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Modeltheoretic imaginaries and localisation for additive categories
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Title  Modeltheoretic imaginaries and localisation for additive categories 
Title of Series  Topos à l'IHES 
Part Number  20 
Number of Parts  28 
Author 
Prest, Mike

License 
CC Attribution 3.0 Unported: You are free to use, adapt and copy, distribute and transmit the work or content in adapted or unchanged form for any legal purpose as long as the work is attributed to the author in the manner specified by the author or licensor. 
DOI  10.5446/20750 
Publisher  Institut des Hautes Études Scientifiques (IHÉS) 
Release Date  2015 
Language  English 
Content Metadata
Subject Area  Mathematics 
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00:02
around the 2nd and we and in the end
00:15
cable thank the organizers of the invitation to take part in 1 of the funniest very stimulating conference was so hard to be talking about Abbas's all the other categories never talking about this picture here which I will explain what's on good gradually all of these are all out of county races and daughterinlaw canola also and although it would be absolutely stable there on so this is top diagram these of 3 2 categories the first one this is the category of small silkily small abelian categories with exact functions as a 1 natural transnational emissions evolves into this is the category all look to category all of locally coherent growth abelian categories so those are good and categories with other durable declined the presented the good generating satisfied that objects and those orders actually coherent In the 1 definition in this context is that every finitely generated some object what should be itself on presented 1 and the maps are among their their their who of called coherent morphism their largest unlocks all of the dojo in Paris where the left joint is exact and preserves coherent objects so what it's really about using the like this in the talks on Monday and Tuesday offer of gerrymandering top assistant models will work and and the same situation but for other kinds of logic invigorating the logic of it's the most relevant 1 here what they are and then this 3rd 1 all the objects are the socalled definable additive categories which are defined and the maps or they can be regarded either as modeltheoretic interpretations but or to simply us of functions which commute With direct products and directed cool OK as of these categories are basically equivalent around here Coughlin but under the this underneath is sure knew how to get from 1 category to the other categories but I will explain various notations that that's missing a little too long this is our missing so this should be exempt from tourists from a 2 . 3 out of groups yep something that picture has been developed to essentially within a month .period modules and representation theory itself 1 module feeding itself but really is analog of something that can be seen in the top of the world there also supplies a portal the broad band also got 11 10 committee also we things OK and that's a positive picture will be 1 of the special case of the regular logic picture but there are some additional of features in this case air is a duality that runs through the whole picture "quotation mark right so I will begin on with original example occurred the final category which is the discounted modules were fixed rate so we of fix ring With 1 orders again scalp small Piazza category also during his case without this was just 1 object so that's a category enriched them into believing so he could have stayed in factory this notation means a category of finitely presented right arm the great somewhat opposite category of all right arm modules words added if under the rolls of this denotes a category of accountants who are all 2 of the groups and a little more orders for the gas from the dissenters funds generated from Internetrelated alternatively animal more emotions fighter jet entered the general definition if the Corvair traps and perfunctory commutes with directed coolants OK so on yeses logos among the modules of which started with a onedish meal as proof of side ability for the evening groups on on they were developed in 7 days all particularly of the Lao approved pp elimination of bonfires which state I'm not vastly simplified in the existing proofs under a lot more developments to take place In the context of the model theory of modules but I don't know what's even particular used on to dislodge Davis a lot of new ideas and results at the end of 7 to 8 months did the work of the bodies in 1988 OK and then at this sunlit continues to develop our making real connections with representation theory of with the kind of representation theory that was started by Icelandair and I the right hand of use
05:45
