Operator algebras from toposes

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Operator algebras from toposes
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Abstract: Grothendieck Toposes and C^*-algebras are two distinct generalizations of the concept of topological space and there is a lot of examples of objects to which one can attach both a topos and a C^*-algebra in order to study there properties: dynamical systems, foliations, Graphs, Automaton, topological groupoids etc. It is hence a natural question to try to understand the relation between these two sort different object. In this talk I will explain how to attach C^*-algebras and Von Neuman al gebras to (reasonable) toposes, in a way that recover the C^*-algebra attached to all the above examples.
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the end you so that the Jarai the that's all in the same way that that was his or some kind of generalized to political spaces daughter brawl soul object that we want to think of as generalized toward cost-based optimizer locally compact space and there is quite a long list of examples of geometric object at good in generate either at sources daughter Russell for example formation of dynamical system and graft certain financial demands and and so on and so quite natural idea to try to understand that there is some marginal procedure to go from there was serious talk .period order brought all the the other direction and the 1st Way to answer this question is to note is that both are closely related to the political were Boyd so I wouldn't use that aspect concretely Michael but did somewhat told us that the easy directions to go from the post was daughter about to try to attach a starter Roger would oppose and that's what I'm going to explain to them how we can do that why the ozone direction is at least very difficult and more probably completely impossible without additional structure and algebra so all adjustment explain a bit what I'm talking about soldiers to satirize just manager brought with and then on to 9 Mauritian satisfied the sections of the motion will be noted by stopped and the typical example of the algebra of bonded the accelerated does basis for example the birth order brought a house or in this case the objection would be the Zastava Archer will just be the junction off operator and of course you the example of Community larger rock sole under blobs of France's 0 extra for X looking compact of off political space it's 0 exist for the other Rolf continuous function of complex value which tend to 0 at infinity so in this case the star will be the complex conjugation Seoul and it's a very testicle result that I guess we all know that all committed to assist on rock artist form and you should define morphism parties in your equivalents of category between the Caricom backed the political space stalled off the political space and committed to .period are so let me give a 1st baby example of noncompetitive algebra which is another interesting but which will be helpful to us that near the end of the book so we will start from the political space and we will assume that the the the hoovering subset which all locally compact and although that AEK's might not be 1 of the officers so for example the typical thing I have in mind is the start from that the political space thanks so for example the and a lot of time 1 minus 1 its compact in this case but
Weaver but the bullet points as the region's so in this case you of X which is able to Union 1 you program was sizable freak June the closing Delaware but the intersection the please Delaware at 9 0 0 region so you have to origins so in this case you can associate this is not a discovering so basically you construct some kind of magic underground so it doesn't have as many role as a member of opens etc. using and as a coefficient are given by function of C 0 the intersection his Road intersection of the corresponding opens at that and if you look closely to that you'll see that magic petitioners was defined for this and this former Nigeria so you can define devolution by taking most complex condition and exchange in a row and column and that is that there is a uniquely defined on that make this into a distorted and you can do something to realize the covers in finite but you need to do a bit more amenities as the Matrix product of his metrics will be defined general so and something you can say that if X was actually although itself then Edgerock Brock is more a type with an to seize 0 that's all define what more like most men forces eradicate of it's very close to Radikal of his soul for example in this case he this construction you will get function from the end of our 2 2 by 2 mattresses the that of diagonal coefficient vanish and 0 so specialized at any point you will get to buy 2 mattresses which are worth aggravation to see so he basically gets 1 point for each the 0 the of this but if you guided 0 you get the square which is correspond to the two-point space and see you can see into the enterprise in the book .period had original so this is not a very interesting example from the point of view of competitors from attributed it can show all but I'm committed to rock and someone those states that are not the answer falls off soul given more interesting example to take basically any Group 2 disparate group acting on the locally compact although after political space then there is the construction of across from a cataract so is generated by the functions of and that's 0 Rolex and by formal and corresponding to the of the group which satisfied classical arts vibrations solar generator from the group the by report raw India in Uganda exchange besides news site between function and generator from the group using the action from the Group of the space and so you have again you can put the norm on these can generate assistant role actually you don't have a unique number you can choose 1 geneticist erupt and the what happened that's if the action of GTE's free and improper and then decided abroad is actually more equivalent