Probability sheaves

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Probability sheaves
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In the Articel "The dawning of the age of stochasticity", Tao observes that the probability theory concerns itself with properties that are \preserved with respect to extension of the underlying sample space", in much the same way that modern geometry concerns itself with properties that are invariant with respect to underlying symmetries. Reformulating this in category-theoretic language, probabilistic concepts organise themselves into presheaves over a category of sample spaces. In this talk, I observe that they further form sheaves, and I consider ramications of this observation. As a suitable category of sample spaces, I take the category of measure-preserving measurable maps (modulo almost sure equality) between standard (a.k.a. Lebesgue-Rokhlin) probability spaces. In this category, every cospan completes to a commutative square enjoying a universal conditional independence property. As a consequence, the category carries an atomic Grothendieck topology, whose sheaves can themselves be characterised in terms of conditional independence. Examples of such probability sheaves include sheaf representations of standard probability spaces (given by representables), sheaves of random variables, sheaves of probability measures (given by a general coend construction), and sheaves of orbits of ergodic group actions. In general, I argue that the resulting atomic topos of probability sheaves is a natural category of generalised probabilistic concepts. Moreover, as a boolean topos, it models a mathematical universe in which random variable occurs as a primitive rather than derived mathematical notion. I believe this model has the potential to inform the development of an alternative approach to probability theory founded on primitive random variables, somewhat along the lines envisaged by Mumford in.
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and if a top and a 2 I think you know it's quite an honor to get to give the very last talk of this meeting so I want to thank the organisers for that opportunity and also more generally fall the whole wonderful meeting we've had this week as we thank somewhat with hundreds took about 2 years but said the works stemming from an an observation and the observation is random variables was considered in probability theory naturally form sheet and served those that observation which I'm going to begin the talk with having made that observation it's natural to look at chief within the context of I'm a category of probability sheaves and look at some other constructions that exist in that category and see what they have to do with probability theory so that's roughly how the talk is going to go on and said
probability theory is 1 of the areas of mathematics this has been Little touched upon this week we've had connections with many areas mathematics but not so much With probability theory signed on to remind you what a random variable is a random variable simply a the map packs From sample space Omega 2 some nice space of values and where for my talk I'm going to look at well-behaved random variables but well-behaved means that I'm eager and I and II spaces and X is a nice man and so specifically on don't assume that some only the sample space this this has to be a probability space to do probability theory at all said and it assumes that it is and it's given by a Kurdish space said that a topological space but those topology is given by a complete separable metric space structure I'm so that we just interested in topology the fact we not really interested in the topology because I'm going to consider it than with the classes of Burrell sets sights have given by Paris-based with paper Rell probability measure which Cokie and again and the values space on on center likewise soon is and I space which again I mean at the Paris-based and I met to consider the Sigma Rouge's MySpace together with the Sigma algebra Burrell set and FedEx it is a Burrell measurable functions the inverse image preserves sets Sunday to drop the rally in future may talk about measurable so in probability theory 1 often considers somewhat more general definitions for the general theory of when you look at many
books on probability theory almost all the good examples fit into this set so is not a
significant this kind of well-behaved must restrictions I'm not such a significant restrictions from the viewpoint of probability fairest another thing that I'm going to do which is not always done and
identified random variables will bald Sarah which means modulo almost everywhere equality and probability theory Stanton not identify the modular 0 that's still the real relationship of almost everywhere is very important in probability theory and so this is a random variable has In my abstract and mention a blog posting of of towels some introduction to probability theory images as well as yet this is all very well but is a very concrete definition if we say this is a random variable convictions state all sorts of and nonsense properties about it that they make any sense for all but from the viewpoint of probability theory that you could say is an element here hit by some particular chose an element of the samples spies as not a probabilistic meaningful statement and terror observes that all proposes that the notion of probabilistic meaningful statement should mean some property that preserved by extension of probability space what extension of reserve by extension the probability space it means that 1 should consider random variables as being appreciated over a category of samples spaces introduced now the category of samples spaces I'm going to be working with this is not quite the same category that tell users but it's better behaved for my purposes and so a please of samples spaces before probability spaces but and so the objects all kurdish portability spaces as above and the maps of net measurable Burrell
