1/4 A Crash course in topos theory : the big picture

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1/4 A Crash course in topos theory : the big picture
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About half of the topos theory of SGA4 is devoted to categorical generalities. They are now subsumed by the modern theory of (locally) presentable categories. I will sketch this theory, stressing the results that are important for topos theory. The category of complete lattices and sup-preserving maps is a toy example of this theory.
Statistical hypothesis testing Geometry Group action Greatest element Functional (mathematics) Multiplication sign Modal logic Materialization (paranormal) Maxima and minima Student's t-test Mereology Regular graph Event horizon Theory Power (physics) Lecture/Conference Term (mathematics) Green's function Series (mathematics) Nichtlineares Gleichungssystem Algebra Social class Matching (graph theory) Equivalence relation Category of being Ring (mathematics) Duality (mathematics) Object (grammar) Local ring Spacetime Directed graph
Axiom of choice Multiplication sign Real number Kontraktion <Mathematik> Open set Product (business) Power (physics) Hausdorff space Frequency Latent heat Many-sorted logic Meeting/Interview Lecture/Conference Complex number Operator (mathematics) Kinematics Square number Addition Process (computing) Theory of relativity Regulator gene Uniqueness quantification Surface Moment (mathematics) Coordinate system Basis <Mathematik> Incidence algebra Sequence Equivalence relation Category of being Ring (mathematics) Duality (mathematics) Summierbarkeit Object (grammar) Family Resultant Local ring Spacetime
Area Group action Lecture/Conference Operator (mathematics) Open set Musical ensemble 5 (number) Spacetime
Area Lecture/Conference Meeting/Interview Mathematical analysis Maxima and minima Subset
Point (geometry) Computer programming Greatest element Functional (mathematics) Interior (topology) Direction (geometry) Maxima and minima Kontraktion <Mathematik> Inverse element Inequality (mathematics) Discrete element method Element (mathematics) Morphismus Frequency Lecture/Conference Meeting/Interview Operator (mathematics) Analytic continuation Set theory Descriptive statistics Matching (graph theory) Surface Poisson-Klammer Physical law Lattice (order) Category of being Funktor Order (biology) Right angle Musical ensemble Family Maß <Mathematik> Spacetime
Point (geometry) Functional (mathematics) Group action Existence Multiplication sign Characteristic polynomial Materialization (paranormal) Maxima and minima Open set Student's t-test Mereology Food energy Junction (traffic) Element (mathematics) Morphismus Thermodynamisches System Lecture/Conference Modulform Stochastic kernel estimation Series (mathematics) Focus (optics) Closed set Counting Independence (probability theory) Maxima and minima Mortality rate Evolute Limit (category theory) Category of being Ring (mathematics) Funktor Duality (mathematics) Right angle Object (grammar) Local ring Spacetime Maß <Mathematik> Thomas Bayes
Group action Interior (topology) Modal logic Water vapor Insertion loss Parameter (computer programming) Food energy Many-sorted logic Algebra Injektivität Theory of relativity Process (computing) Closed set Staff (military) Measurement Connected space Category of being Ring (mathematics) Order (biology) Quantum Figurate number Resultant Spacetime Classical physics Point (geometry) Standard error Existence Similarity (geometry) Event horizon Rule of inference Theory Product (business) Power (physics) Congruence subgroup Frequency Morphismus Goodness of fit Inclusion map Lecture/Conference Helmholtz decomposition Operator (mathematics) Reduction of order Selectivity (electronic) Gamma function Condition number Addition Dependent and independent variables Incidence algebra Mortality rate Equivalence relation Algebraic closure Einbettung <Mathematik> Local ring Maß <Mathematik>
Point (geometry) Group action Free group State of matter Logarithm Multiplication sign Direction (geometry) Sheaf (mathematics) Insertion loss Food energy Product (business) Wave packet Mathematics Meeting/Interview Lecture/Conference Term (mathematics) Operator (mathematics) Algebra Set theory Condition number Addition Theory of relativity Generating set of a group Surface Prisoner's dilemma Polynomring Physical law Moment (mathematics) Commutator Computability Algebraic structure Mortality rate Limit (category theory) Connected space Category of being Ring (mathematics) Funktor Right angle Object (grammar) Family Arithmetic progression Resultant Spacetime Directed graph Maß <Mathematik>
Computer programming Group action Sheaf (mathematics) 1 (number) Similarity (geometry) Insertion loss Student's t-test Rule of inference Product (business) Power (physics) Lattice (group) Lecture/Conference Meeting/Interview Well-formed formula Term (mathematics) Analogy Operator (mathematics) Addition Closed set Consistency Physical law Model theory Infinity Mortality rate Complete metric space Sequence Connected space Category of being Funktor Right angle Object (grammar) Maß <Mathematik>
