 Thank you. It's quite an honour to get to give the very last talk of this meeting so I'd like to thank the organisers for that opportunity and also more generally for the wonderful meeting we've had this week. Thank you. So what I'm going to talk about is essentially work stemming from an observation and the observation is that random variables as considered in probability theory naturally form a sheaf. And so there's that observation which I'm going to begin the talk with but having made that observation it's natural to look at that sheaf within the context of 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. So probability theory is one of the areas of mathematics that has been little touched upon this week. We've had connections with many areas of mathematics but not so much with probability theory. So I'm going to remind you what a random variable is. So a random variable is simply a map x from a sample space omega to some nice space of values a where for my talk I'm going to look at well behaved random variables where well behaved means that omega and a are nice spaces and x is a nice map. So specifically I'm going to assume that so omega the sample space this has to be a probability space to do probability theory at all. So I'm going to assume that it is it's given by a polish space so that's a topological space whose topology is given by a complete separable metric space structure but we're just interested in the topology in fact we're not really interested in the topology because I'm going to consider it then with the lattice of borrell set so it's given by a polish space with a borrell probability measure called p omega and the value space a I'm going to likewise assume is a nice space which again I mean a polish space and I'm going to consider the sigma algebra this space together with a sigma algebra of borrell sets and x is a borrell measurable function so inverse image preserves borrell sets so I'm going to drop borrell in future when I talk about measurable. So in probability theory one often considers some more general definitions for the general theory but when you look at many books on probability theory almost all the good examples fit into this setting so it's not a significant these kind of well behaved restrictions are not such a significant restriction from the viewpoint of probability theorists another thing that I'm going to do which is not always done I'm going to identify random variables mod zero which means modulo almost everywhere equality probability theorists tend to not identify them modulo zero but still the relationship of almost everywhere equality is very important in probability theory um so this is a random variable but as in my abstract I mentioned a blog posting of of tells which was some introduction to probability theory in which he says well yeah this is all very well but it's a very concrete definition if we say this is a random variable we can state all sorts of nonsense properties about it that don't make any sense for from the viewpoint of probability theory like you could say is an element here hit by some particular chosen element of the sample space that's not a probabilistic meaningful statement and tell observes that or proposes that the notion of probabilistic meaningful statement should mean some property that's preserved by extension of probability space and what extension preserved by extension of the probability space it means that one should consider random variables as being a pre-sheaf over a category of sample spaces so I'm going to introduce now the category of sample spaces I'm going to be working with and this is not quite the same category that Tau uses but it's better behaved for my purposes um so a category P of sample spaces P for probability spaces but um so the object polish probability spaces as above and the maps measurable borrell measurable measure preserving functions again identified mod zero and then it's easy to see I think I've got yeah it's it's easy to see that random variables so for any for any polish space space a we have a a pre-sheaf um rva a pre-sheaf on P of random variables so namely rva at at at the sample space omega is just the random variables omega 2a so by random variables I'm implying already this identification mod zero which is important because to get the re-indexing working so obviously re-indexing is by um composition here okay so the first observation or the the the starting observation is that um this pre-sheaf of random variables is actually a sheaf and it's a sheaf for what is going to turn out to be a growth in diktopology but I'm going to verify it's a growth in diktopology later I'm going to put that aside um but it will be the atomic topology which means that every map in the category P is going to be considered that just the singleton map itself is going to be considered as a cover so I want to now state the main lemma that is used to see that random variables are a sheaf with respect to this family of covers that I will later show is a topology um is it bad form to run a lemma across two boards um maybe it is so let's let's start a new board here so I'm going so in a sense this is an observation but I'm going to do this observation in some detail because there are some technicalities to it um that maybe could be could be avoided I don't know but anyway this is this is the way I see it so so the observation is going to be the lemma is going to be the sheaf property and it's going to be just about the pre-sheaf of random variables valued in the reels in the first instance so so it's the following given a map well actually so I was using a sort of epi symbol there this is just an arbitrary map in the sample spaces in fact every map in the category of sample spaces is is epimorphic um so given a map in p and a y in random variables a real valued random variable um on omega could you pull the blackboard a bit down please again so we can see