measurable preserving functions again identified a lot clearer and then it's easy to see I think I've got yeah it's it's easy to say that random variables said Friday for any further space space we have a the cliches Of all the time at the of the from pain of random variables said namely all of the a at the simple space a meager is just the random variables from them and again 2 set by random variables I'm in applying already misidentification what's 0 which is important because to get the re-indexing working so Odyssey re-indexing is by composition OK so
that the 1st observation all the the starting observation it is that's this breaches of random variables is actually shape and it's chief for what is going to turn out to be growth and the topology but I'm going to verify its aggressive it topology later admitted but that aside from that it will be the atomic topology which means that every map in the category he is going to be considered the Justice Singleton map itself is going to be considered as a cover so I once and overstated the main landmark that is used to say that random variables are achieved with respect to this family of covers that I will later showed is a topology arm Is it bad form to run the dilemma across 2 boards maybe it is start a new board so I'm going so in a sense this is an observation that I'm going to do this observation in some detail because there are some technicalities to that maybe could be could be avoided and I don't know that any witnesses this is the way I see it said so the observation is going to be the dilemma is going be the chief property and it's going to be just about the PRI's chief of random variables valued in the rails in the 1st instance said I states the following given a man well actually starts using its of at the symbol that and this is just another gene mapping the sample spaces in fact every map in the category of samples space is is at the Norfolk so given the matter in pays patterns the Hawaii Brendan variables of real-valued random variable on same again the people who live along this confused because this is a story says right what you so sorry which
was on the air From
Connecticut the shadow here this project writing like this clashes it occurred to me just before the talk that this might be the 1st ever talking on top of Serie conference where Omega is a sample space and from then on the probability space rather than the subject classified never mind I want to keep my notation the saying this as and when it's here so said given this the following are equivalent but he has following a recruitment so firstly why this man what I medical invariant which means matching in the usual sense matching family for a sheaf from relative to school this fall so the terminus for atomic topologies the terminology the definition of matching family works absolutely fine the definition matching always feels a bit awkward to me because of a family to be matching you imagine there have to be several components to it to match each other and it is just 1 component that has to behave well with respect to the single men so sigh prefer to use invariant but I mean the same thing as matching family said that what this means is in other words Max Avenue Q Q prize into middle Chaparral present much like this it are composed of Q equals pocketed the Q Prime so if they essentially intuitively to land in the same fibers over our than they reinvent reinvents right to the same thing so why please understand to write this for the weekend re-indexing why but there was a lot of time and why should be from Prime Minister thank you so that might to be confusing and it's indeed because of family up here Saturday and a variable that's supposed to be sort of invariant with respect to mark said this is equivalent to regular forces for those who still need to be there of yours sorry while another once a completely well .period of not starting that was fit preferring a more votes for his dedicated pushes up now I'm sorry about this right so 2nd property so in a sense was saying that why is somehow In variants on fibers over Q intuitively the effect this has a probabilistic statement which is but why it is almost surely surely constant on almost every flight also what this means this uses the concept well regular conditional probability which exists because we're working in a nice category of probability spaces of the whole of the original the model what we think is reasonable Ramirez along with the value of his last fall rule of lot of pressure measure the samples spaces carry probability measures in Europe who are very random and the and the maps in the subcategory of samples spaces preserve measure but B random variables themselves there's no way that we not the added that have polluters chosen probability measure here to these adjust the emotional functions that were be the 1st time this year yet so what this means is so we have the using here the regular conditional probabilities which I'm not going to define give the full definition but understood intuition which is we've got the map I hear from and applies to dome again and so we can't do this we got sort of a meager down here and we've got and we get prime up here we might have a little .period pointing which has defied the it's like this 1 then again here and