To his Adderall functions of caution so it's the same idea that if the cushioned by the action of the group went well then you actually don't get anything more we should see the starter Robson Radikal but in most situations because this is across political minerals more information than just the abroad over aggression which in general which revealed this is 1 then you will get to the bottom of the group of people with some kind of 2 more of revolution of our so it's a good 1 will give the seized on your list 1st of all this businesses should you have 2 choices either you can take a year or long which is now in the morning of old norms that make room Superman of all known of coming from all representations all you can take the long coming from the arrest triggered our representation on the tour that's 2 different interest in general you have sometimes intimidatory choice of so this situation of a group acting on the political space again sausages at the water so it's a doubles of equivalency that has been introduced the that of the scanning site this morning so I and so it corresponds to the cushions of the deposed by section of the groupings of 2 category if that was his and people so idea that this is still pose of a crime she will be equivalent to the post of chief on the question basically the also is the action is trampled also it's that we have the same the same situation Is the boss that's because in this way is actually that posts of supervisor question then Diaz arising type of run the edge eruption the question so this summer suggested it should be possible to define the role of summary die directly from the book and so I think that's what's that's what we are trying to do so the 1st point for this construction is that using internal logic of the policies that we have a very good notion of continues to off in that space into starter Ryan Minor space over there so we call them continuously because that's the historical DeMille but morally there are things that we would like to call for the lifting of space chief of the space and she wants to start a branch of a ban on space and concretely they are defined just as a young adult spaces starter Brian Bannister a subject of the internal logic say explicitly what it meant in a minute and so on as the internal logic of the deposed as a Ontario explained that unless there is a general interest as technologically need to be on a bit more careful what we call into space because I mean you don't have the office indeed although all the definition in all 3 of them basically all different objects so there it is a lot of things that need to be made precise so firstly that the conviction Bowie on Monday when the ideas as scanner answering of Scott was itself Michael space really cost base in general so we use what we call the continues remember and continues compact thus also called King the real number and the president side decided that it can get so we define a real because it decided it can go so but there is a subtlety that norms cannot take value in continuous number because there a little programs that don't satisfies the supremum peritonitis said Bondi subsets of continues with me on Monday does not always supremum and they need to be defined as supreme on general so weird see freed .period known to the valuing continuous numbers and minnows lot of example for example the other Robinson the birth of the Indus space won't be about 100 Robert restricted definition like this so we did what we call a person he continues remember which offer but that they can get so one-sided that can get and those of to by property but they have also bemoaned so we can define normally and they contain the continuous real numbers so all lack non-compete defined without problems In the last point we need to make precisely what's completeness Maine because it has been about aspirations .period running to be complete so we use a decision like cushions or because pushes sequence doesn't work doesn't work for you very well without them toward the production of trucks yeah and I salute to mention it's written that space and unlock continues because they come from of biting off Allman to write down by narrative we need to take valuing continues complex continues complex remember but also continues complex number but unions so when you defined everything like this and you look at what the burning of fossil bubbles of Shiva wrote portable space you get the previously known motions which are watching the order to fill the mandatory Of all continues field of space and what we call 70 continues field of the starter Brian Bannister so now we can make for example speed of space objecting to a category morphism between them will be bonded also 1 that mean soldiers are now not which made in a joint because without the office in the low it's not admitting the truth that bonding I'm not between MySpace joint and go bonding just mean that sold out internally Bundaberg doubled the numbers bonded by an external Constanzo goodbye defined constant so if we have achieve few that space over to Portugal spaces means what we think it is at the enormous globally bond and all the space and not just look at there which would be the meaning of internally bonded and so the result is that this category of the of space and nobody bonded Alberta between them is what the cost stock so it's a several objected rescission of his daughter Ryan roughly what it means is that if you take 1 of those who space and get sides are often the movies and it is actually started because of this yet considered estimates so this producer 1st Way to take a double as an associate all bunch of the start rock but which is a good starting point but it's not exactly what we want others to problem the 1st is that all those about that reporters are not worried I provisions much of related to each other or look as precisely as we would want and the other is that if we look at all the example we had of a situation