and if you do In the light tanks the organizer for asking me to do little costs in to pursue on that Of course shared with because you don't for us and we believe that we have some discussion we had some discussions it is here and we all know what to do about the player source insisted to try to get the big picture of the proposed to me because I think our depositary has evolved quite a bit since its creation burden on hundreds of Dick and his students and maybe it's time to understand what was going on I would be found separating might cost in 3 parts that are hours so it's a bit of a problem because it's difficult to devise for victory and a requirement to my best that this was granted to locales and so I was spent this 1st hour and a half an hour before they went to the tune of locales and then I will begin with cable series for the 2nd half of our 2nd part of the legend some goggles during tomorrow and I was finished with tired of series on the man isn't it presumptuous of me to believe that I can actually described in this mystique series here in 4 hours surrounded us make some kind of sketchy and some sometimes would be approved when the group is really easy if it approves of the difficult but will just the theory of discussed the idea of and approved I'm going to the tedious because was invented in the 15 actually sourness Monday introduced notions of sultanas and um and many of the things that happened in doubles to meet can be added to the already seems to have gone the theory of locals is a good start for locals to read because it's very elementary is quite Elementary in much the same pursuing and many of the of the phenomenon that that happens until the city can be seen there and if you know the dutiful crowd sometime in the event of hostilities are hard that's something I know that you were power and is just the same thing and so I'm very much when learning the subject very much on the idea that you about the bottom up approach not a top-down and so on unfortunately books and opposed to renounce the start straight to the top was theory and became a chapter maybe the chaplain lions have pretended to have a palace of the soul Ivory much recommend you read these books to go straight to the tune of locales in which ousted Kendall compromise and then go stop To raised at the beginning Back on the estimated at up because of the the stock to begin trading about geometry and algebra on the way here where undergraduates Rees besides it is subject to about interactions between the interaction between the between underground and we know that somehow I have lots of surgeons political geometry at least the so-called community of your materials pose and uncle reasons is about this in setting rings community rings and let me just look at the and it to start but officials reported that it's a disgrace then is a good we often this summer understand or studded displacement looking at maps from pigs to itself it is this is a ring objects it said Greenspan if you like it a object in the contiguous spaces but whatever category especially an whether we studied match from the start well that's the historically this is the hard things happen you want for example to describe some spaces the offer of a fixed by using Equation so you take so this is a map and so you take over many of them have a and a you described the questions you you look at 0 intrusions and you'd you pull back and you get this summer subject for some money of budget regularity by describing some of the very questions so this is a big discovery I guess this is due to the cast their terms somehow a lot of what we think of of the geometry is completely understood by the or described by the algebra of functions or something like that In now turn their if you have a space so you look at the matter from the tutoring and this itself is a ring but so this is just give you a soldier from it's a of that it's all From spaces too the category of friends and they happen very often you Lieutenant described by some kind of fragile which maybe you could call this the spectrum so that you would have done a better action between on necessary on sharing a then you have respect for and there is a better option between emerged from its respectively and and agreeing that to of its into that makes it harder than getting it right OK and in now this a duality between the maintenance and space does not through war perfectly in the sense that very often it's not an equivalents so when it is not an equivalent from trying to improve it this added by extending the test of greens you consider or by extending the class of spaces trying to adjust the 2 sides so there is a kind of dialectic you have the notions of space and ocean
0 spaces the and emotions 0 offerings and you have a sculptor maybe some kind of federal submitted it to and you want to transform it into an equivalence and maybe you extend the notions of friends for that to you have a notion 1 and this leads to a notions of space which is a bit more generous space 1 man again this this things that fit completely so you extend 1 of the 2 sides to make it and it keeps on like this and then I don't know what is the end result and it seems that on in kinetics we have been doing that 6 or 7 times and it keeps on the sole maybe I was 1st used the data comes from the start with the examples of the problems in the case of Macao this place is evidence the political space is so I will have to start for the category was too because spaces to political and and the rain in this case is undertaken of course the contract to be taken to be the year of complex numbers the real numbers but turned 1 also has a choice of picking the serpents In the therapist displays has the complaints this is 0 1 1 and the opening of this it is time of whom you have you have listed 1 has opened fire a 0 In addition the nice things about