the regular conditional probative probabilities we have a full blown promised the media In in the bank I hate probability measure Burrell but it isn't probability measure the was by like this the consent of conditional probability measure so Our equals Omega on the fiber into the minus 1 vinegar and these what in such a way the that serenity the rails that ass up here you can get the probability of that space bye integrating the conditional probabilities here over by varying the only get down here so friendly that any morale as game and it's a subset of immigrant crime then the probability I'm here to compete prime Kovacic is integral immigrants downstairs all of the conditional probability measures on that fight all integrated with respect to the measure downstairs which paid OK so that's the interest in using this week states the property of wiping almost surely constant on every almost every fiber of cute the 3rd equivalents now and this is the reason for the taking it valued in off is that the conditional expectation you feel is not enough the commented of basis of our here sorry so I had paean my notes but then I thought when I write on the board my little piece is going to get confusing my big because of their rights are carefully such the 2 are often and I'm not running into problems from that saw them anyway so conditional expectation of acts given our ranges over the base space to this exists which is that this is a major function from go to our His exists as a little Sharia that normally conditional expectation is defined assuming that um axes Intercable here it need not be insuperable but it doesn't matter using regular conditional probability you can make sense of conditional expectation would generally in this setting I don't want to go into such Nichols in any case it exists and on if free if take that conditional expectation what's and we really index at along our planned this equals acts will maturely that so that's the landmark and this conditional expectation it is so this is the madam Mehmet other gives us the random variable which is going to be the amalgamation began amalgamation is not really it's like a descent property you really consider stopped 1 lap but we're so an image the desired blue of his until they still won a yes the Axa's life story of I'm really sorry and the Soviet Union ,comma balls in the right place start this is a composition which is also re-indexing in the appreciate send consequences all this is forever a referee space the day of the position and also for every Samples space and a sample space and again why of Omega that there is a chic said so when the chairman that the and atomic covers former topology that's apology will be Sept canonical but in fact the baffle canonical because it's a it's an atomic topology and so these follow because every parish space In beds in the real numbers so that was that quite easy but we needed to use the real numbers here today to use the argument by conditional expectation
right so we've got this she property I side have attended many pay attention to what time I started and I know we're running late anyway so what do you think the 8 and then question 7 minutes and then 8 minutes the and and crashed yet again for his love of music videos and more every she come from around the world the Romario who have lost going well and for every Paris-based aid have appreciative of random bag of a valued random variables and that is always a issue that's vessel that means so well so far off the underside of her this was no of course not his agenda because we have a whole she top also things and going to be random variables that provision is no way they can afford something like that the last thing you want to know why I think it is the nature flight is the representable of the representable preachy generated by by the generated by the use of the media rights serve I I had high ambitions for the rest of the talk and I didn't realize how slow I am on the blackboard and Harold and how dangerous it was to depart from what the notation and so so maybe I should just give up at very high levels summary of what I was going to say said the 1st lady that I well we have the condition that means that the atomic topology is a topology so as she maybe this is the lot more important thing to go into a little bit more detail but I'm not going to say as much as I intended to say we identify while recent Maryland will was saying To say that a square I To commuting square they paying says problem using only different things you have to put a lot of dashes but some say let's have all tried I keep trying to kill or something like that and we say that it's a and Independence Square here so we can consider this a measure preserving maps between samples spaces but we can consider them as random variables in their own so if those 2 variables thank you and you and you primers random variables are independent conditionally on the composition random variable down to the bottom side has collapse park use this is conditional independence that I'm not going to have time to define that as a woman concept in probability theory so this isn't as an independent splash on the proposition is but every case bad let's keep it the same as above there is is a universal an independent completion by which I mean we had already and over it they can completes 2 could obtain Independence Square well let's call this only get prime tenants only did double crime South only again so it's kind of like a pullback but he enjoys the pullback property just with respect to conditional but doesn't just mean respecting independence squares but never like best characterizes this up to buy some office on so essentially this is constructed as a as a pullback from in the category of power I mean it is a bit fiddly really is that such a quite severely construction and you need to take fiberwise products of regular conditional probability is up here in order to get the the measure on this basis of integrating it all over the conditioning space but anyway there is such a construction 1 can find it in also in the literature there quite hidden never seen this universe of property mentioned before but it has just really an observation from sorry in regional you there should be no direct over for this property Back is it's in the paper by that somebody could end Eric debate capped From about 11 years ago riches he was and the same property is considered in infringements book a measure theory but he doesn't play doesn't have existence result for Polish spaces that but he's got that is discussing the general properties of this construction and a nice construction and it gives us the property that every case panel completes to describe commuting squared the something or a property always forget left or right or what have you done for color but anyway it implies that the that the