from total approach fund category approach to all understand modules of the rain yes I mean the idea is that you you want to understand the defiantly presented modules but so relevant investigating them directly you action you take board are to be the small Piazza's category and you look at the category often of modern are modules of hunters From what ought to be the groups that result from falling much of the of world looks as if it should complicated OK and then again on so more connections being made with the some total approach to model the also under an earlier had Inc Our realize that the duality which had been seen apartheid extended farther into the picture and he also introduced this category of people imagine all of which are identified and then again changed the viewpoint all OK and so it says there's been a gradual process with respect to that part of the beginning as a mayor urged OMB and is isn't middle became clear not that all long ago but still little bit about model AT and this result elimination quantifiers OK so long as the future is gonna structure nurse and what we're doing is of fixing a ranking order could even be a small Piazza's category making modules from to select 2 out of and then we're looking we're investigating the structures we set of inappropriate firstorder language and it will be a finite 3 language took it because I want to say we want to use the back here but this is a finite 1st oral language for 4 or more jails what you actually evidence that this is setting that up but it's been a really but we need only 6 was labeled a writer so maybe you're interested in definable subsets of structure you're interested in the structures of modules you want to know what the definable subsets of so what do I mean by definable subset so we we must missions as equations OK and then he is as busy with emissions hasn't quite as normal Asians in language so let me say this for landed without relations solutions as equations finite blooming combinations intersections unions compliments of those and projections of and there was a definable subsets mineral operations cost wanted logical the the ruling connectors and or not and the existential quantifier so you understand what you get by on look at solutions as equations and "quotation mark of the item I'm going to look in particular the the socalled propose affirmative but they just regular daily terminal to be solid and we formulas that forms of this kind so the context the free variables are from the tuple X dollars must be excellent directs tuples variables of and so you just look at a litany of homogeneously news the homogeneous system of linear equations are linear equations revolves a new projector to reject some of the coordinates OK so all projected systems and homogeneous the equations of solutions that's always going to be subgroups it was your module the powers of the module 0 1 under the Basic result among the modules that provide above is that every definable subset of molecules every subset you can get by not just using this conformance but also incontinence unions all is actually find booming combination of the simple these PPE definable subsidy 7 and we should also love we should allow ourselves in which a new systems or promises from molecules to to appear celebrities such a course sets of such BP different subgroups or bases definable sets OK all yeah under the the the results is small office indicated that read This is potpourri as natural ordering and beautiful so it is in 1 of these formulas of the defines a functor the front from the country of abelian group from the country of all modules to the category of abelian groups it's use it again formula undue defined path across frontier from the category of modules to the the module to solutions that In that also that formula and because of the the formulas pp all it does is this mission says opposite 1 office themselves this is a function so we order our formulas according to the audience the Commission sets will from terrorists so essentially Nunavut Phelps salary is listening and fight all of this the In every Modular Solutions sets order like that modes of sources some from cheerful fall right so
11:15
corollary is that there among the fits well the Algebra I. so what I mean by that is model the you the category that can actually work in it's usually has the object to Tennessee as the opposite objects than the maps tend to be the ones that preserve on flight the definable sects in other words the elementary and better and there are very few of those in general all but because here the RPP formulas are are we defined the whether they're preserved on decisions as preserved by morphisms it does mean that we can stay for most of the time in the ordinary algebraic category to stay out of the normal country nationals so far sighted since if you if you're going to do the multilayered modules you have to work in the category of modules with elementary betting on but in fact it's wasn't on work in the ordinary country mortals if you do what you're not stoop to reflect solutions sets apiece before because then you need that Putin bearings he should walk in touches period that for you said the betting modules fury if another form the following you have the solutions that the 1st modules history of decisions that in enlargement into 2nd with this is the 1st major rivers universally injective just as illegal is yet another blow to the Yeah Yeah this is the hazards as queen as equivalent to that diffident if he's right motions that he is a good thing in advance of any alleged module you get offensive this not that left much of the gettin embedding of being to him as of yet OK and he has some of the changes all become simpler sensible will constantly algebra because even the elementary embeddings even with someone pure embeddings and among the users look at socalled saturated modules and here it's enough to give me pure injected Cultural Center for quite a lot about you injected mortals so the Jetta models are near kind exaggerated for set sets speedy formulas if you get a bunch of course sufficient his emotions give objectives while 2 definitions of the conjecture renewing its injectable relief maps or you look at stake in the filter and any time if the filter of course that's a PP defined subgroups so that they insist section such