this sentence disparities is that if you is an open subset of her office space In relation the register people like you have subsist by then this is a unique map consumers snapped from business separates space such that the following Square is a pullback take 1 here so on this matter seriously undercut to supply of of the open said if it's continue in the role of the office and so there is a benediction between map on the it makes into the surprise is it and the open set fixed and the sole mapping to discuss space into 2 separate space give you all the information about the European subsidiary of a fixed so the idea of a tearful Powell to concentrate on the sort of fearing that the open sets off except his constitute sold all world so what are the real operations on the separatist sympathies spaces you followed another ring I mean we are interested in operations visit operations coordinator during prior to the addition of an end product and specifications of the operations that exists in the surface he states where general operations where I can give you a couple of them and their fingers justice on 0 1 is a poster to train you can look the femur on this process so that for example the disciplinary supremo 1 of 2 things with you the operations targeting a DNC money would be another 1 on the supervision of the Nunavut much like sums on maritime history you have for example the we the are of course you have much more relations and many more regulation just distributed along but also you can consider and Senate operations because you can take care of incident powers of this opens the spacesuits and is also the political space next to the product apology and incomes contrapuntal as much ground as studio at best as basic operations on open sets and 1 of them is just a supermodel where I disapprove of what I offered the family time period is just what it is a problem the excised so this very moment where I actually always produce 1 only list of the sequence that you have here is completely 0 and there's operations electric consumers and this is the basis of the Treaty of Council the local can be defined to be the 1 to be actually it's close friends because a win but when we talk about locales that we are in a tough talking about the ring so actually the derided the picture that rings schemes so only know that there is a kind of duality between rings in skiing 1 year and went to college friends this is garnering like objects the to book lookouts this is just the opposite category the considerable powers would be the opposite category so what is your friend the definition is that the frame it is complete and that is maybe witnesses described what is the latest in a moment which is
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where the supreme unlawful assembly said of course it is the Union but this so that is also a complete and indeed as in Finland to India and the amount of the family is the the mother of a family of before consent it is just the intersection and then you take the interior of the exception OK so we will see here that the consumer has a kind of further with the help of the we're description because you need to take the interior and this is why I the opposite politics or is not a afraid that is is spirit date food a trade union or intersection which exist in the past and visits 2 D also because of the modern law does not to survive when you look at all said so the opposite of a friend is not friend and the map of the political space give you a map of the Indian addiction which include thought that opera star just the inverse image map and if you look at the properties offered this matter is that it is amorphous and the now that amorphous and frame it and in a manner so you have to friend and feet and the meeting it is a matter which preserve the operations arbitrary superintendent and family and fuel or meat and we should not forget that the world uh there is a unit of payment for the purchase of which I will write 1 persistent top element so 1 in the case of French music success is the unit is the unit element for the intersection so this is how the notions of morphisms friend offices is an and that he is so straightforward to check that the inverse image map is on the amorphous and a friend were sold do the you have juventus stronger from the category of friends what right and our and the category of friends in amorphous and when you get something that goes in this direction from Taft and it's got it all there is to it .period condominiums sometimes I don't believe a word about the technologies have here because some people will say that a contractor in front of him to say that this is a contaminant center late I'm sorry this is a cooperative contracts because it's programs so there have recorded Santa from the globalization the Government Center is the same as a contrarian functor from C to D so when you say that the founders contend that had you need to write all located so because I prefer to do things like that and also a four-percent 17 if you have a map of post sets and will use that madam who sets and called preserving the partial order you you have to have versions of continuity and inequality at 1 and so if I want to say that in the match between 2 concept is a map of Kosovo's status quo but have not been our conference that there are 2 possibilities OK so here you have an Congolese Franca it turns out that this function as it has this country hasn't as another joint 0 which associated to a friend that the political space on and that the mosque's because of its so essentially the points offer from but I have to apologize for its home indicating their friends after a into the EU the initial Friday so bad bracket 1 I mean this crime with 2 amendments 0 this is a friend this is bracket 1 is actually all of the other before .period and a new defined in ocean surface .period also framed by looking at morphism