atomic topology really is aggressively topology I moreover we've got this in a really nice way and that really nice way I mean we have the completion in a sort of universal sense and that's what really nice universal sense was going to give me a general current construction Serena got the category although she however the samples as she's ever overpaid and it is going to be a general trend construction on this as a cow and which and give a mother and that has the following problems so is justified purely category-theoretically but it has the following property that's end of random that of the sheaf of random variables of is isomorphic 2 random variables In the space and the fight this is the the gerrymandered all all of them well this assigns a set an end of is it's the parish space all of the probability measures it has advised that the space measures on pay so we recover in a sense of the gerrymander had in the chief Category 5 registered general construction that I afraid times is running out as I wanted just say something about where I will once again with the sergeant said that like pointers that sheaves of P I think is a nice category although Cheney's within which doesn't probability concepts of probabilistic concepts lives
so we've already seen some random variables we've seen probability spaces that the representable we've seen a more the monitor balls and probability measures the sense that is also and equivalence relation between on any object of the top defined which is just identifying those elements that reside within the same atom in the lattice of subjects for this exists always for atomic top also has nice properties in this case it coincides With a quick distribution of random variables so we've got an equivalence relation back distribution is a general property in internal logic of anatomic top also but if you have a century of "quotation mark of Olivier a hair band for all X Y in Olivier if there equivalent and made 1 of them satisfies the property in any other ones satisfies the property to this says in a sense any definable property in the top boss of random variables is invariant under like redistribution of of of of random variables as denies it saying We can only state probabilistic things probabilistic 3 meaningful concepts in the top Austin some in some sense defined holds important every start sticking free variables and morally speaking than to condemn the property is going to go so this has to be a subsea I'm sorry that the property needs to be a subject of its billion calls because it's an atomic top lost I am but I know that doesn't satisfy the axing of choice but it does satisfy we do have dependent choice and the proof of this users the proof of this is basically users from older office little extension Fareham earlier in the week of the month where is the justice sense of and measure that will use the analysts among college managers and enjoy and it's very different from those of a model and so yes I was so it at me like it we have we don't have writer choice but we do have depended to but this doesn't satisfy that every set as every subset of Real's is measurable resisting the ,comma this is in quite a different direction because somehow the important thing here this is like in David Roberts talked he mentioned permutation models this is very like those in that you have found that the use of the the random variables will not live in the cumulative hierarchy sets of that basically random variables in the stop-loss or somehow atoms so what I think all this stuff together sigh as she suggests using the internal classical logic of this theory to do probability theory with random variable as a primitive concept and actually I mean I thought of this as a kind of pipe dream when I was playing with a stop-loss but it think about it morning it actually seems to go quite a long way developed it to some extent some I'm really quite excited by what's coming out well what really excited but let's say so anyway it is it is kind of it is that those same thoughts that had to be interesting to do so and interesting setting for probability theory theory and then some probability theory ?question mark on and this connects with of paper cited my abstract served in the this summer volume of million mathematical from visionary papers and there is an article that cited in the abstract as article by David Mumford that in which he's saying what it would be really nice to have approached probability theory which we do probability theory with random variables as a primitive concepts like some ideas that this topple I should provide a model for that kind of endeavor and the working of the details of future work so thank you very much the get relentless unit of the responsibilities of the problem here is that the name of the right so I so it seems to me like a fundamental you have to do greatly yeah yeah so it it seems to me like a quite a fundamental property but it's curiously absent in the problem Dougherty literature and so In in Finland's measure theory if you look at maybe he calls them relatives products he has aid and to concrete definition of what a relative product should be With some statements about don't exist in general and some existence results but none of which are exactly what I need I said the same property was all she needed in computer science of all places which is very where I came from far aware of some of the computer science literature and again it's not quite in exactly the same for all but the timing is a bit of history there but that the nicest exposition In computer science is a paper by DOE can instantly became 1 and of the dead members Sonora narrow lot which I can I don't remember the title this from their legal remedies from 2 thousand and 4 again he's working in that sort of different setting motivated by Computer Science Applications and the result I need more or less but not exactly is found somewhere in some remarks there but that doesn't history to that they'd actually mean the construction is not terribly difficult when you have the right