1/4 Introduction to categorical logic, classifying toposes and the "bridge" technique (IV)

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1/4 Introduction to categorical logic, classifying toposes and the "bridge" technique (IV)
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Introduction to categorical logic, classifying toposes and the 'bridge' technique The course will begin by presenting the basic notions and results of first-order categorical logic, with the aim of reaching the theory of classifying toposes by Makkai and Reyes and illustrating the general techniques allowing to use them as unifying 'bridges' for transferring information across distinct mathematical theories. The exposition will be accompanied by several examples and applications. The lectures will require a basic familiarity with the fundamental notions of topos theory, as reviewed in André Joyal's lectures on Monday. Lecture 1: First-order logic and its interpretation in categories. Geometric theories and syntactic categories. Universal models and representability.
Category of being Computer animation Logic Topostheorie Multiplication sign Series (mathematics) Mereology Event horizon Resultant Connected space
Group action Multiplication sign Discrete element method Grothendieck topology Order (biology) Sign (mathematics) Many-sorted logic Meeting/Interview Lecture/Conference Velocity Focus (optics) Theory of relativity Consistency Algebraic structure Thermal expansion Price index Lattice (order) Term (mathematics) Limit (category theory) Category of being Logic Universe (mathematics) Order (biology) Right angle Object (grammar) Game theory Directed graph Maß <Mathematik> Spacetime
Functional (mathematics) Confidence interval State of matter Multiplication sign Insertion loss Rule of inference Fraction (mathematics) Order (biology) Duality (mathematics) Goodness of fit Mathematics Lecture/Conference Meeting/Interview Well-formed formula Term (mathematics) Natural number Modulform Series (mathematics) Devolution (biology) Condition number Newton's law of universal gravitation Area Process (computing) Basis <Mathematik> Maxima and minima Line (geometry) Incidence algebra Term (mathematics) Category of being Arithmetic mean Exterior algebra Right angle Hydraulic jump
Polar coordinate system Logical constant Existential quantification Building Group action Multiplication sign Decision theory Equaliser (mathematics) 1 (number) Junction (traffic) Grothendieck topology Subset Many-sorted logic Meeting/Interview Arrow of time Social class Area Theory of relativity Special unitary group Price index Term (mathematics) Variable (mathematics) Binary tree Directed graph Functional (mathematics) Observational study Similarity (geometry) Heat transfer Inequality (mathematics) Hand fan Product (business) Frequency Morphismus Causality Lecture/Conference Well-formed formula Term (mathematics) Natural number Operator (mathematics) Genetic programming Set theory Condition number Multiplication Algebraic structure Numerical analysis Quadratic form Network topology Free variables and bound variables Musical ensemble Allgemeine Algebra Table (information) Maß <Mathematik>
Free group Theory of relativity INTEGRAL Equaliser (mathematics) Multiplication sign Lattice (order) Term (mathematics) Urinary bladder Junction (traffic) Connected space Frequency Radical (chemistry) Many-sorted logic Lecture/Conference Logic Well-formed formula Finite set Thermal radiation Genetic programming
Multiplication sign Decision theory Direction (geometry) Moment (mathematics) Modulform Variable (mathematics) Hand fan Numerical analysis Statistical hypothesis testing Lecture/Conference Well-formed formula Musical ensemble Lie group
Axiom of choice Complex (psychology) Computer programming Haar measure Multiplication sign Sigma-algebra Expression Einstein field equations Algebraic structure 3 (number) Sequence Urinary bladder Theory Quadratic form Inclusion map Arithmetic mean Positional notation Many-sorted logic Lecture/Conference Well-formed formula Musical ensemble Arc (geometry) Chi-squared distribution
Process (computing) Physical law Content (media) Limit (category theory) Arithmetic mean Regular graph Film editing Voting Thermodynamisches System Lecture/Conference Meeting/Interview Modulform Object (grammar) Series (mathematics)
Area Addition Process (computing) Observational study Variety (linguistics) Lattice (order) Incidence algebra Event horizon Theory Arithmetic mean Mathematics Regular graph Voting Many-sorted logic Logic Selectivity (electronic) Spacetime Newton's law of universal gravitation
Geometry Existence Group action Multiplication sign Sheaf (mathematics) Food energy Theory Mach's principle Inclusion map Sign (mathematics) Regular graph Meeting/Interview Reduction of order Genetic programming Physical system Area Sequel Concentric Weight Physical law Division (mathematics) Line (geometry) Price index Cartesian coordinate system Sequence Computer animation Factory (trading post) Sinc function Maß <Mathematik>