the definition oh sorry is this oh right sorry sorry which one um so now you get the shadow here um let's put it let me write it like this so actually it occurred to me just before the talk that this might be the first ever talk in a topos theory conference where omega is a sample space in probability space rather than the sub-object class if I never mind I I want to keep my notation the same as in my notes here so so given this the following equivalent or t f a e the following are equivalent so firstly y is what I'm going to call invariant which means matching in the usual sense of a matching family for a sheaf relative to let's call this r so the term for atomic topologies the terminology the definition of matching family works absolutely fine but the definition matching always feels a bit awkward to me because for a family to be matching you imagine there have to be several components to it to match each other and here there's just one component that has to behave well with respect to the single map r so so I prefer to use invariant but I mean the same thing as matching family so so what this means is in other words for all maps q q prime into omega to parallel pairs of maps like this if r composed with q equals r composed with q prime so if they essentially intuitively land in the same fibers over r then they reindex y to the same thing so y q I'm just going to write this for the we reindexing so why sorry why should be from omega prime yes thank you sorry that might have been confusing earlier so indeed we've got a family up here so a random variable up here that's supposed to be sort of invariant with respect to r so this is equivalent to a right can I push this up now or does it still need to be there sorry well I know but I want to complete the the whole point of not starting there was to fit the thing on one board so is that okay to push this up now just I'm sorry about this I didn't right so second property so in a sense we're saying that y is somehow invariant on fibers over q intuitively so in fact this has a probabilistic statement which is that y is almost surely surely constant on almost every fiber q and what this means this uses the concept of regular conditional probability which exists because we're working in a nice category of probability spaces but what's the what is the relation of I mean what are the morphisms here because you work with things with with a probability measure always so then you work with maps module the sample spaces carry probability measures and and the and the maps in the category of sample spaces preserve measure but the random variables themselves there's no we're not we don't have a probably chosen probability measure here so these are just the real measurable functions lost everywhere in the confers condition yeah yes um so what this means is so we have we're using here the regular conditional probabilities which I'm not going to define give the full definition but I'll just give the intuition which is we've got the map here from omega prime to to omega and so we can view this we've got sort of omega down here and we've got omega prime up here we might have a little point here which has the fiber p to the minus one of omega up here and the regular conditional probability probabilities say we have um for almost almost all omega in in the base um a probability measure borrell but it doesn't probability measure p which I'll write like this a can sort of conditional probability measure um so r equals omega can on the fiber p to the minus one of omega and these work in such a way that you can get for any borrell set s up here you can get the probability of that space by integrating the conditional probabilities here over by varying the omega down here so for any but any borrell s in um which is a subset of omega prime then the probability up here which I'll call p prime of s is the integral over the omegas downstairs of the conditional probability measures um on that fiber all integrated with respect to the measure downstairs which I'll call p okay so that's the intuition here using this we can state the probability of y being almost surely constant on on every almost every fiber of q and a third equivalent now and this is the reason for me taking it valued in r is that the conditional expectation your p is not an r it's strange it should be r minus one of r this is uh oh yes sorry so I I had p in my notes but then I thought when I write on the board my little p is going to get confused with my big p because I don't write so carefully so I changed it to r but then I'm now running into problems from that so um anyway so the conditional expectation of x given r ranges over the base space so this exists which is that this is a measurable function from omega to r this exists there's a little issue here that normally conditional expectation is defined assuming that um x is integrable here it need not be integrable but it doesn't matter using regular conditional probabilities you can make sense of conditional expectation more generally in this setting I don't want to go into such niggles in any case it exists and if we if we take that conditional expectation oops and we've re-index it along r then this equals x almost surely but that's so that's the lemma and this conditional expectation is so this is the map from omega to r that gives us the random variable which is going to be the amalgamation again amalgamation is not a really it's like a descent property really because we just got the one map but we're um so it's anyway it's the desired gluing of isn't your x a y yes the x is y sorry oh i'm really sorry that's and the the last the last formula you mean you compose it on the right by r so this is a composition which is also re-indexing in the pre-sheaf um so the consequences of this is for every for every polish space a rv a is a sheaf and also for every sample space polish sample space omega y of omega so the is