results about you have just said "quotation mark it was there was when was the last year of there always is made in the development of this square problem you it was a question Is there is a lot it is indeed indeed To have at working with the Paris-based structure appear which itself is not easily found because these immeasurable that's not continuous maps Citicorp pullback of Polish spaces using measurable lapse between the among can still 1 can actually find the perilous along the way this follows the community who formerly held up well so it's in fact what would the whole so I she said the start the whole choice of categories in fact so you would want to wear standard Burrell spaces of the ones In which case in which the Borrell Liberals censoring by ejection with the standard practice of of the so the measurable facts earned by the action of this the stunned the canonical authors Abril such I do want to have a point measures and things like that inhibit the other as she is a very natural category to work with the category of water-cooled stunned the probability spaces here which is which the Saskatchewan in fact is equivalent because all the identifying maps up to almost almost every Republican simple refusal to work with a lot of murder remark Yukon or always is describing the nurses yesterday usage to worry disjointed cockroaches were removed ,comma this is the worst thing is that Jean-Marc becomes conscious right is in the world the merger of right so that the latter that also might help with this problem
and but in fact this just the sort of pullback thing can be constructed just directly using the Structural of so even without using properties rather than just directly used the accountable presentation of the Sagrada structure on the on the war rather ,comma home yeah but that that was going on in November that 1 point I should've made at the start of his but that as soon as you identify maps
to almost every or equality so very many different categories would coincide with the sponsors as a very economical
so it's also is the is the category of this is also the category of Canterbury presented measure algebras opposite so customers there are also fears that the US the in question was not only 1 of your teaching quality it's going to be very difficult to teach in a lesson because usually we have not affiliated unions that have a lot of attention to what is going on on the desk somewhere between what he would be convinced that this shows that the solution to remind students that when the find that only yet someone still there and still has very good student hasn't contributed to the number of was morally death in contravention of the smaller only within a week and pollutants in the air the legal corner for the point is that he has not lost this was all that was going on and I think it would be a question of influences so but actually the only goes I just here in the construction of the category 30 properties .period is once you do this there no visitors nowhere makers at all neither big nor small said this is actually 1 of the 1 of the nice we have I think 1 of the nicest features of this approach the idea is to to basically use this together with the calculus of independence give conditional independence it and come to you this is the beginning of the school year if you so will in a sense you can't but the other was collecting radically and and just so that some of the lessons my feelings can be heard over the objects in the water he said there was evidence that good for the rest of you kids in the office of the father the rented variables are not the atoms so I mean my idea is that you wouldn't characterize them you would have to those of the other about you would he would add b object of random variables as a new set in such a hurry arm and then so serious I say it's a sort of let me have a random variable Axum any random variable and any other random variable you can find another 1 that's equivalent to the 2nd independent of the 1st and then this sort of thing for the B these kind of conditions that will fit together with depended choice to allow it to them to do more and it's that kind of the SEC kind of reasoning that we're going to be a bit of public it is clear that this side all the failure of the cost of lake and just think of it as the subject of pairs of all friend and I said that's a meager you've got random variables from the GOP functions from only and you're equating those pairs of of such functions that I could distributed so that injured Accra distributed to induce the same probability law on also on on a series of equivalence relations so random variables x is is related to y if if the probability law of Exxon on pay is the same as the probability law of honors the probability measures dangers are equal but the point is it is actually it is just these this is just the instance of a general equivalence relation that exists because this is an atomic top this exists for any atomic top of the Soviets in this equivalence for there is an agreement relations exist for all objects on on every on every object to have the city having equivalence relation due to the fact that we've got some atomic topless if you have an atomic topless you have such an equivalence relation and it just so happens that in the case of random variables it is a very meaningful thing the losers in I'm so you wish you had some of whom quality so safeguards the book she asked only only we've got the the best use of X Y today we go and you can see several way of saying this is that the this is this this homes before the only if there exists Q so there exists a prime and notice queuing Q Prime from many different such that you said that extra indexed and Q is quite relaxed and the indexed comprise but also it amounts to it amounts to we look at the subobject lattice of F and its