Axiom of choice Point (geometry) Computer programming Sequel Variety (linguistics) Similarity (geometry) Propositional formula Ordinary differential equation Regular graph Rule of inference Theory Product (business) Prime ideal Inference Sign (mathematics) Regular graph Lecture/Conference Different (Kate Ryan album) Musical ensemble Algebra Set theory Social class Weight Model theory Physical law Content (media) Horizon Algebraic structure Morley's categoricity theorem Grand Unified Theory Lattice (order) Cartesian coordinate system Complete metric space Sequence Equivalence relation Connected space Category of being Algebra Computer animation Logic
Computer programming Functional (mathematics) Existential quantification Hecke operator Maxima and minima Similarity (geometry) Product (business) Mach's principle Many-sorted logic Lecture/Conference Well-formed formula Genetic programming Set theory Social class Newton's law of universal gravitation Theory of relativity Algebraic structure Incidence algebra Sequence Quadratic form Equivalence relation Sign (mathematics) Category of being Film editing Logic Object (grammar) Freezing Directed graph
Point (geometry) Functional (mathematics) Group action Existential quantification Divisor Euler angles Equaliser (mathematics) Correspondence (mathematics) Quantification Maxima and minima Rule of inference Event horizon Food energy Theory Product (business) Element (mathematics) Mach's principle Many-sorted logic Lecture/Conference Well-formed formula Term (mathematics) Operator (mathematics) Gamma function Position operator Directed graph Social class Condition number Lemma (mathematics) Projective plane Algebraic structure Price index Variable (mathematics) Leak Substitute good Fast Fourier transform Category of being Arithmetic mean Logic Order (biology) Object (grammar) Musical ensemble Directed graph
Axiom of choice Group action Presentation of a group State of matter Multiplication sign Decision theory Axiom Parameter (computer programming) Mereology Junction (traffic) Grothendieck topology Explosion Many-sorted logic Analogy Physical system Social class Area Process (computing) Theory of relativity Special unitary group Price index Lattice (order) Complete metric space Sequence Higher-order logic Degree (graph theory) Category of being Arithmetic mean Order (biology) Classical physics Geometry Slide rule Existence Functional (mathematics) Observational study Image resolution Maxima and minima Discrete element method Theory Product (business) Element (mathematics) Mach's principle Regular graph Lecture/Conference Well-formed formula Natural number Operator (mathematics) Modulform Genetic programming Set theory Condition number Standard deviation Matching (graph theory) Forcing (mathematics) Model theory Algebraic structure Morley's categoricity theorem Line (geometry) Axiom Equivalence relation Quadratic form Length of stay Algebra Logic Object (grammar) Maß <Mathematik>
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hostilities and and what you get the idea and conclusion you can call it fearing inside the European OK so and now we don't about that these which could move from the American public he has exactly exactly yes I mean when you when you walk into thinking that logic saying only don't give out their regular sequencer which means the sequence of the former when you walk in a dramatic logic will only consider or you will yes yes yes yes yes yes I mean the old then was that you couldn't continue to come up with a product that is an important exception that there will be excluded in don't How on "quotation mark we will have to be the United States has not yet presented in a slightly different way sees we have presented any inference of sequence so they might not be recognizable easily or hiding in the linear sequence that means said you needed India India presented in this way so I didn't answers to their in the program so you can continue to work on this and deploy a strong today with so this is completed this year among is something that the affairs of this Saturday's Bazemore there's always something here now and it acquires the actual choice if we have a great advantage over their dead which is the fact that we we can consider models of our fear is not just a sets up but did not any categories of provided that these categories processing enough categorical structure for us to be in the interpret all connectors and quantifies the horizon you know you know a fragment of this to to prove it is still the completeness of the which is even stronger than that of the affordable strapping therefore through the notion of fine back category that I'm going to introduce an hour because the I'm on the notion saying that the category well now maybe it is indeed the premature to talk about this because I haven't even introduced a categorical semantics matzo balls that I am then at you could cut the number because of the developments from the question and that there is a positive sign that the refugees in a certain categories with an appropriate structure that the notion of saying that because I think it is very nice because it allows you to lead them into son all your feeling inside and sign that the category which would end an old like that so we will have a universal model was headlining a universal it is universally in many senses that 1st of all it is universal because as we shall see in winning a little thing all the other models of the fury by pulling back