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to the formidable education and so on and so if you have it's the political specialists and a end but then you can look at all of the and all a into politics since this is the category of frames on this would be option with a conscience masked from the from fixed To this space the point of being apologetic so this is junction between these 2 categories but that the junction is not adequate evidence that that he and there is a there are more the objects here but then there are the worst because this is actually there are friends not gonna have any point so this is why I materials will cars has been a called sometime .period apology because of the points In a local and not that important to institute a focus the unimportant but some locals have once and they are an important for in that series so he was maybe about 100 time to give you discuss examples but registered units summer councils have points on face the located so what is happening what is it work dutiful accounts well had a friend all I forgot to tell you what you is in this picture the Lookout is just an object in the other category of becoming a friend so locales don't have existence except mine being afraid someone so this is kind of fixes the the notions of is of fiction the reality is is that you what you have is a frank about this fiction is very important when you think about it too if you want to have intuitions about do displays that you're looking at and so we are going to look into the opposite undercutting your friend visited it like in Azerbaijan retreat sometime in the tuna Friends it's very good to think of firing as as enough finds came on so we but in this case there isn't an actual notions of general notions of his schemes which is partly independence from well not exactly from the 2 communities being sold and so this function here which was the continent that once you use this duality you get actually a pair of functor the future energy needs quality and secure a place environment proud of what you have here is the comment Franca gets the nod the of the world well in principle are defined here I will put call that a character you see out of the political Lookout is lecturing and this is my database ring and you will you look at morphism from pay into the bays rating and in his intention to his ,comma character so plentiful ,comma is a character there might be a good student about it because this is a war on you and your guests right but maybe I can say I about the notions of points in the form of count by looking at the morphism character of a friend so he captained needs to send the element 1 you see it must be the characteristic functions as something here there are only 2 values possible so it must be the characteristic functions of something it sends the the elements won the top limit you to 1 by notions of amorphous and amorphous and a friend says the units to the unit sold the only thing which is interesting animals as a visit is a colonel colonel thought not this can't go on in its close on the Supreme because if you will higher is goes under and in respect supra at the supreme hour show around the country of Friday's close on the so you can take the union of everything that is in the current so let's call that 5 should promote the 1st of all it takes In the kernel of fire and it's so this is the maximum element in the current and if you know what this maximum amount you know everything about the character and a soul you may think of this as an open subset of the space so this is open open in the sense that in the crowds it's an open the OK so what what is so special about these open says is that there need to be deductible fool so that is if I told you so that you can find a maximum element the kernel of you is needed intervals and this may what this means that if you have if you express you as the union of 2 other elements of the the flight and then it must be that you you want or you "quotation mark mutant so that's I'm sorry mate yet but think you can and do you need that you know it is different from the 1 this is called amid it ideological said was when this property comes from that comes from the fact that this uh mortars and 4 proud respect intersection if you'd just play with the actions you discover that the image you hear is meteorological so the complement fuel which nominees should exist at a close-up of some kind I said nothing about close from the becomes fuel which is in the closed part of a and this is just a opportunity and this and that is of close subjects in account this is just the opposite Gladys opening is joined irreducible so it is it is joining the neutral should join in about means that if any is if union if you you'd didn't if or if you