Section 5 amended and is non empty is non empty than into the sectional what should be moment that's the kind of completeness promptly but certain that these are these kids more to the relief of the important of but in this context yeah are examples appear injected modules on locations was working with abelian groups take referring to BZ the so the the the opposable Putin directives of you've got the integer the finite and decompose will want to this but the sooner than new possible so it is not a look at the end their consent of all work that so over the led heads peak and decomposed conjectures of good vintages mold Peter all you've got perform groups you got the padic integers undeveloped rationals for good and in general of many more to decide dimensional which has an underlying space structural issues such as finite dimensional than it's pure injective of Jules emotions appear injected home jewels of modules all of if you reject deals but they so quite common right so we've already heard about margarines of Seybold like in this context about how we want to adapt them in this context right 2 imaginary is all use among Bosher locked in the 7 days and they gradually they were regardless of it's strange to begin with but people gradually began to accept them and use them so users on the site of the module to to the ones using a oncesolid language all you elements they could work there in the same place missiles and kind of but is useful to odds not just tentacles of elements which Yukon regard as new sort but little things you can take out elements of Yukon quotient these why definable installations Of I don't know sir the powers of the home sort factored by definable could relations as new thoughts new kinds of elements so that this has become the common wisdom among the 2 not just of the original structure but all these homogenous assaults about as well all of them and I modeled on the of modules and became very natural not too are just a few of them once you need but actually to the whole category of all sorts of colleges of the world wants to doesn't realize scattered OK so what are they the because we're working in the size of context we want to do we want to see other destructive to be preserved by when and that no new sorts slot does forces to use only 2 of the New Source of me from Pepi formulas it's essentially you take if you before formula you take that set the cut side to be new or all of this in fact by definable "quotation mark insulation all of which again should be PPE defined in other words it's equivalent to factoring by some groups 1 these fights are defined subgroups forward so essentially southern New quotient subgroups the module there but the basic sort but you have all these extra salt surrounded by "quotation mark and tho it so that we can be useful to you to have those thoughts there are but in the category of sounds nice so examples of sorts of but you get you could take yeah so if a is defined presented module then home baby plank the new song but extra staff with various F 2 meaning is gonna projected presentations were at 1 2 3 terms defiantly generated chapters the next 1 page like also was a new sort and if so for example it is a little ,comma logical so something as if you know promoted when they're not modules basic sort of attached to it you have homes many 25 modules these all new sources next elects groups and so on so role some of their implicit in the model the official launch this is also the place yes yes but I'm not really very quickly but you gave his side a projection presentation and you can get quite then formed expressly from here to there yeah OK :colon yes so what listen to the category of these pp source all this is new and much areas of so the objects are speaking fares so take to be formulas of following continued so I thought about such issues Hubertus quotient remaining caution is what grabbing news or the maps of from 1 such categories sought 1 such object to another party given by the relations which defined functional permissions from 1st sought to the 2nd which a functional OK so again those on of
19:33
Monday at accused on a continent where the Korean this contacts in the top topples context OK all and the musicians use is of look into getting the rain this is a category of PPE enlargement is and the EQT accused last New musician for adding the margin race and plus indicates we're just working there possibly structure just using PPE formula OK right to turns out all that this category is equivalent to the category all the presented from tourists on file at the presented modules the sold in all of them so I mean more than half of the began as a locally presented category CTU hitting extended objects on the new machines of the depreciate cataclysm that this Italy just category might exhibit Avon's sect that Korean the artist of world so you taking fight dissented preaching solidified presented objects 11 and so this is having something this problem there already in some of the Senate versions of us in the north and the set based 1 so you it the Londoners who is the fact that the eurozone as usual movies who as we all know what I'm not in the interest which audited the printable doesn't matter just hours just definable Wilkinson just of formulas which defined it could installations while the judge's approval the evidence was knowledgeable of the theory of modules approved yet defied a police radio here it will lower regulars idea miraculous time relative to the molecules of uttering the these before when I say something defines an equivalence relation or the ventilation is functional