where we actually want to attach a to the the don't give the greatest tantalizingly larger was touted to be generated with his start brought sentence about A-Rod's also that there is no law apparently no nice way to recognize them so in and there is a we get the other rise actually very natural to understand so we should get get the other or or browse over the space of a few dimensions then this is the extremely complicated to start about which is not married I couldn't a complex number it works if you have all shown a finite dimensional best with something more it I couldn't to collect number but find the individual case what you want to do is to look at the ads Ralston Bhaktapur at Witchazel son or Barroso which can be approximated by finite rank operator and so this is roughly why we get way too much amount in order of because we are looking at arbitrary overtones investors that can be infinite dimensional so I'll make you a small burned this time so today I am concerned about the what I call the topological gays with his daughter rock respond to look I come back to Portugal space this there is a completely porous story which could make another thought in which all summer in about 1 minute so there is also the serial fund manager which are specific to start around which correspond to measurable space so Community Fund Manager rises something the floor an infinity of new for some measure of space in new measure which is not a role almost a bonded function but resisted moment With the essentials of but so late and the point is that the is often the manager Rock and it's a good fundamental rights more than in the sense of moral fundamental rights nor a type of see so in this case generated which shows that's why the story works better individual cases so if we take a born under Bush's Mendes's docket agree we get by looking at you that space was deposed is what we call the moment and complete the stock category and with so it's almost the same as the fundamental which are W. stock category and relieved reasonable assumption of existence of local visuals born and oppose and we actually gets the respect it was so we get from the manager brought this way so and now the point is that any doubles admitted covering the by the way onto busses and if we choose a covering by wouldn't the postage at the Met is injected we can actually seen dazzles space overseas would uncover as enjoyable field watching the space was a balls and this is way that to attach fund manager right to any deposed by choosing a bland covering which correspond more into a some sort of Janise new and we get a measure broached which can be thought of some kind of generalized as brought measurable functional roles at the post the Soviets this atrocious visual 3 to get brush can be applied there To basically any divorce but uses older all the information of the political nature so it's probably interesting for all the talk was that come from Margarite geometries so it doesn't oppose all the scanning side or and also to deposit our testifying to pose for 2 reasons .period you right but I don't mean In the comparisons with other dodgy right in nature the workers boss replied right apology and Xabier fight for example 1st spectrum and we cannot expect to have anything interesting by looking at continues function among them with complexity so let me come back to the October logical space so this situation the idea that we once restricted to positive that have nice the protocol providing abounded was to look a compact housed off to political space so let's forming will look ,comma battles of the political space so it's context and assume we have a you build space of ranks which is denoted by age so we can look at them again now that she a set of global section of installations issue so she went on the structure of an about-face intimately and so it's obviously come with a structure office Yonex modern West elected the article or complex that it function Onyx so but because of age is you don't space objects also of scout product that can extend the when on global section can be seen as a parent should take 2 cross-section of again outraged then you can compute their scalp product and it's going to be a complex valued continuous functions on this basis so you get some sort of a scout products with value the and then you can consider space so give me again I 0 of age all 1st sections says that the the norms of the scanner products with themselves is actually in seasonal sold with a 0 at infinity and this is what we call rights you that's Euronext modules lower right to pursue what it meant seoul because his Rolexes community there is no real difference between right and left module but so are like 2 definitions of what from unconditionally but with what precise investor rights in this so 1st it's a rights is erotic module also he died this scarf product which stake valuing the related check that if you have to women's Indiana 0 then because of British Walter Scott product is also that was all we have also had right actually so it's been on the 2nd and I actually mean in our additively to an issue that needs a viable than this you applies a star operation so it will be a symbol on on the 1st viable so when you observe 1st free reaction then this actually find homes on this space and we ask it is complete for this the lonely usually means the it's grown over all but 1 of these yes yeah yes it's a phenomenon added Russell existed long close to the people of Hong noted this is external I was thinking were the norm of the start of the World Youth shows this is the because this is the most desirable that there is also the reason was willing to work if I have the choose Provencal wanted function for example but for what I will states in 1 2nd it's important to us as you so the bonds that we have an equivalents of category between the category of it that space over X and of that's 0 next module wife we have chosen to take bonded function then will not get some bets the continuously lived in the space of roses stone church compact invocation of the space In so what is this is actually the kind of resolution was the way we want to attach study to articles we want