could have to so there are 2 conditions and there is a is so there is a better option if you like between the points of a lookout and indeed the individual at the close a subject of the up so that's 1 way to understand what was a point and that the political status of our if it's is the political space 158 Canada except that the political space and then the closure of the Singleton is is that visited notable is a close-knit social so that's the connection between the classical notion of quantum because they that the morphisms out producer the closes the subjects that are irreversible and Idaho before this incident in question OK the Bank of England held her overlooked it will unit of President spaces like this is to get the Karunakaran example of a powerful tools of reducible that there enough to because space sometimes closer to the notable that 1 had better have a generic .period which is called for a generator yeah I'm OK now OK so what I want to and developed here is the fact that friends are rings and a really mean I'm serious and his friends are arranged in a sense that the exterior phrases is very much like that to two-year community rings so but if somehow this it is simply it's a kind of a simple reference to community rating and so on indicating friends is another break category in the sense that this is like the category of grooves on 2 categories of rings or a category of opinion rules a category of the algebra universal applies tho what is made amid for example if you have a morphisms you can the composite into 3 more visible the government of fire likened it to your friends or whatever will be the 1st event the image the morphism you here as some friends the emotions of the administration on a decline the use of congregants that is emotions of congruence and interior friends of what is economists in the 2 refrain prior to include the installation of of previous relations and aggressive measures is always as some of their own the subject of the product that it should be said Friends of the Prophet that sort of thing and they couldn't selection is a comprehensive is a sub-prime approach it tells you everything about the Congress so again waters and the printer is a congruence which are right here has included so you can take a caution to that and here and here you have this as an office so in other words designations of subjection and the compositions of every man has assured action followed by an injection injected more often working well fortitude for Palace it means something when it's over because Universal everything so you know that consumers may have problems was an error and the injection he becomes the quotient locale look the discovered the coaches so what is the notion that at the notion of good questions of power comes from the notions of of subprime that's just straightforward reversing an earlier in the day cautioned Friday leads to an embedding on this year this is embedded in the floor so heavily more dozens of friends has a composition as a caution followed by an it said that it reduced something center is acting because the amount of fuel palace emotions and therefore this is far from intervened has has already drawn an independent the process the Kennedy is Doretta joint is not indicative of the record went is not amorphous and the forecast is just as in the order preserving that poor quality of my friend of between 2 per cent in the county processed and this this is just the finished response from the fact that preserve supreme and a map preserving supremo between 2 complete lettuce arrived on its own Missouri which are entitled to our staff from B to a little or a lot of all necessary conditions friends now look at how it wasn't I and the existence of this joint makes things very simple about the described the composition because its educational tours of the of category theory but if you have to oppose the until and that's a M map this way that's it answer the are mapped the where this college each of the opposed that as it is destined to Entergy that means that we have a fixed we have disagreements then this is well known you can compose so you have composer Jean and college signal that this is a map from around the world a matter from Peter between the is a close repetitive that
on on the way so strongly Britain means that it is of course equivalent companies or preserving so you have learned that in the year ended you have seen must be fixed before its and a decomposition getaway gene that role which is the from Putin humor is right to call that a common medical director it's the jewel of the enclosure because it tells the operator can because of such a move magic the so Solberg called Connecticut parent musician who owns it's another preserving the parameters that satisfied and on the other hand you have the compositions effect if I look at the center of the city battles Kingston and a few role as the overall plan payments if you like it was just a fixed point then you have the composition of the offer of energy hazard a map 1st of the signal in order to cure and inclusion here and a hearing out of office and that some the other conversations like this and other similar composition Delaware for G show for you have inclusion in the same manner as morphism and a Q 4 0 for the 1st 6 points I mean sometime in the abstract that to read the suggesting good because again what connections and if you have a Delaware connections to the Internet about action between the close things on the 2 sides and that's really no more than that from now on all that figure that's why I called monitored and except that in the years situations of having to get a good reason to fund and this you have more medical went on and on and it is the composition there is a composition with the new judges not another mortars and injured so that makes things more complicated than the health it is the biggest loss in case of what all of the the the only way out of this and 1 of the story on other hand if you want to follow so find this way the the but in addition to the the Commission will the result is the the it was the 1st of the villages and we have been working on sentence of the 15 years so right most the army this is a list of and because is also called for there was this I'm going to go Of about the directly in the the New York Times this is the start of the year before the office of the political commentators said you will find that the you know what it's like what is this 1 of the most formal said it was to have been and the author that sent it into and so it is not going on this portion of year is among the people the 2 With the people said we don't so OK and