atomic and where it was sticking to X and Y in the equivalence relation if and only if they're in the same active in the in this atomic it's about the quality of of the good of the National League but most of the differences the use a room with he had suggested said that this can't be empty that suggests the problem that the probabilities bases as part of unlocking of lymphocyte has not yet this is in the site this is 2 elements of the externally or in the equivalence relation defining the defining the equivalent relation has a said well it said it is assumption anyway over time that we have to go home it's going To those so of go in the book stores and use of the all them of which you have a lot of interest in this investors and that it would be great to see you on want the inside of his name In fundamentals the field of money you sitting area well "quotation mark so emigres all that she objected in the in the continent using a particular site and it was just another tree object in the category that reform intimate positive that this is this this is just over the northern Luzon region close to those of us who will have the last fall because of no reason was given for you that so that there is there is no and there is no end to probabilities based on all of you all you will that this is the only yes it does if you've got an initial object in in the category need food for the atomic topology on then you get when you get something you just get a battery of secular or was called in to see if they can form a government that be have seen the atomic atomic topology is a very trivial in categories that have initial objects from use of all mess with some of the we're on identity but in my case I don't suppose it suggests just as the the case I had the highest overall ,comma side of it is not necessary for atomic sites but if if you're going to get it a few it's as if you if you want the topology to be sub canonical than that'll be acted on this year this slowest who will love the euro's value of some golf but some of the concepts and wearable the company's growth with make like the center of the School of Law and Order Number Number of other than those who really was building the someone structure on the basis of images and words like you ladies like the British schools London's also would the events that year because the post-race news of the war itself With this space of quality images on a police station in the center that of we the square from model of the this is the 1 that hit you yet he put yet you did put the weight topology of a topology some have called on the space of probability measures and indeed it's affair and that if the start of the package you get a Paris-based back you can fight you can find that 1 in many textbooks on probability theory initiative yeah at the same probability theory don't like to more and also in the area of only 3 you can do it so can you not only that but to sell between Maryland has and also have sided with the use of higher and yet I don't know if there's a natural higher thing to do that said to avoid the muddying by now to begin with 1 can use the the measure algebra presentation instead in which case automatically does voice that she doesn't much nicer way of presenting at the magnitude of what he could probably do on the and suitably bulges in following years he has to say honey what I use the fact that up by 0 To get the chief property because the condition expectations not defined otherwise I don't know if it's true I mean and I feel that something would fall down if I didn't want by Sara but whether there is another way of doing it by bringing higher category theory I don't know and firstly I'm not going to look at the site some the of view on whether I was born ,comma our excellent questions so some flashy had a conversation last year with Bolivia in
human image conjectured that the top has no points and she was very skeptical and rightly so and on the this this meeting and I've been spending most of the meetings trying to calculate what the points and so so I have to have a conjecture but will not result in the family will not like to work with me on this contention that the very happy to have a collaborator so I mean 1 strengthen the conjecture is that I'm so this is the stop-loss is actually kind of accountable distributive top also in the sense that there distributed the T property between countable limits and Carolyn it's in that context it's natural to look at invests geometric Mofaz and to have a stronger property that invests in image front has preserved countable limits and I believe that the points that have that stronger property on exactly the normal separable monotonic measure algebra I believe is also a point which is the separable the unique separable atomic measure our which is not a point that preserves countable limits I don't know if there are other points either but I mean my conjectures but those to that this other 1 is said basically nonmetallic measure algebras all points I believe you only you have a lot more if you then want then exhibitors as a category of and continuous actions for example of the use of liability there has been he said he loses some of you be there in very viewers he is lower than what was that was how I got the conjecture about the I'm accountable and preserving films of the year yeah you have slightly Russian home random violence progresses of internal you can't touch because of and is still in the picture if the state of affairs that next year we'll will vote on the articles on you want well so I would have real value rendered valuable as sick thinking in my in my internal language so just select XP that but look at the locals fear relation between that and and Nice subsets of office space so they can have them has probability relation between random variables and that and good subsets of the of the of the space and but under again going to be some other things too and also to say this conjecture about points said the model in the in the top forces of random variables of the tournament set which is intended to restore possible made looking at it like it is they have surrendered their labels of 2 carriers the structure of the enormous separable atomic measure algebra here so that some internal structure that exists In thank