the remedies for and saw that there and I'm going to talk about this whole stated 1st and then explain what that meant so it's what I call that a poster at a gradual Theron so we start with the deposit satisfy a suddenly stuff condition that I'm going to expand in a few minutes so it has to be separated this to be your carrier decidable instigated perfection has to be you can come back then we actually ever since .period about the wilderness cheap said that the category of you that space of equivalent to the category of events is rose to modules for most of the no space with Scott products but taking value into rough cheap and so this is were defined 2 Marita equivalents of 2 unique Florida equivalents so I still undefined what's technicians forces thought about that no I can do it To start a bomb or a takeover that category of the Dutch modules microphone and there isn't a history as sorcery of rain we can describe as assassin by module and so on and so this last statement is completely topological because I've described the category of the Dutch modules the other Russell it's completely useless right-to-die actual so then so yes so this is isn't there and I'm I want to talk about this this nice way to get us to start a roar from the sum of its form certain deposited by nice the portico commission song going to expand this Commission but 1st I want to mention that there is also world reversion which coefficient but so we can also take the to be he started brought over cheese or to study ways in the logic and then we Allergan constructor and all their resources testicles .period Ryan said of this season the director of the those either category of death would seem module over Cheney's equivalent to the category of that the crossed the module these the man was there is a real crime creates the external on right and they unique of 2 unique Marie diagrams for the exact same reason why my will focus on the version without conditions because of proof is basically the same With whether use coefficient a lot
I was so let me explain what's was but those conditions separated
UK decidable Maine stop
decidable so an object of the bosses say to be desirable if we have generally that for all X Y and X we have exit y or x equal to 1 which is a form of Norfolk Southern middle which of course does not hold in another treatable so externally to mean that the initiative for the objects expand exiting the decomposing took a productive tombs of Shields of objects 1 which is a diagonal and 1 which is lot something as a complement of diagonal but also for example an issue of issue although locally connected to political space then these decide ability conditioned responses the analytic continuation properties of for example a sheaf of or conscience of all senior over some complex manifold is decidable so decidable object are important because that's the 1 where you can actually count the amendment or the consumer index of rose and a body that is shown at a level to their lead given to women and I call on all different than it's completely impossible to calm the amount of the song indexed by the elements so for example if you want to define a zeal but space and 12 for some settle some object with the post X well you need extra decidable so either because you will define it as a set of questionable sequence and to define suitable unions is said to be desirable or even more clearly it will make sees space which has 1 generator for each element of X's and O's generator satisfy this condition that desktop products please equal to 1 if the article and 0 otherwise In although this this makes sense to me the said to be decided well that's exactly what it says so we said at the bosses look any decidable if you have enough decidable objects so if every object can be covered by the decidable objects I also look at for example will that was what was achieved although at a political space automatically decidable or it unusual to poses over equivalency on .period also located desirable and if 1 Conter examples and pursue still pulls out most of the time not to carry decidable but also the allocate decidable exactly always Yaro of the corresponding category item his so sources some actually very With British industry like from the point of view of uncommitted tree because most of the time when we want to attach to start a right to know Citigroup along category we wanted to be left conservative which is exactly what this condition of the morphism Maine and yet also I actually don't know any example of doubles which is not the case excitable and which as you space that don't come from it's looking decidable pots so I actually think that this construction of looking at it space over the divorce is only interesting for Oconee decidable but I haven't been able to prove it I just stumbled to little more examples so this was for the look decidable bottle it's a radio weak condition that is absolutely not most important the and the important conditions separated so it's a condition which is due to more I can summon but also the geometric morphism between 2 say to the proper if of internally in the sewer along every and explained yesterday that when more feasible you can think of it as proposed in the indemnity Article which is morally the idea that you want work fiberwise as if you want to work preserve the fiber by fiber and so we say that a morphism his broker if this intelligible as compact welcome back and look any profession is compacted so it's just the usual finite growing conditions so it's basically will saying that something's is about it's finalize combined with you know look any uniform way and felt political space so completely equivalent as usual definition of upon and so we say that the Bush separated diagonal Matisse proper which isn't averse to his usual definition that something is separated into their skills with the exception that for purposes as a data and does not doesn't have to be here and inclusion of soap opera and come back and closed became different and that's why we're using proper which is a bit of condition