that they have to adjust to the that this is a summary of case is closed only in the sense that we have here is not the result of years of health and not of so it is the Union of the family of the victim it was In this is the 1st of all of this is anything it also includes a bit of an agent is surface .period so what this is the right to choose the use of however because of it was not a lot of criticism is something of a Yukon this is the way to start with because this is going to have to do the right thing for all of what is going on there no this is not a qualified currently the danger of losing the United Kingdom all of the of on the substance of this space because the floors of the reserved section the interior section of the rest of us this is they want people to know of the college this is 1 of the qualities of the apology is exactly that later that that on 1 of the reasons for this this is practicing with the of so I understand size of the the rest of the on the whole thing and now we have this from the point of the American Friends of the the and then there is not 1 and used by the and news attorney here and wishes that advantage in between me and coach friend of a friend is about that in the course of time and energy use as is the a jewel terms saying that the direction between some friends and the quality of the 1st completed we will need to know how I want to talk about being free the frame the city is friends are really ranged from the condition if we want someone who was that because it all community mozzarella polynomial rings playing an important role so you would like to know of 1 of the 3 friends now that I should say that to you know In community underground and free algebra history community has agreed on a set of genetic is constructed actually in 2 steps so you have the category of commuted the rings and you can forget that as an addition and you get into that category of Community money so there's a forgetful fungus from community ring to 101 you will you forget unintentional that again that you like and indebted to the sets for that now he each of these 2 has the underwent went 1st you can generate free commuter demand from a set and and you can generate a free commuter ring from the community with 1 with this 1 "quotation mark they said its on this could be the final like I think it's sometimes do not at the expense of cocaine that certification work inflation against for before community of monitored research and so this is a community of memorials in weeks and then and a 2nd 1 you take turns and something here great and if you take the group actually the monitoring efforts In end so what may be a connection rate convinced so the free community on the set that can be obtained by combining these 2 steps now I want to stress the fact that only the 1st step is just normal but circumstance as something peculiar about it because the end is using denser and actually it's just me and then group generated by the set and you forget that and has when I structure you take the view that we our eyes on it and that's it this object is a group the construction while you could and did not have this is the to the end if you like on you so you can
take a fear that in group and then it happens that this set of 3 had been group has has a product we all know that and this product give the federal ring structure and that's the end 1 of the 4 vessels from now this is a situation which happens a pretty often in mathematics it is 1 that is distributed the law Sea visits to attributed Taylor between the 3 had been from and the three-minute Franca and should combine them you get exhibited a lot of maybe a distributed the state I'm new to the level of or commutation I have had done could make it a more precise analysts say they think of the free and the group the following day as a functor from set to set just because I want to see it as an end Franca you differently grew from terror and you forget is that a set of keys this is examples of someone soul the president has committed occasions as government and its units and they also defend the community monitored from God Association again I forget this summer and I see it as an honor from source at the start and you have and will structure on the planet that's come simply because c and is the complicit affiliates satellite with the right out on the adjustment moments after is like can the community that it before now it happens that composition asean if you compose cm with added better appropriate because this is just being committed during Franca and so we must be on must must have a move that structure so how is it that encompasses just to learn that is mother had and this was a observed by John when