a sheaf so so when we've shown that the atomic covers form a topology the topology will be subcanonical but in fact they're therefore canonical because it's an atomic topology um so these follow because every polish space embeds in the real numbers so that's that's quite easy but we needed to use the real numbers here to to use the argument by conditional expectation right so we've got the sheaf property um so i don't i don't i didn't really pay attention to what time i started and i know we're running late anyways so um and then questions seven minutes and then eight minutes and then questions yes okay sorry but here's one question just quickly so it means pretty good that not every sheaf comes from random variables and then so random variables who are nice no okay well for every polish space a you have a pre-sheaf of random of a valued random variables and that is always a sheaf that's that's all it means so what i was trying to ask was are all the sheafs on your side random variables with respect to some porous space no of course not i mean they're going to because we're going to have a whole sheaf top-offs of things and they're going to be random variables over polish there's no way they're going to form something like a sorry why of omega is the nadir applied to is the representable of the representable pre-sheaf generated by the um generated by the sample space omega right so i i had high ambitions for the rest of the talk i didn't realise how slow i am on the backboard and how and how dangerous it was to depart from one's own notation um so so maybe i shall just give a very high level summary of what i was going to say so firstly that um well we have the condition that means that the atomic topology is a topology um so actually maybe this is the more important thing to go into a little bit more detail but i'm not going to say as much as i intended to so we identify well we know we'll we'll say we say that a square um a commuting square in p um so the problem with using omega for everything is you have to put a lot of dashes but um so let's have r r prime um q prime q or something like that um we say that it's a an independent square if so we can consider so these are measure preserving maps between sample spaces but we can consider them as random variables in their own right so if those two variables so q and q and q prime as random variables are independent conditionally on the composite random variable down to the bottom so let's call that rq so this is conditional independence that i'm not going to have time to define but it's a well-known concept in probability theory so this i say that's an independent square and the proposition is um for every cospan let's keep it the same as as above there is is a universal um independent completion by which i mean we have r and r prime and over it they can be complete to a to an independent square well let's call this omega prime tensor omega double prime sub omega so it's kind of like a pullback but it enjoys the pullback property just with respect to conditional just with respect to independent squares but nevertheless that characterises this up to isomorphism so essentially this is constructed as a as a pullback um in the category of I mean it's a bit fiddly really it's actually quite a fiddly construction and you need to take fibre wise products of regular conditional probabilities up here in order to get the the measure on this space sort of integrating it all over the conditioning space but anyway there is such a construction one can find it in also in the literature though it's quite hidden i've never seen this universal property mentioned before but it's that's just really uh an observation um sorry in which book you mentioned which book you mentioned for that property for this property though it's it's in a paper by somebody called Ernst Eric Dubbercat from about 11 years ago which is um anyway uh and the same property is considered in Fremland's book on measure theory but he doesn't he doesn't actually have the existence result for polish spaces there but he's got that he's discussing sort of general properties of this construction anyway it's a nice construction and it gives us the property that every cospan completes to a square commuting square the something or a property I always forget left or right or what have you people call it but anyway it implies that the that the atomic topology really is a growth in the topology um but 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 that really nice universal sense was going to give me a general co-end construction so we now got the category of sheaves over the samples sheaves over over p there was going to be a general construction on this as a co-end which um gives a monad m that has the following proper so it's just defined purely category theoretically but it has the following property that um m of random of the sheaf of random variables of a is isomorphic to random variables in the space m of a where this is the giri monad of um of uh well that assigns a so m m of a is is the polish space of berell probability measures which has a nice polish space of measures on a so we recover in a sense the giri monad in the sheaf category via just a general co-end construction that I'm afraid time is is running out because I want to just say something about where I want to go with this so I didn't so so my point is that sheaves of p I think is a nice category of sheaves within which so probability concepts probabilistic concepts live so we've already seen so random variables we've seen um probability spaces that's the representables we've seen a monad a monad of um probability measures in a sense there's a there's also a an equivalence relation well actually between on any object of the topos it's defined which is just identifying those elements that reside within the same atom in the lattice of sub-objects so this exists always for an atomic topos