Toposes are commutative rings
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Title 
Toposes are commutative rings

Title of Series  
Part Number 
26

Number of Parts 
28

Author 

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. 
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Publisher 
Institut des Hautes Études Scientifiques (IHÉS)

Release Date 
2015

Language 
English

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Subject Area  
Abstract 
Abstract: In this talk, we shall develop the point of view comparing (higher) toposes to commutative rings. We shall then see how the corresponding integral and differential calculus are related respectively to Verdier duality and Goodwillie calculus of functors.

00:00
Computer programming
Mass flow rate
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Student's ttest
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Lecture/Conference
Topostheorie
Analogy
Circle
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Mathieu function
Gamma function
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Topologischer Raum
Infinity
Set (mathematics)
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Ring (mathematics)
Recursive language
Maxima and minima
Category of being
Computer animation
Object (grammar)
Local ring
08:02
Logical constant
Point (geometry)
Group action
Finitismus
Connectivity (graph theory)
Multiplication sign
Sheaf (mathematics)
Embedding
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Automorphism
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Morphismus
Goodness of fit
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Topostheorie
Square number
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Subtraction
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Military base
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Infinity
Grand Unified Theory
Set (mathematics)
Binary file
Limit (category theory)
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Maxima and minima
Category of being
Network topology
Universe (mathematics)
Charge carrier
Right angle
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Annihilator (ring theory)
Resultant
Local ring
Directed graph
23:35
Correspondence (mathematics)
Multiplication sign
Distribution (mathematics)
Complete metric space
Discrete element method
Rule of inference
Number
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Morphismus
Frequency
Sign (mathematics)
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Wellformed formula
Homotopie
Analogy
Diagram
Selectivity (electronic)
Gamma function
Subtraction
Position operator
Condition number
Covering space
Product (category theory)
Spacetime
Morley's categoricity theorem
Infinity
Ring (mathematics)
Mathematical model
Functional (mathematics)
Measurement
Maxima and minima
Category of being
Proof theory
Boom barrier
Network topology
Equation
Object (grammar)
Local ring
32:34
State of matter
Multiplication sign
Decision theory
Sheaf (mathematics)
Parameter (computer programming)
Mereology
Food energy
Grothendieck topology
Insertion loss
Analogy
Circle
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Area
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Infinity
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Maxima and minima
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Vector space
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Equation
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Finitismus
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Ring (mathematics)
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Table (information)
Coefficient
Local ring
Spectrum (functional analysis)
50:58
Complex (psychology)
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Grand Unified Theory
Distance
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52:22
Point (geometry)
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Divisor
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Sheaf (mathematics)
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Lecture/Conference
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Profil (magazine)
Operator (mathematics)
Theorem
Position operator
Condition number
Compact space
Shift operator
Spacetime
Product (category theory)
Forcing (mathematics)
Exponentiation
Content (media)
Algebraic structure
Infinity
Incidence algebra
Mathematical model
Cartesian coordinate system
Bilinear form
Local Group
Equivalence relation
Maxima and minima
Flow separation
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Proof theory
Voting
Logic
Duality (mathematics)
Object (grammar)
Spectrum (functional analysis)
Resultant
Local ring
Separation axiom
00:03
we Over a worse and I
00:16
so thank you very much and I like to think the Europe organizers for inviting is working and so it's a pleasure to be here and where particularly to talk about the process because this is his the very place where the officers were created some quite happy to be there the so I don't know yet if we the golden eyes well with because when I give my title and undecided on my talk didn't always way I was going to have to do the same thing as I said was that appear well I guess I guess is for the best because now I don't have to justify the analogy between the octopuses and committed to algebra uh soap and this is this is this why change them under the title and so instead of explaining the are committed during all yes that's on using interest of mine a bright topples inapril constables but but but has sold this has been done for our review of rings yes No no this that makes you can print that this woman unit with the messiest so I want a 1 0 . 