this as long as certain funk but back that follow from the point of the completeness and is important because in this modest satisfies the property that would his money that he is probably the fury has already a the and ordered that bikes said fertilizers completeness and this is constructed in a completely insane told a logical way like a among Pascal Djurovic proposition fearing his see when you have a proposition hearing you while you can take the algebra consisting of equivalence classes of all the proposition formulas of its language and so on and this conveys a model of this hearing and in which what is valued at these are precisely what is probably the future and so these analyzes completeness and use the difference between what you had in this framework with a single differences apparently wanted to take all all this Saturday's models of the future and more or less last letter to suppose the choice here we have always said that the 2 articles in the modesty of Atlantic Avenue in in this isn't that the category where we find that this model and then this is modern English syntax and semantics of the fury meets it the thanks to the recent meeting with Italy's unification you can get them back and people living in here and this is 1 of the main advantages over from Tonia semantics it is about administration of the fact that they're looking ahead to what the evidence suggesting the Senate's is married and has very limited because you don't get the faithfully image of the fury as the main man in the fight I mean you see the forefront of any beneficiaries need national choice and when I make a wish that some of which you can do it but there is a major issue having initiated a theory that the natural choice will not suffer as I have from my to political structure the fury from more you will have had her and then you are you are lost if you want to stand in Philadelphia is revered a classical technique similar funerary lost the weight yet even days is such as you might if you had these things and there is only of this is called a stronger completeness because it is not just because it this is something more because everything is concentrated in a single mothers OK so I don't know how to find money also saw what is what we saw the
1st of the signature signal what we can do is to consider the notion of a sequence structure the accounting only 25 whether
or not you need a justifiable because the only thing you need is there to be able to take the certifying products here and if you want to interpret computer logic that we should know when I can't find it in because we have rebound from the absolutely justified infiltrated this would be useful if not today but don't worry confusion similar conclusion hearing so I can inferior here you require a decent formula as currently there are about only to this hearing so he have 1 has a really believe that with the notion of probability inside the fury as it means that the government so far formulates confusion if it is needed the from atomic formula my only using the final 30 conjunctions and existential quantification switch provably unique in the fury this is not logical McCarthyism categories In this sense that I should take a a few of the infuriated that meets the confusion saying that the category will be confusion and conversely any character generic category can be seen as a disinfectant take a category of such a few this is a new findings from today and yesterday this is the community that you lose yourself in your office otherwise you constrained finally completed the CIA but I Kincaid will use of confusion and misunderstanding always so suppose that we had a cut including had grown scene and we want to the define interpretation for this back the conventions that we have introduced so we have to interpret the sorts of a functional singles and the relations singles now sources we interpreted that as a objects all other categories function seamlessly with interpreted that as a proposed that supposedly had a function similar to that this is going to be commencement of this is going to be interpreted as a I'm not only going to from this kind of their products this is no appeared on these generalizations with evidence 90 classical dance dancing 169 because you don't understand semantics assaults on interpreters as a sexy and the replaces sets up a we've objects because we are going to do so in the custody case CD is equal to set by the category of sex and functions between them religious symbols such as will interpreted as a assembled just equivalence classes a one-off incidents you know what I think we are seeing in the 2 no adjustment freezing so so this will be among more feasible under the cost of this there will be interpreted as a 0 I functional symbols of sodium and will have enabled the um product that and I believe that can a depending of also on the category so and equal 0 also stunts so this will be a thing OK so we know how to interpret all of these UCI everything that goes on there is no free from um from whom it is a very small visionaries solution from a weather Benson sets of tools to continuing and economic adequately funded programs but yet another is that we have just interpreted very very busy competing survivors we just interpreted on function was and relations health questions and you want to go along with an interpreter complicated formula as we needed to require some categorical structure