it has nice properties in this case it coincides with equidistribution of random variables so we've got an equivalence relation of equidistribution it's a general property in the internal logic of an atomic topos that if you have a sub-sheaf of say of rva here then for all x y in rva if the equivalent and they one of them satisfies the property then the other one satisfies the property too so this says in a sense any definable property in the topos of random variables is invariant under equidistribution of random variables so that's a nice it's saying we can only state probabilistic things probabilistically meaningful concepts in the topos in some in some sense definable is important here if we start sticking free variables in morally speaking then then then that property is going to go so this has to be a sub-sheaf sorry the the property needs to be a sub-sheaf it's Boolean of course because it's an atomic topos but no that doesn't satisfy the axiom of choice but it does satisfy we do have dependent choice and the proof of this uses so the proof of this is basically uses Kamolgorov's Kamolgorov's extension theorem it's very different from the solove model so yes so i mean like it we have we don't have axiom of choice but we do have dependent choice but this doesn't satisfy that every set is every subset of reals is measurable this is doing a it's in quite a different direction because somehow the important thing here this is like in David Robert's talk he mentioned the permutation models this is very like those in that you have um the these the random variables will not live in the cumulative hierarchy of sets so they they're basically random variables in this topos are somehow atoms so what i think all this stuff together actually 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 this topos but i've been thinking about it more and it actually seems to go quite a long way so i've developed it to to some extent and i'm really quite excited by what's coming out well moderately excited let's say so anyway it is it is kind of it is it does seem that anyway it'd be interesting to do so an interesting setting for probability theory and there's um probability theory question mark um and this connects with this other paper cited in my abstract so in the this volume of millennium mathematics visionary papers there's an article that's cited in the abstract there's an article by David Monford there in which he's saying it would be really nice to have an approach to probability theory in which we do probability theory with random variables as a primitive concept so so my idea is that this topos should actually provide a model for that kind of endeavour um but working out the details is is future work so thank you very much yes yeah yeah you have one question you will mention the reference for this uh the existence of the product you in fact this doesn't catch the name of it yes right well so i mean so it seems to me like a fundamental friendly friendly you mentioned the textbook by fredlin yeah yeah so it seems to me like a quite a fundamental property but it's curiously absent in the probability literature so in in fremlins measure theory if you look at maybe he calls them relative products he has a a two concrete definition of what a relative product should be with some statements about they don't exist in general and some existence results but none of which are exactly what i need um so the same property was actually needed in computer science of all places which is where i where i came from so i'm aware of somehow the computer science literature and again it's not quite in exactly the same form but the i mean there's a bit of history there but the the nicest exposition in computer science is a paper by dubbercat Ernst Eric dubbercat um i don't remember if there's an r in there or not um which i can i don't remember the title it's from the early maybe from 2004 and again he's working in a 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 there's some history to that that actually i mean the construction is not terribly difficult when you have the right results about it are you just saying that almost every fiber you take the product probability space and then you just integrate this over yeah oh my god this is just the naive yeah so up to analysis question problems it is clear what you do so the question is to is some conditions to the analysis it is indeed but you need to have it working with the polish space structure up here which itself is not easily found because these are measurable maps not continuous maps so we're taking a pullback of polish spaces using measurable maps between them but then one can still one can actually refine the polish that was the so-called standard yes that was the commutative for normally algebra well so so in fact one would the whole so i should have said at the start the whole choice of category in fact um so i wouldn't want to work standard borrell spaces are the ones in which in which the borrell the borrell sets are in bijection with the standard lattice of of the the sorry the measurable sets are in bijection with the standard canonical lattice of borrell sets i do want to have the point measures and things like that in here but actually a very natural category to work with is the category of um what's called standard probability spaces here which which embeds this category but in fact is equivalent because of the identifying maps up to almost almost every very quality the important property is that uh with uh when you have a measurable map you can always discarding a null set yes and reduce it to a discount countable union of compact visible space in such a way that your map becomes continuous right that's the classical property of right so so that that's also might help with this property but in fact this this sort of pullback thing can be constructed just directly using the the the structure of of the so even