on mean more units provable within the arts troupe was every module that it does that it wasn't latest approval from the city of Mosul said of OK and yelled at this time discovery is also the beefy abelian category which of them the notion of fried of always feel category all in all since not the original previous small Socialists wall just category in defiance of the billion category of the group was ejected along with all this is not the motives yeah it's so it's illustrated amenities such as the reign of then it's pretty in the sense that so while there is little here you have something here during his history of 1 object the other category with the elements of being the the and morphisms so the Tribune category of all is the ability of of amorphous amended about an appealing cast small abelian category With property that if you have any morphism 2 of here heartrending category then no essentially unique exact the function of the chili .period but haven't fried proved it more Jennifer small pianist categories was an exact structural this should that there would be preserving these acting exact traction there to be normal the structure of the OK so yes what we've got a of then further said for starting with the beginning case something's got modules scars on definable category all we got this associated scalp with small giving category here and then here we go the county of all functions from what 2 of which adjusting and completion of OK so that's what that initial tried to put up 1 example and from stick all them thought categories that point and we get the general case by localizing of billion minders are indeed that there's enough to be a with 1 object to another Mariel jets so is really just localizes at all distortions while excessive categories that are too often the Newfoundland time let us say hotel transport wouldn't like here we go about but before doing that underdog but a couple of associated structures of but we have coming from the Wallstreet 1st on misleading spectrum right a topological space you look at descent does for me out that that is a set of eyes Morton classes of indie composable pure injectable objects OK so in the case of integers taking the submitting reading this allegedly taking these these we need to pull Johnson said of David basesloaded sets the mood given by the PPE theirs see you take all the points not space without appears open without support is not just a 0 groups 1 of them the basic allowance that and to see the show this gives year at Paula J. and a compact something positive that space is something that's nothing like her style this is a combat Spacey's are actually the compared on the ushers in a vase of agreeing with what before 1 of fundamental objectives compact otherwise of intuitively many objects in orbit the compact sold but the of this will still be the basic set still will be compact I understand .period amusingly useful space From longfeared motion in various ways OK I should also say on a codiscoverer comments about space means a lot abided destructive thing so what is out of the door right modules of Moses in the solar arrays Commissioner of soda category left modules which would actually quite different at all and it's kind of the conditions and so on but that is the reason the duality of here so you can look at the this wasn't even spectrum for right modules as a signal spectrum for left modules with us tonight to have proved that but at at least as locales if you look the open sets you get out utilize morphism between us so on under inside and it may be that the didn't level .period subtle true well maybe literally home office from right to left that only be on some conditions on the ring there is also another topology on the space which is 1 of the hostage UAL also that construction which forced their into you spaces which these are not but you elixir construction so you take visible when sent to dictate of so you take the competence is continents of and you declared was to be opened new new topology on CD that and you get To the ticklish and you get a new topology of put it is much more like of this risky spectrum of the conditions of the city and could bring on a site it is the Gabi you take the definition of a spectrum of committed in the theory ring and may make state in terms of country of modules so placing Prime's boy indeed was injected sector but then all siltser Gabrielle spectrum on then it is Gabriell spectrum of the founder country although the that's universally the although the area immediately fired hold any of your readers
27:55
letters saying the absolute last yeah yeah yeah yeah yeah yes I mean all this country citizens model categories in which the government will have to find ways to work with us to find the best way to fairly fractures his it's so Missouri of the various levels of of a the year there around 1 of the people of the country functions yesterday the country from tourists from extended but left or what used to have a look at some point in the space of indeed was 1 by some of his appointment and the topology is given by the take presented objects that fund is in front of and you take the horns from those you know 1 of those blank to give out free at the Super closer to on you take those 2 to give you the assessment all the basic closed costarring consensus forgery the the army it said so if you apply the construction if you if you start in fact was just both all there are 2 the but category of modules innovating range then this exactly gives you the normal primed with the price spectrum with these escaped Folger so I think this is is it is the