present in order to be able to do this because all the places that you want to interpret that as a conjunction of
formulas while he had to find that the products in Japan and will not accept this anymore because I would like to take at the intersection of the interpretation of this we the position of the and such intersections formerly next year category and a source of finance for events we know general subsidies and we have to acquire at least that uh and the affected communities can and of course there are the more complicated formulas are pleased you can and do you might want to interpret things like the use of it .period and although quantifiers and it is also possible to particularly in the doubles to interpret these things but you you need to use that particular structure in your category to be able to do this you cannot avoid it in every category and there is a solution see each of these deferments as a corresponding class of categories in which it can be interpreted but their lives lost a wheel on the OK assault was that we have defined decent in the preparation of this basic ingredients and then we will describe how Altintop before talking about foreigners will have to think about the terms although interpreted those Wendy's is completely stifled 1 because you see functions have interpreted in this way those are just the things that trying to slightly the variables functions soon applied that be amended so it is isn't completely clear how you interpret ending will be as ROC will really be interpreted as idols of the terms and he said you don't need anything more than a fine its broadest Lucas and the whole an informed instead if you want to interpret that the leak formula was a they're not this only if it is a constant it is a valuable including its projection the projection or just didn't think of it depends on the context is larger than previous projection otherwise it just didn't think salt and now In terms something really now of course the authority says in these formulas of this former imagine for instance that you'll want to interpret the formula was only signed at Formula One year union likely won't winter formula kind when you necessarily need them more than capabilities as this formula then you will have a better the interpretation of the school terms these are this would be true of Arizona a certain things and the interpretation of the Senate will be the equalizer on display so you again if you need them that your category also sees this equalizer and yet will need the backside of the that world will make a point where you'll considering the Provisional IRA which is a civil jet and an adhesive the stones will give you an hour and then reform for the pullback and this will be the interpretation of your formula now as and after you have clarified all to interpret the meaning of the as all the other acts that are formed the foundation by using introduced a logical next season In Florida the quantified and the local police later and this is stopped so I did not feel it is you don't know anything more than financially in itself so that you can also use laser Sony's means that the pages well actually just already final that of his because of this fall the accidents are all this former Soviet agents in the categories refined products he had abundant you want to interpret the ball in the final that a conjunctions and existential indications of source for sure find that the conjunction also you need the full Republican pullbacks such enormous and that the real objective so it means that you need a contingent category but it doesn't stop this because in condition category you might not have the possibility of interpreting music is essential qualifications in in a meaningful way and now you have right there when you come and when you will wonder about halting the reportedly so you just try to to find out that the categorically devotees of interpretation that situated in the provision of the old existential quantification and US authorities as with Museveni's there when you have a lot or just the function and when you consider its image this underrated as a total about admitting that he is the cost of energy so this operation is very important that this operation although the effect of rising the manager and hazardous objection afforded by you a more feasible because it is the key to the whole Alcantara but these exists there shall be quantified because they can look at all this is defined you see that there is a sort of existing quantification him so that he is an element of such a thing it was OK so much of it might be latest the relevance of such an attitude that exist something which is assigned by fluently and that's all basically yeah you need them the notion of ignorant of the ways of a cabinet I categories which are currently and English tomorrow that any harm as a factorization is an image of a viewing the managing general is characterized as least some objective for which you are factors and require moral that these images to be stable and the pullback a disobedient Quebec is related to to the fact that you knew fragment of logic you wanted a day the substitution rule to be sounded and therefore you need also the search for america's Community College now form a coherent theory is unique on because you see you this being here and there so unique that what is called the poignant and cutting I'll