without using properties like that you can just directly use the countable presentation of the sigma algebra structure on the on the problem yes yeah but the yeah but with that one point i should have made at the start was that that as soon as you identify maps up to almost everywhere equality so very many different categories coincide with this one so it's a very canonical so it's also it's the it's the category of um it's also the category of countably presented measure algebras opposite it so that's a nice thing it's not a question but just a remark from the point of view of teaching probability which is something there it's all right it will be very difficult to teach i think i like it because usually we have major theory at the beginning and then after a point we tell the students yes we have an omega somewhere but we won't really look at it we just we compute expectations and so on and sometimes i have to remind the students that when you define a stopping time there is an omega even somewhere a small omega and sometimes very good students will ask but can you give us an example of one small omega in a computation of t of small omega that you know we never look at this we have the omega in a big omega in a corner and we forget about it after a point and that here you have lots of basically because they're all the possible omegas i think it will be it's quite a change of point of view i think it's very nice so but actually the omegas are just here in the construction of the category theoretic properties the point is once you do this there are no omegas this is so no omegas at all neither big nor small so this is actually one of the one of the nice i think one of the nice features of this approach the idea is to basically use this to together with the calculus of independence of conditional independence on it and so you let us begin the beginning to prove that this category exists well in a sense you can but otherwise you can work axiomatically and just so that's the you know what the axioms might be like can we characterize all the random variables as certain objects in that purpose you said they're almost the atoms is that what you said because that's what i guess that's what the question is after you you would say let's just go to this dot what the non random variables are such and such well the random variables are not the atoms so so i mean my idea is that you wouldn't characterize them you would add the idea where you would you would add the object of random variables as a new set in set theory and then so so you would say so you'd say sort of let me have a random variable x and for any random variable and any other random variable you can find another one that's equivalent to the second independent of the first and and this sort of thing so there'll be these kind of conditions that will fit together with dependent choice to allow you to to do more elaborate i mean it's that kind of it's that kind of reasoning but we're going maybe a bit off topic here but that's this equivalent this sub object lba lba cos lba well you can just think of it as the sub object of pairs of of random variable so at omega you've got random variables from you've got functions from omega to a and you're equating those pairs of of such functions that are equidistributed so that's equidistributed so induce the same probability law on a on on a so it's an equivalence so on random variables x is is related to y if if the probability law of x on on a is the same as the probability law of y on a so the probability measures they induce are equal but the point is it is actually it is just the it is just the instance of a general equivalence relation that exists because this is an atomic topos it exists for any atomic topos so it's a this equivalence there is an equivalence relation that exists for all objects on on every on every object you have this you have an equivalence relation due to the fact that we've got an atomic topos if you have an atomic topos you have such an equivalence relation and it just so happens that in the case of random variables it is a very meaningful thing so what is the general thing for atomic topos um so you have a sheep on an atomic topos and then well let me so so we've got so we've got a sheep f and we've got let's say we've got x y in f times f at omega and you can so one way of saying this is that this is so this this holds if and only if there exists um q so there exists some omega prime and there exists q in q prime from omega prime to omega such that q such that x re-indexed under q is y re-indexed and re-indexed under q prime but also it amounts to it amounts to we look at the sub-object lattice of f and it's atomic and we're sticking x and y in the equivalence relation if and only if they're in the same atom in the in this atomic in the sub-object lattice but of course here you have assumed something because omega prime empty is uh is uh no but both empty no omega prime is a little empty it's probably with the spaces yeah sorry this home this can't be empty there's yes the problem that there are probability spaces that's got to be are you working in the topos or in the side yeah this is in the site this is two elements of the sheaves externally are in the equivalence relation i'm defining the and defining the equivalence relation as a sub well it is a sub-sheave anyway of f times f okay if you consider so if you want to work with the topos you consider the site of atomic guys in the yeah top of something like this and do this condition okay well okay i mean intrinsically can you describe what you do in invariant terms sorry can you describe this in invariant terms without using the side maybe in i'm not sure no i mean you said in every atomic topos well okay but so omega is an arbitrary object in this in the in the cat i'm not using the particular site omega is just an arbitrary object in the category that we're forming the atomic topos