is it true that composition of Truman and is among years ago prompting so you have a Category C and you have fled since the Truman ads and we want to know that if a it's a composite composer being disseminated so what do you need to get the product on a composite you would need to say Well I'm going to do that want to exhibit the map like this From the mitigation of Eyadema's petitions been butts on the composition of the thunder from prisoners committed there so you cannot switch to stay in the you would like to switch a and B and a place that diet and I am and then you would be in business because you will be able to use them invocations of a and medications and me to get back to there but the need as switching operator and switching operator with a goal from the edge 2 babies and this switching opened it and put it detail like a young that certified trashy like but it's it doesn't need to be useful Oh really does need to be fully reimbursed over but it's a it's accommodation of our time which is saying that is you want to do this construction the other way around in the woods suppose that you're not by accident that section you will want to construct this thing and the 1st state in the city had been group followed by disagreement you won't get what you want tried but you would be able to rearrange everything there would be a computational Beretta and assistance contributing submitted loss this is so constituted by John back and the on uncertain as it implies a precise notion is not only a peripheral progress but it must satisfy some competently relations with limited editions of a B OK great so I don't have the time to residents who could not In the case of community ratings severe is exhibited London too it president "quotation mark and impede the kids are offered on the train also is exhibited along the words you can construct the 3 the frame into steps on a on a post-Sept so I'm going to consider from became friends and I forget that this is in addition go from forgets Brimmer so we get into lover sending that and
I were simulated suggests that while there the mode of operations climate and seemed to not have any risk and promoted general fitness and have a unit 1 so this is what defines a loss similarities so you can forget that the frame has in addition and then keep forgetting and goes into the 20th 4 no this country has invested 1 which consist into having fun it needs to oppose indexed is very explicit and this sometimes also has left everyone and there less leverage is very simple because it coincided with the rest of the warrant from sunglasses rescue branches the former the category of complete lattices and so preserving maps and then there's a forgetful functor here understands what is called a debate on the of opposing the Business Center for Law sections of the the rest is In the downward scholar also downward sections or approval of peace so it means that if it's isn't as an what is smaller than its what is next for now discourse said is is as supreme Amin Hassan pronounced and it's so it's new left-right owing to the forgetful from draw from the category Of course complete lattices consistent with Supreme around 2 per cent the repertory and she uses saying this is the same so there is a natural finished with that there with the analogy we know that show we have sex education groups but and OK so that's a sequence and we know that if we want to study commuted arrangement to Andrews 1st or model something like that but now we're going to extend that medical science and instead of having work but take some lessons In here at the French on and educating of some is realized that any good of having groups we know that the category of having to this report or call new intercepted in the end it a all the candidates have been rules means that article product finanical product and a product "quotation mark scholar on the supervisors is assigned thing that's added edited except that here it's even stronger any Cobra office of lettuces is also the prime it's very strongly that it is the presence of infinite operations that make this song and it still is it is significant monetary don't close playing and Frank is a community rating assumes greater kind encouragement of community rating indicating compliance so there a assisting here that there might the as suggested in the paper but that may be the right unity could be generalized to anything the situation where you start with the pessimistic Mandela category which is edited in some sense and you can't community monetary object in there and the formulation and you take the opposite formula schemes and this things just follow this pattern completely in some sense who could say that to read at the funeral for power is really like to me of skin but it is the case with the assistance of the there will be a great friend of something yes of course yes that's true friends our Pettigrew object 4 people on that students through the through the juvenile hearings here the whistle-blowers is our program ,comma practices that development of the community and the community 1 microcomputer the Quantock was 1 that exactly through but is a nice connection between the 2 the intrusions of frame into Community Quantum has both the left and the right it and that is that it makes a lot of the realized if you the Israeli easy to transport clearance about community .period goes to terms friend using this this and options on medicines would stop here thank you very much composed terrorism our earlier by half