call you indicate agrees on the categories in which some or all of less and you have to find that the although some objects and this spine unions are against the war and Quebec wife because you want to that but it is Junction although I do what it's supposed to hang on hoping that day and Agencia like this defined his job just because the clearer and logic and I know that the interpretation longer feature formulas we need a subject all of my my final product line that all of these would be subject to using and also in doubt these junction of these well you to take a the unit of the soldiers nothing and this is something that you can always meeting said that by the end 1993 "quotation mark categories so you need not to impose on the existence of the state of the union also understand that unions of some of GE's annual moreover have required them to be stable and Quebec and businessmen emotional coherent and now the shape that notion of that there is an awful lot of geometric and what is a particularly important for us is that they don't think that was his argument that the catalyst and so on we should be able to interpret any kind of Jamaica fury unit and a study of 10 years before OK thanks the whole case sold no but I think you have to to great think a wedding if you want that have not yet you don't get any you just under false and whose you know what can you do you think we can turn on you know the definition of matter and in what you can get going here is the same the following me you can write things like it's something you can take a step forward this you can find that this is the only kind of allegation that you invite you can look like there was something completed in more complete all on no the the yet the sequencing the sidewalks said the only problem is being treated the staple classical nature your system because the usual stands for the start of the logic of the classical so they had little excluded me and this is the precisely and this is the only thing agency that there and it doesn't want the snowy investment market in another critical positively United 93 category because the sun in order for this to be told whether this means that the identification diesel is a complementary to the some object and in general and you don't have the uniting the category which is what you would need to interpret the oldest litigations and implications in any meaningful way in uniting category on it categories William and Neukom under Houllier categories about 30 unaltered frequent anybody today are the majority of couples is Arnold Williams saw and if you want to interpret this anatomy that it opposes so you'll you'll you don't want to know excluded those who is to lose according to the report it I used history to think that this is not of things for suggested this season has sort of what I suggested to Bolton earlier when and personalized the media our what I what again I believe I want to introduce and just the 1 particular obligation for instance I might want to introduce an indication of the quality so if I to introduce this I can do it that way remained inside there are and logical and the way I do always assumed that I call on diesel let's according to any foreign additional the quality of the Sweeney it by adding relational similar and what I need able to the of a book that axial which ensure that the center would be delegation but litigation in the sense of a compliment all the younger 1 of which is that in the provision of the yellow cards this season adjust them to death stock and the other 1 is that when you take the intersection of toward usually against the forces of using these completely coherent Axelsson because he and I have a seat at this juncture he and I have the the fullest and this is going on so we have inquired entomology and this is an operation that there willing going to talk about the later thousands will do that to you as the need to understand that anticipate because tomorrow the simulated with three-year-old because it is a natural thing to do when I when you have a fury to and through on injected 5 8 in the sense that there you want the category of modestly the same except for the fact that you want the job rose to become a werewolf seasons and this is the way to do this you will add to the signature of your fury this thing that automatically and you will and you will get there and in times of Saturday's Mandela said the models will be the same as balcony areas will become the injected and a lower fees and so forth for your theory and of course you agree to work with the recovery of cars without a hearing through hours of course you can always so what they said to me What concerning the in what is said earlier concerning the modernization means there is a general procedure and that doesn't look at the fury as specific adding faced an invoice for interfering in the driveway and solve fraudulent mortgage in most situations such as you do this way I mean you just take a specific formula and the conceded his knowledge you try to make it through metaphorically reinventions of the whole mess and of course you have seen an illustration of these indicators of litigation which can seat unhindered implication is more united than that because litigation can be seen as implications of forcing and you cannot make a decision 3 years went 4 for the implications so embodied in situations where you just do the same relative to the genetic purity don't to need the tool to win floppy life disability hate the machinery of modernization OK sold all match and all Californians know what to do what I can say OK so what