refused to hear the story of the fatal from the house was allowed but it was really hard to find those urea yodel Munitis defunct of my talk was from 2 of his visit no yellow sign that they don't have to be exact here you just hours in front of some of them want to such as these take the formula and evaluated and they don't have the burden of its along with the magic level for because the government from yeah you take a look at continuity and have fewer and and you have to get is yeah security of the yellow lights that so you have you have this on with them in the reasons is the embedding of the more to catch the left portal category so molds are embedded into the country from transformed as evidence bedding motions of mentioned earlier on the beds and not from the category while adjusting module and you take it to the function and transatlantic so that's see even imagine that makes this work you beds last or modules into Funchal on file center right on modules and that takes on the other side punitive appearing a sequence he pews act if and only if the senator's exact said earlier on Monday the pure rejected see exactly correspond injectors so what's pointed to the candidacy of the points of the the spectrum you got through the GOP you injected points of the appeals process is a lot you with the move to the area was his view that the review there is very little power is available who about there in the villages of this you get a lot of points within you will have to make lot closer to close settlement of the original injected but you get a lot more points it adopted got initiative this Oh yeah I'm indeed the inject the injectable modules are pure injected stood up .period see the general along so it was only lifted the definition of the Gabrielle spec Oct 1 representation level OK yes the other structures like the love looking at the auditing pp form city a lattice on the intersection and some of the soul if anything can that applicants this is duality between right and left happening there as well all you sold yes a lot only the performers are ordered lawyers inclusion of the solutions that sorry if you like just I mean every key before the gives you a some frontier of the forgetful from 2 1 of its and the variables in from of powerful forgettable from from promote or to our socially just let all funds generated some from tourists all of the forgetful from 2 from 1 part 2 of the that's a coherent objects the action from the presented functions the government has also couldn't let and pointed out sentimental modules so so far and blank slice category from foreign to move on to little motto OK so there's a duality of us that is excluded unit you got of BT formerly Yukon for right thinking that for right modules you can write down pick formula jewel applied to left modules and all that gives you an anti morphism of leftists for right frontal somewhat infamous from Tunis but you twice and get back where you started off to cook so this was the younger 1 against the place we the duality in all this was noticed 1st of all but actually extends through everything on it is obvious that opened yet final .period Pickens that 2 categories of a small being categories as an obvious onto your boss in the northeast of automorphism order to namely just take the place each country by sources but was less obvious elsewhere at the center of the year he is the right thing and in the this is the president of the United States the thing about the yes yes yes yes Y'all here yeah yeah yeah right on yes is talking about what happens of given you localize from 2 categories of what happens the final category on that side of the diagram of the triangle on Seoul's was userdefinable some country model are it's given a lot having some extra conditions which say that 1 pp formula all DVD there if you are the condition that that followed since that also is less than equal try to use insisted this sort to be 0 all of these were mentioned again early in the week that just of sequence so guess with an efficient from then you'd be doing something like having a condition to get set up here is closed In you have knowledge of the Vatican to close at subsequent with regular for this they decide these extra conditions of undiluted modules which satisfy the was enacted defined the form .period got produces a definable the country on Montreal's under the algebraic opposition of countries closing art forest area columns and Crucible jewels and these are really the right contexts for doing little things under the model the original license results and way things work well so it and it's at times I saw me in that picture of you and yet sits on multiple pictures of soldiers in the final category all yeah the terms the defined categories correspond to the close subsets of people spectrum but I'm just so obvious no 1 of several the but chemical substances given by sings and there's a closed so the subcategories what awkward cost close subsets of the 6 spectrum of yeah again so that
36:14
