have to be Boolean? | |
| Feb 3, 2016 at 0:13 | history | edited | Taylor Friesen | CC BY-SA 3.0 |
added a closely-related additional question
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| Feb 2, 2016 at 2:36 | review | First posts | |||
| Feb 2, 2016 at 2:41 | |||||
| Feb 2, 2016 at 2:34 | history | asked | Taylor Friesen | CC BY-SA 3.0 |