that India social topological spaces is almost the same as the ordinary also conditions by slightly stronger as a reason why the differences because the product of the policies is not exactly the product of the political space but for example if your are restricted to Republican-backed space and it's exactly the same and all the chief 384 and was positive conditions implies a certain to posterity to separated conditions but so far locally for the worshippers presents a very reasonable some condition before general deposits and extremely restrictive conditions so for the divorce of sets and we've action of geez and that the policy separated if and only if the group is finite and Hugh consider a process the Portugal group then it means the group compact so it's the restrictive and for general reported the correspond to the notion of a pop it's all reported was action on this program In so this is it is a very restrictive condition and I will explain at the end of the Togo can't applies it's a into non separated divorced in an interesting way on focus
on its comeback to fair and saliva the you what separated and look at site it will mean the end of locally perfection is locally come back to the floor and Princess overlooking perfection is just that so it was 12 you have the inclusion of the category of recounted a category of 2 posts in the desert Leicester joined with the it's right it's not all the Eldredge isomorphic 2 the morphism from the reflection of the 2 questions at the boss and his lookout and for example if you are considering tube was attached to a dynamical system as I mentioned mentioned the beginning when the Saugatuck reflection is the cost of the vertical portion value consenting Spain by the action of the group as the portable space so if estimation is also vary because for most interesting example the organic profession will be actually simpler and there were lots of presents space we are starting from seoul ice-covered regional fear and because the range of and is something that basically saying the same thing for .period instead of all problems of the weather's so for program believe we have an agreement of category between in that module although reduced to start a rather employed and continue sealed off in the space of Rosenbloom avoid With an action of .period could I am continuously within the space of a . so the point here is that because they're that most of their energy both for Boyd fall the boss we know that's when the ball supplies balls Jesus and results because they all want to marry diagrams we have the same category of India models out so this is what all that this addresses worst is a correct has a right to actually all example that we know so there is also a constructive version of the deposit agrarian so it made it to the Senate just need to add some Hagerty around their souls separated status look at decidable world we could give this and that we can actually weakens the conditions and only that the category generated by decidable objects which is equivalency physical loading which is weaker in constructive logic on the site of the Commission close this year says he's he is less than all of the economic but in the case of the they actually were every reader changes mainly to add 1 condition of that profession completely get out which mean that there is an continuous function of its soul in classical mathematics consequence of separated and look at combined by the Roslin others constructively its doesn't work very well and there is another thing that we need to change that's 0 50 cannot exactly be constructive as a society rather we actually did it as a stub category so we can turn it into a sea stars are 90 per cent of the time for example as soon as the base that was used several can decidable but in general it might be a small problem has all point of having a concerted version of course to other related version of the so now we have a dramatic like this which show indemnity satisfied as a condition of the era I can relate the index would you over the 2 the best Samodurov usual subsist on Iran internal to be sold and the reason why I was reading his insistence that mission it's actually wall however sigh I was spending to prove that Iran's 1st Way by using this as some kind of the exaggerate the Weaver and intermediate oppose improving the 1st for this morphism and what is more present In the way that will expand minute but actually I final proof I didn't need to do that so we could prove the classical form directly and also filed explains roughly osetra works Our because Italy it's actually really helpful to understand what is the size of Rice's you to so the 1st parties that have proved that I can extract characterization of stock category which allows the former category of you that module reverses so this characterization is analogous to the fact that the and category is a category of more durable submarine dazzle project to generate fun at fiscal complete so basically Scheunemann category and an objected cited to get a front or by taking on a from the subject objects and from your category to the category of more durables Iringa from the morphism of the object age you assume that the subject is a projection Donato finance types then you get an equivalence of Category I think I should have check that this is true replied sounds like an easy results so I approve this is stuck at American on also in order to have something senior officers that category that there is too difficult step but the 1st is that we need the notion of good completeness which is adapted to stock in figure because well I think and knows stock at American Eagle competency the ordinary categorical sense the anyway the category of best modular system here or not but it's actually possible to um works at the outset nice convenient notion that it's enough to prove that to remove this kind can then we can define an abstract notion of compact also combat tubers are supposed to be those which are approximate life financial prepared or any that given