this is a relatively this is what happens when you localize vision had more than accountable on mountains there was no party could find subcategory by adding 7 of sequences all such accidents then the effect on the other categories use this to localize it To factor was source category consisting of all from tourists found the presented from 0 0 or 1 of every object to a few definable subcategory Dean so you look at the interface of the final subcategory you look at all the fun tourists after although efforts said the speaking divine and find presented are modules whatever motives directed Calment of these so this a unique Linux extension of after all mortals so you ask for the Athens 0 on every object in daytoday supervision cells and that's equivalent then this is a locally cohere these categories here all locally coherent without being kept that forget fortunately the model absolutely coherent category in fact is nicer than the general Prince the country mortals so the undergo that so torsion theory of says prosperous here justify finite title from the the picture localizes alone and this gives you the general case in fact as long as you start to buy yourself to start with many objects To get old it all small categories of you .period Beyond the Internet you can also think of leaving much recent temptations story of this category this gutted pp marginalized but also localizes insensitive just have the same sort objects on undoes was discussed before you go same objects but now you have more conditions for sequence some relations which were for just defined relations not functions now may define functions to get more functions but so of the localization of the country of margin for defying the subcategory deal but the idea that every small abusing categorizes all in this way right Celestial Pictures again on the other yes or briefly held you get from 1 to the other is of a small wooden Category a all of you cost 1 indefinable subcategory but actually the exact functions from 82 out on the kind that then from Gates is just the video while giving notice flat right in modules or is just in in the face of the decision completion all Ireland yeah you can't see systems for the absolutely pure objects but over the SFP injected the ones with the next 1 5 percentage of expediency look at all of yesterday's a couple of things about that picture of point out about and that is what is undefinable category can be recovered from the country of marginal so this is the then this would be the artist takes the young syntactic category the researchers from but regular logic accusing beatings attacked category and then the effective ization but of it said the same thing in the month OK so yeah you can get the the final Category D from the discovery of a margin rates on and in the other direction if you have this less clear of this the final captain indeed you Yukon actually recover the and and you know not you just give given a few is a category enough that you're not giving Uganda's representation of the 5 subcategories can get the final category it's a country of enlargement by localizing but he was diagnosed as a category hurricane but the that wrapped the presentation of it how do you get custody of a mountain resort towns up to just a functional From at 2 of which can be done for its indictments and these are determined that this again is less than the obvious that all of these are just of always been in the general context is more clear on the largest interpretation from considerable theoretic cents some more generally shift to definable categories C and D then the moment the erratic interpretation from tourists from 1 to the other all all are just the functions which commuters daughter products indirectly started :colon on Monday unveiled a natural to control whole by budget with exact functions from the corresponding much syntactic categories or which about you want before they come from 2 countries there is a widening the battles since the thing there is to know about questioning the of so what else would but just as of this moment in the whole year of saying you have there is smaller than that of Honourable Wyoming yard you know I think you need something to get other I'm of what because of you know what I mean we know that in the front row of up to EUR small abelian category here and because the exact functions with 12 will be a final category so you want exactly this construction you going to get the final countries I mean this doesn't even do include locally finally presented hundreds of categories includes definately accessible mordantly financially accessible categories of products almost final countries on but due to the source about casting there are also categories pushing up from successful included on under "quotation mark defense factories OK so yeah I mean this is the general framework of once a little bit about just how use our practice so all on the way of life the developed development happened within 1 year of modules on but all of the applications have been too about 2 modules on Monday in to representations of modular Final dimensional entrepreneurs of so what on the the course that joggers this .period jollity runs through the whole picture on but all just flip back to the picture for a moment there you're not least
43:24
jollity there on taken a beating category Prince that transfers into the picture but 3 because conferences
43:35