should I say now whether it will also called his 1st on the way talking about the soundness and completeness labeling already alluded to completeness earlier laughter I OK here's said analog Kaneda told little something missing in virtually all OK and hour OK we had said the what the what he user and a structural is seeking a structure in indicating that he will find it proponents of course we are now ready to say what it is a model of the fearing when I intend to define their own refuse sense of structure this is completely stifle warmer than in a structural fees and is there something that has the nation and behavior with respectable for functions Engleson relations symbols so you can write to System for 1 who commanded the condition yourself and he doesn't present any surprises and so I guess the police and the standard I'm sure there's something in the diffusion of mother of hearing so you fury achieve all that is certain a signature and the I supposed that the In a fragment of the refusing a fragment that is interpreter in the last year that the utilization of categorical structure for 1 to interpret these things so and suppose that the series in fragmented is intolerable in a certain category C and then once a and that this is the a structure and is a model of the he will always the axioms all key trends on the actions site of the original sequence of these former soldiers would things of this kind I'm satisfied mean and what does it mean that the slide in the Udinese cemented their interpretation of the unprecedented ease contained has of objected until the preparation of the conclusion this is a clear because any man and entered the intended meaning of a sequence was that for all the excess and fighting and various this so this is the same thing as saying that there would be another element state accusing his interpretation is also increasing show so now we know what they're in modeled fury and thereby conceding today the category was objects are he has said this with the layout of a widening actually not you are there to make the cinematic about that and if the substance to check the Axelsson because we had the soundness and I am about to think about the soundness so solid tender the only business year that lucrative we were in government is just a part of the logic in which you make the choice of restricting your attention to certain particular classes of forms the Socialists tightening security tools will consider just some the not is just the way you define infuriate you say if you use a set sequence this sequence of called the actions of a few major image might have initially along that anyone can non-aligned in Kenya within a few days and presentation sold within the whalers to degrees and then just east of the sequence electrical accident on the of the unknown not at all not at all also on otherwise it could even be appropriate soul analyze it is a set of sequence in and there and then and there is this very important for now just the location location is the 5th of Jean Monnet the This is an occasion for the category was the objects are the mothers of the inside of a Category C and the gyros are being on a structure or more seasons so the national movies and he had a little more than some of her were reasonably search is exactly the same Seoul you will hear the finer than than ordered the category of structures in the season .period got the OK this is just the former patients now soundness and completeness season precisely what the answer is that there question has something me is that there any see plenty which she will and in seeing his body can embody dissatisfied insisting on that in all modern some of them came in categories the product the categories all the appropriate kind of a relatively fragments of course although the appropriate time and knowledge of the managing the sees a feeling that it is very easy to do because the only thing that it's quite a sight toward the only thing you have to check user will eventually need for resolutely the brain uses a satisfied and also because it because the whole body the Committee said that the interpretation of the understand and he's conveying the influence the seasonal notional value the Indiana OK and completeness of his adjustment in the economy yourself I mean the expressed sequence is valid in all the knowledge of the fury in in the categories of profound and then it is probably and I have already given up on the director usually don't adjust and completeness but we had strong the completeness of thanks that exist themselves and that the cataclysmic this it would be for after that tax time mean