the combatants can be characterized with some kind of dark categorical property related to this notion of completeness and we can define abstract notion of Timbuktu Brazilian local convinces that ignorance and then it's possible to formulate of securitizations to the of so stuck at area would be a category of that moderate or something if it's go completed as sufficiently many compact Oberto injured as a generator and the proof of the terrain is then refuse sentence for the year a beyond category case so that the 1st step up the 2nd step is that we can it's not really how we can prove that I need to post a category of the space of virtually satisfies stock degree coal cooking that his condition I not make the difference between could compete and compete because in this stock degree or descendants and that's really has a duration of morphism so I actually have no reason To find that competes in different from completes let's think isomorphic Oakham fitness condition In so we can prove that those category of space of a cheese Occoquan Britain's stands and that was all there at the which I indemnity compact paradox and there was no intent to 0 at infinity so it's been under a reflection of shovel continues stamina for graduates non-executive functions of any perfection if it stands as the right infinity than those of birds often back in the sense of the above characterization still there is a difficult thing to which I have done away think more about the joys of truly post-surgical think we need a constructive proof that if we have at the post which is separated located inside the boat and I definitely detonated means a local professions trivial a reduced to a point but then that was actually admitted generating from of object which are finite and decidable so is finite and edible falls intelligence also accumulate which are internally isomorphic to something of the former 1 and to accept rider to win so this is a purely the poster a which is in the paper that I put a marker from and actually at this point at the end of the proof is relatively easy so when can use this constructive results about separating look inside the good and I definitely to post so it's come to this diagram so we have to we have perfection and we this because this 1 is a super-G the height connected and presentable so we can apply the 1st .period until Monday and we'd get objects which are some some of finite and sister pros related to this 1 and finite object all we need to construct the financial rank Gilberto which Orenbuch back Alberta and so it's just switch of easy to by using some partition function compatible on this space to actually construct explicitly and also back berth on the category of investors and we can also from the generator and that is actually relatively easy interests showing that everything looks to be as well then so proved told us is that the center Russell 0 50 is basically says the ads are often Bhaktapur at over something that space of the polls which is big enough will begin of Maine it's a generator in the sense that I had to construct a kind of 1st .period I by compact Oberto lining up in the abstract and sold Gaza which I entirely compact and was known then 2 0 attention this is the season 6 yet securities and external 3rd Sadirov global section of on the way of accelerating the morphism which internally after compact and was known to the rest of the day think that this is a place of NH sub yeah that's repertory the go out of why unions the services the area single there is 1 in the face of the boss says that this is 0 FTE is the giraffe Bhaktapur at industries but that gave her the other classical of the use of the castle of the vision this year after year he added rather derived from severity but with his or her very specific on the hour what I can say is a just political types 100 I don't know always have this we for more stressed but then I will explain how to use this for months separated divorced and again well out of this very specific situations I think this is just 1 of his guests yes we also all the other shows also this is that it's actually a very important if we want to try to reconstruct the topless from this from this is that your we need this money search of the home I will not talk about it today and so we are not yet so if I have time I will come back to the cell to readable surgical was at the end In so I want to explain how we can generalize to notice person that was sold at this time and this result really require separation to work to you there is no way to know 2 were honored in fact they should take basically in the example of the books which is not separated then you can show that there is basically no 1 no back to the verge of format and almost every example you can think of so we really really need separation conditions whether 2 of the regional Fareham watching you
so but there is something we can do but is it the Bush is located all the fault of the theory so buying locally and that we have an object which I would prefer to use as soon as it is desirable and inhabited indebtedness of local fiction everywhere but but because it was a way to protect you and your pick of course the I said that this is a site that was the ranks of the adults space X satisfied I put it is of a theory maybe so we by the overexcited just mean that the category also objects to general director we want to so include so it's a double and said about space of the chief excellent teams the portable space X is achieved then this is a category of she will about space of the ship so it's a space which is no Kenny isomorphic I was about to the base so that's why we all this look any year order form of the fury and so is so this situation into immediately ordered example I know of .period was where it's seem interesting to try to construct a system here a cannot prove this of course but 1 example I knew all this fall and in this case and I wouldn't do it can actually constructed to start a riot that Stewart was which is locally of this fall they think buys using a generalization of the metrics trick I talked about in the 2nd side