and so on for every defined category is a dual definable category which right modules and left multiples all under the eyes and the soliloquy coherent categories that was called its conjugate category the island and on the amended floor only the abelian smoldering category side if you think founder categories this was already use a lot by of slander and writer this equivalents of from 2 categories the government and to reassure onions and also developed dozens walk Jenin and as underwriter news slot for for international tourists followed this audience and develops and the general feeling there OK and that there the duality it doesn't do a lot the 2 of you locked the door and having the general case "quotation mark yet so using the sense of the word contexts of stake the recantation fundamental entrepreneurs somewhat I just and algebra field which was a vector spaces 5 dimensional understands all the comments will mark for Arsenal to press which are all jurors who centers ring in which a family generated as molecules Wilson OK so what Putin directives of the point the indeed most wasn't Butler from points spectrum so all onedimensional molecules to the old or field field here if you just feel just OK so on the other side and measurement of the pure injective if 3 of the artifact representation taught me only 5 men and that's it but all boys summoned the spaces compact of the world and uneven fact of well yeah I'd say no the needs .period 1st isolated in the open .period spectrum fundamental points in this case also there will be some more points .period compactness of the reference that would be a fundamental OK so so this is an obviously just the fundamental algebra class of them and see what what is expected to assume some of these already can figure out what fundamental representations are often about the the limits of feels because essentially the infinite dimensional pure Jenkins will be limits of folds in the 6 spectrum portions of sensible OK all another problem was in this is to determine how difficult it is to classify the modules over the range so of these notions of faulty representation .period wild representation ,comma which inventories fine it's a fact all the idea being that you if you're not retain representation type you can public far modules some useful and if it's wild Wild while the saying that the classification problem fluttering contains the classification problem for lots and lots of things essentially Centennial ring so looking at how you of compare looking at measures the complexity of mortals of .period the people of the country with right so let me 1st give you some measures complexity of so 1st is the end dimensions of and for the will of the lattice of PPE formulas look at what a stake in PT formalism entries variable in what stick forgetful fronted more from its entomologist abelian groups and take its final journey himself from the from much of the former leftist supporters OK I'm so the general procedure for demand for giving dimensions emotional lattices dimension emotional lattices so you take your lattice you look at the intervals which are Atlanta new collapses OK you look at cautions lattice but will again be emotional artists you repeat the process transfer nightly case like his new and eventually they may be platitudes to a single point so it is emotional dimensions for an emotional the recall of and dimensions and replied Well intermission when applied to velocity before it it's a bit like of cool dimension and sends off Gabrielle and French law on people who developed them but it's got to New directions but unlike the collection write another letter of
48:23
complaint status from the fund country so we take this fall father presented from category on online again we say would collapse with excessive category of funds was a fine captain act you you abelian category take the optics find landfill factors might keep going tossed finitely some .period you may end up with new strategy of I'm from justified that went on to say that emissions infinite undefined otherwise it stops at some point not the count how far How many statutes taikonauts dimension kg I should have said as well as last as he might have got a point where the caution flag had got it was a fine outlined knows it contains a copy of the rationals Lawrence then said emissions and undefined saying this very quickly just an idea OK on another 3rd dimension is kind Bendix and rank with more familiar with the topological space so here goes topological space and you look at the isolated points the points that are open missile them away because process left you look at the isolated .period varies on the way and you keep going on Monday the original space was compact you will actually of stopped a lot of the OK so I don't you reach a point where there's no isolated points than the dimensions infinite or you will reach a point where you've got thing and give space and also points francs on according to when they were fully right so these dimensions understand Frost similar some say session 1st that's because they are the same so the could go beyond the mention of all funded category is the dimensions of the PPE license but Ireland conjecture late but well it could be that source also the KowloonCanton Bendix and rank of optical spectrum I'm not cemetery refer these dimensions and find under are also in fact lots of cases but it's mostly from General so this seems to be censured onedimensional on so I just want to compare the it's something different results comparing that dimension with me complexity of the fact of the modern representations overfunded dimensional ownership but because of
51:02