over so this is this this is no but here is not a good definition because you can take omega prime to be empty so i suppose no but i suppose that you take for that no so but there is no empty probability space no but you are you sensitive make sense in an atomic topos yes it does if you've got an initial object in in a category and you put put the atomic topology on then you get then you get something you just get the category of sex cover covers in this case they can only go from it cannot be empty atomic topology is a very trivial in categories that have initial objects if i would you suppose that all maps in this side that a piece anyway right i don't well in my case i don't suppose it it just it just is the case that's not necessary for atomic sides okay it's not necessary for an atomic site but if if you're going to get if you if you if you if you want the topology to be subcanonical then they better all be epis okay yeah yeah this this looks very cool and i'm very curious by the you thought of some calculus based just on primitive concepts of random variable and independence where you could maybe like derive like central limit or strong or strong large numbers or something but i'm really interested in this moment so the space structure on the space of world probability measures i mean what's like if a is like the best space what's the dense set of probability measures on that sorry the dense set yeah because the polish space needs to be separable right so yeah but the space of probability measures on the polish space is itself a polish space yeah so that's what i have to wonder what is the so this is a natural topology yeah yeah you put yeah you put the weak topology or the vague topology as it's sometimes called on the space of burrow probability measures and indeed it's a theorem that if you start off with the period space you get a period space back and you can find you can find that one in many textbooks on probability theory and measure theory so um yeah uh so you said probability theorists don't like to mod by null sets when defining the maps yes uh category theorists don't like to take motion see there because he was doing it yeah so could you not do that but instead say like two cell between measurable maps is a null set outside of which they agree and do some higher topos version of this um yeah i don't know if there's an actual higher thing to do there so to avoid the modding by null to begin with one can use the the measure algebra presentation instead in which case that automatically does that for you so that actually that's a much nicer way of presenting it but maybe not to you could probably do von Neumann now suitably well chosen von Neumann yeah um yes so i mean what i used the fact that i mod by zero to get the sheaf property because the conditional expectation is not defined otherwise i don't know if it's crude i mean i feel that something would fall down if i didn't mod by zero but whether there's another way of doing it by bringing in higher category theory i don't know and personally i'm not going to look at that so um yeah so do you have any clue on whether this topos has a point or not ah excellent question so um so i actually had a conversation last year with Olivia in Llewmania in which i conjectured that the topos has no points and she was very skeptical and rightly so and um this this meeting i've been spending most of the meeting trying to calculate what the points are and so so i have i have a conjecture but not a result and if anyone would like to work with me on this conjecture i'd be very happy to have a collaborator so i mean one strength of the conjecture is that so this is uh this topos is actually kind of a countably distributive topos in the sense that there's a distributive property between countable limits and co-limits and in that context it's natural to look at inverse geometric morphisms that have the stronger property that inverse image functions preserve countable limits um and i believe that the points that have that stronger property are exactly the non-separable non-atomic measure algebras i believe there's also a point which is the separable the unique separable non-atomic measure algebra which is not a point that preserves countable limits i don't know if there are other points either but i mean my conjecture is that those two that this other one is a so basically non-atomic measure algebras are points i believe we only need to find one point for this kind of toposis if you then want to then exhibit it as a category of um continuous G actions for example maybe you could try a logical analysis that could help maybe finding the points i mean to to start from theory of appreciate type and then that would be homogeneous models for that well that was that was how i got the conjecture about the um the countable limit preserving bonds but but yeah yeah i have slightly naive question about how random variables and probability spaces would look like internally i mean you kind of touched it but but um i still don't have the picture on how it would look if you have a statement like x let x be a real value random variable how would you interpret that internally well so i would have real valued random variable as a basic thing in my in my internal language so i would just say like x be in that but there'll of course be a relation between that and and nice subsets of of the space so so you're going to have um well it has probability relation between random variables and the and good subsets of the of the of the space a um and there are going to be some other things too and also so so this conjecture about points so the the model in the in this toposis are random variables of the two elements set which is internally to this topos one way of looking it is a um so random variables of two carries the structure of a non-separable non-atomic measure algebra here so so that's some internal structure that exists there