so we had a situation where we had the space which was not said separated economy compact but which was he said Britain can come back and we well-built upon circuses .period of forming some sort of metrics and of tumor it was exactly what we wanted so that's exactly what we're going to do it so we have altered post which is so we have an object X which as I say I prefer to assume that it's excitable but we assume that Eurex and satisfy health around so then we begin viewed that space in interact so we can think of it as a finally opened the space HEX index by X Germany the goodwill on the space overstated Just a Friend of Bill space and so we take action which is a generator on the face of the rights which is big enough so we can actually has shown that it is the entities consultant excludes actually ended up speaking to you that we have pulled back security from considering importance so if we look at them and the morphism Of the direct soon so we can construct a space which is a direct diets and me Author will also more for this age facing the islamic and on the morphism this thing that I wanted said It's a Mad between 2 direct sounds like another metrics the composition so far so I think we have on then by composing the injection in production that Maddox immense which operates on its front page 6 and so this is something that he had been this deserved to slice the pose of experiments independent to Barnett servers in excellent natural even Sno and I'm considering the algebra of such unions the message that all this borrowing from a text which express annoyance only come back to Britain and surge that's it's 0 outside combatants soap space a list of so I'm some looking at metrics with compatible and so it appears that this sellers also said such a move was compatible naturally from Niger brought that I can for cities city of change and so then I need to construct Normandy but that the critical theory of Roger about buoyed work exactly the same I have several choices these of the maximal Norman Els arranges known as we talked about for group over here and in both cases I can't exist condition was going to be assessed on rock sold at either reduce or Maximo Sistani right 2nd as we have for for solar cell producer on the way to attach this not right to any deposits in look period of this fall which is which will definitely 1 example but is not as nice as we could Of course because for example at 1st sight it's not clear out all the decided rise was the final 2 more died in Sweden that its around and that was what I mean is that it would be better if we have a more universal characterization of In located separated Gates so I will say a word about the proof of the year on this point separately the case recyclable items connected to post security by China to blacks but because it's a really served variously as a most important part of the province reunites results which I'm sure will have all their obligations and considering you go space and started about the process I don't know so the probe was into 2 states so we have at the policy which is the severity of the case decided on an IVF connected so you actually don't need to know what all those conditions to understands prevents coming but also the 1st state is there to solve the case whether that was as a ball so the 1st assumed the other points so in this case there is the very nice results the Kamal died in Finland which is basically a constructive version of
Copenhagen with theory I would say that then is actually isomorphic it to the category of said with so I have my I remind is proved to be constructed the nondestructive prefer this will be a very easy said with an action also look group he Ng is actually a group of automorphism was supported for the automorphism another .period injury of any group so if we were in classical mathematics why then does it this would actually we could actually approve that this was as finite objects as much as we won because as a group is compact look at compact groups sorry and what if you have a combined group acting on said then you can decompose set into his original for it's all as your bid for the action of a combined group offline so we will have all the finite objective there is a big problem with that which is that because this is not a phone sets the political will but the location but if the group does not satisfy technical condition which is called to be open which is always satisfied find physical mathematic by then the albeit also .period actually might not exist outside defined as a of broke out of the sets on which attacked and might not be a so we cannot use the old beats to construct finance above actually fall beating the then we can we are in the case of an intimate was it's it's complete Israel but but there's still something we can do so we've I have X decidable is the 1st to be located inside the please election of genes if I take and element the set then because move all of the Ruby scumbag nice to know that 0 Beach which is not a said but still makes sense is contained some finite set I just cannot construct actually 2nd stable and energy my exact a arguments I can also constructed this prime so that's if I look at the action of G on as this is included into some other financing and then at this point there is a really nice combinatorial organ and so it's still true that constructive mathematics if you have to come back to account Kintyre Prime says that some kid enterprise is empty was back then plays look at prices and something that is still true constructive mathematic and that's what's savers because basically everything you have appointed as prime then you if you try to look at the set of barrel filaments Ng says that the G Dynegy prime-time X is amount it will always be empty because it would be geobag so it will be so there is also a finite set into a decidable sensors are decided will we know if we when we have an invasion of privacy it's in the form of so the products although the token back empty so we know that 1 of the reasons so I the only demands but cannot be returned