engine results watches him work as saying that the numbering his look at many instructor account do profitable or very well on the that this is there's a bibliography of the number means that you find the reference owned in the on the page with reference to new crackdown on and wasn't sure could along the Brookfield Morton across more than 1 frame source in 2 different the long face which is 1 of makes strange right at all what a certified Emission algebra of under the authority of dimensional and possible tools for exactly what doesn't .period spectrum and in fact the danced together dents space so every indicate was will cure injective on business closure the final subcategory generated by finite dimensional points the fact is a strong result is also got some wonderful direct product before intermission molecules what is right and then the whole of volume that will gobble the mention of the rain so this diagram relationship with students somewhere here he was 1st person to look at this dimension of 4 financial algebras "quotation mark right so what can we say of what's known so far so cool down other dimension 0 is "quotation mark the a of sentation tight the engine 1 turns out to be impossible for optional process it's only possible from rings look for all dimension to across the team had failed to press so pathology resident think of extending can quivers following are the whole crew got dimension to but under the results are not comfortable with it was a conjecture for some time before the the possible values in this context was 0 while 1 to infinity all in fact there are find any financial you apart from 1 controller of so you value your insurer Holland do some examples of and then you in fact you've got some more complete results that are so any finite volume Part 1 can occur from whom got Garfield mention all of these articles not that's a dimension that is preserved by these interpretation functions between the final categories soul you also there you can save you up 1 and the temptation frontier when these hundreds commuters .period products .period limits between 2 what of some or subcategories of different subcategories of those then you know the dimension of this once at least this is a nice thing to a piece of sports sessions here Carroll you the mention this once at least all of but in this conjecture about the the volume of calls the for these algebras is known to be undefined for wild algebras and also from Kmart shoppers so there's a conjecture that perhaps find program real dimension corresponds to domestic representation .period OK so as I mean there's a lot of information but just give some idea what's investments like in particular context the pixel on the just they would vote that notion of of show me what Asian betting so you can clear the complexity of 1 module category with another on they conferred emotional truth so representation bedding from From dimension motions were once to the use of the other justifying tour which is exact exactly consistent that sequences preserves and decompose stability In reflect Sizemore systems of 2 things of interest to things over here isomorphic that was already isomorphic to begin work in such a frontal will have much testing of simple biological which makes it a more explicit OK on so yeah under the idea representation betting is that all of the company's representation theory of armor is at least as complex as the representation because this is essentially taking more than embedding it into model are with no most of complexity I was love those conditions can be seen as saying who gave have so it so yeah 1 would like to know that Arabs envision betting preserves these all dimensions of mansions and all that is until some of so but that is the case of British cases were known before the time and general in general representation embedding will induce the body of lattices and hence the mention of these include the issue of representations and banning from underestimate our the true global dimension of on the at least 2 more blasts so I'm of interest revoked
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the new restrictions right
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Cecilia so owners of the given the general picture but analysts say the Maybach you what how these ideas appearing specific context all of the people of the I think this is that there was a measure of what this show but while below I August fuel Irish law that suffered was responsible for the recall Munger is very on because of the funding of the emotional state of mind know you're out these new kinds of elements so this was like the construction much of his parents have more so than there while they're there and I mean the kind of you you thought you knew what emotion walls adjusted 1 kind of element and it will you find all these all kinds of elements that can be no so this over there I the elements but regarding the module in a country with more sorts the result is now on culture with the mission and maybe a man and an Indian man is the I haven't tried to write that the III the idea you should give you yard I know that none of this is what we I was thinking that if you think that that is the case of an emergency and landing was made and I think you think you are that was the year I understood it was solid as ever since original who looked at trying to move the the number of the other 2 stuff to the on Monday and is looking at all of preach offices and so on but misuse finding out that the results we wanted like these dimensions trying to define the dimensions and the number case like Krueger and dimensions of but he's finding the lack of jollity was a problem on the it's in Austria because this got somewhere thought of he for the vault things so did yeah this unilateral withdrawals is really making a big difference in of case so you know 1 of most burdensome said there is a story about 51 has been created thousands of course what we know now find it easier you do have get because what happens is that of a representation Benitez in the final duel which is objective on 1 side so you do have it's it's essentially the same benefits and retail equivalents followed by restrictions Scala's and the reader comes to end by and makes us over of the recent so that's where they from you yeah yeah the theft
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