as coming from and the amount of I so it cannot be acting by and among coming from it so in 1 we can eliminates eliminated from prime and in other case we can actually removed and in mountainous and in most cases we can either reduce the side-effect Brian all the side of f up to the point where the 2 are going to be called by induction and when the 24 we actually formed stable finite sets and that's actually it's always is right point so the general gays will buys what we call dysentery so at post he should doesn't have a point always has a point some covering all the site but it is also on stable by pulling back so we are actually are going to get finite objects in some covering and then there is involved descent argument that I'm not going to explain that although to finite objects into may still pose From those finite objectives of covering yes I think that hold to say so opened conformed at the local level the this people would be the of the view that the National Book so you have to do with the goal of do you know what love you have cry I hope I'm not benefit something with taking they'll with proper because bright combatants were killed he said is that there is a list of the voters you know university of the but this is a man who wrote this it was good for last the living quality that was going to this was the start the of scheme of nowhere in doubles coming from scheme was a it's our sky at times the initial whatever you want with no except for his off-field will never be separated in the sense I defined basically because of the body sponsors path we always be something like a czarist a space in which she would never be separated itself there is a topological spot was Tobosa we always
be a czarist unite the fastest of the 1st question is believed that I absolutely clear yeah but the brother diagonal of agreements the product of 2 was as I don't think it's going to end amorphous available for use when the ground for something is going to bring this point because current of was some of the burden well In or there is some exceptions course is a good thing for of those who lived with his wife who was known to be without merit and have the size of the the provisions of of this is that it isn't that a move which is not acquiring so it is a product is correct then it means a diagonal is not a occurring anyway the product of 2 of title arrested the West does not create to product of skin below the physical facilities of all the other people whose face was overseas Bloomberg the suspect was meant to do those letters as is the moment when was security going who's going to the world is and I see the market is so that is so you will have a basis you so know I I don't understand what's wrong with what you say but I can tell you is wrong because of life you should take it to post which is separated in the sense I says it's a really very easy a result that it's look at each reflection is separated and uh .period was is coming from Margarite geometries ROK perfectionism is never separated or only on very specific a lack of fuel for the purpose of this visit is of the United on officers and state is stronger than others of space so why don't we are not really an expectant grant apostle I'm not and I would go so the the services of so maybe it could hold "quotation mark maybe maybe I was and that I don't exactly know if this is the that I'm not I'm not really familiar with current opposed to investigate cases on the Net :colon then sorry I said something when I should mention that the dissolution of the love of the game all this the regional more than newspaper which of the for the major authority case tho there will be close with the way and covering the the officer separation it's a it's a paper in Finland score put them out of the woods now that's out there you have everything in this and it's also injured Johnson In sons sketches of an elephant yeah this is not about you the question is on the basis of the old 1 was the notion that the contract was saying all of the same thing as the election so of that cost is the the concentration of the sort and suggests that the claims made last year is also due to his knowledge and who I actually thought to be lost migration have idea sure I know that the prevented that from the data back already contained thing that I'm not separated 2 of the city it should be about which of course specialist action inside the city of natural that the usually promotion from the group this is only a short while and but wasn't such a vital side but it is and all of the usual plan to which the construction of the largest was the classification yeah maybe but the thing is I agree don't will be beyond the stress the the the fact that the concept Is it is again maybe but maybe by I really don't know what I mean at some point we look a compact deployed In sometimes at the end of a compact really constitute about the selection of the whole thing you know what I mean is that you can all hope to attaches to starter but were known locally compact to will buy things and there I if you don't put dislocating separated assumption that you will get a look any combined Robertson Bonsall I hope that you will want to wipe "quotation mark which then can you get continued fraction yeah he said this is the last thing you know the an alliance that would include all of the that that the prime of this formalism so you you you lose connected I support people here and there is another side of city the justices on the beaches the system was not good things and that comes from the fact that the losses of John Iselin said that the U.S. and essays that they can go for something you agree that there was no way to tell actually overseas .period Robert we get this Wickham Weaver real structure so of our guys isomorphic you you go home following is the less in what I have an objection to this moment you think a version with quotations we can define them added to added group without about twisted by a good fight against there so that we can say is that needless to say that at all yeah at the start of the if it