2 2 you to describe herself program I which is under development of the analog if we take this analogy with committed to rank seriously then there should be such a thing as a commodity of algebra affordable I almost wrote market Marion should please everyone so this is a joint itself around when we know what this is why we decided to take on the same matter that there would be demand the staring and the manager so some of the thing I'm going to talk about today are approved their conjectural the kind of trying to do develop the understanding of what this committed of ruffled services could be and must be a naive question about that's all students without answers a lot fully I'm going to give up pieces of the of the theory that should convinced that there should exist such as thing OK so the 1 on the network in the setting of infinity categories and Infiniti divorces so let me briefly say that I'm not going to define infinity category I'm going to need to use the on the notion of infinity category as if they were ordinary category because actually the syntax of the 2 theories exactly the same so but if you want to have something else In your mind and investigative is probably best summed simply defined as particularly enraged spaces only proposed as long as you know what I mean by spaces Surbiton vigorous so MySpace I mean Infiniti will parades which can be defined as those with can be formally defined as topological spaces that tumult of pain or contemplates as the Soviet leader so today I'm going to do some of synthetic higher category theory so I'm not going to to to use explicit models for the objects are going and talk about so I'm going to start with the definition of an infinity doubles so only gave already the definitions of a miracle what it was and so on and isn't s does been news a lot of I'm going to use it to to refer to the category of states so it's not sets anymore and all the symptoms space so the category will be called presentable it is an accessible localization Of appreciates category where C is a small if doing so I'm using categories with really meant is indicative the firm Vice President the tools you mean what you would call will exactly yes I'm using this unconventional murdered the in its so a Category III will be an infinity if it is preventable tell for any diagrams from a small category II we have the circle this condition but the objects of the
08:03
continent the category of objective of the commitment of the diagram is something else for the limits of objects of each component of the diagram so I'm going to on and to tell that to bits so actually the final find most of categorized as to abuse from tolls between them so the limits of this category can be described I was hostile to the category of eye diagram in the and I look at all mine over my and acts and then I look only at full subcategory of morphisms of diagram that the competition which means but I'm looking at objects why why I the them Weihai over such that every square like this for any oral in is contusions so I don't think the it's of such a diagram on that produce an object good intellects and here given an object I can't just defining constant diagram by just pulling back a to my object here over each of the exit so the fact that this is an equivalent so the design this is an intention and the fight is an equivalence can be overstated by 2 conditions it's an equivalent to that of to say that the underfunding of constant faithful and the good in front is fully face so if we unravel what this condition means this is equivalent to In the the universality of commitments and this donation which is no is very strength is a very strong dissent condition it's called of comments so in the usual posts only occur when installations are effective In an infinitive opposed where can we can that there a violation of there's a set of facts manageable for infinite Stupples where we where every infinity will .period is effective instead of a Waco insulation and with this formulation is even stronger it's infectivity of any kind so this it is true In what I will call 1 top posts in the old irritable says so this condition is true and that this 1 is very false as all explained you know the nite certainly give you some examples both so have I want you to come to the park there review usually up so approach to vote yes yes also was the sole reason to be grateful if you could exactly exactly Europe will be in the final of the this is the year of the release of 1st of the 2 of you is the view of some of the government sources have so so this is about wanted to avoid so probably the a complete formal approach to these would use moral categories and I will have to use as a result case for categories of diagrams on and the find amusing the right from tolls on but I'm kind is keeping his part and I'm gonna presenting the theory as if it work as if it would work without the formalism of from a look at the guys and without presenting an for the theory formerly with insets recent so I'm assuming that there exists a theory of infinity categories that the national diagram makes sense and that I can construct a category of their own and except for some I'm not describing the basic accepting onto unassuming indicated it exists for this so you can go home mortgage worries of slower growth this in the year the I don't know and that's not what I meant uses the history of the world as a whole was so yeah probably you so so here I'm not I'm not assuming I'm not as ineffective so that is what with the definition of a presentable category it will be the case that some of the Hong Kong space between 2 object will be small but what we can even managers that OK sir ,comma to my examples so only explain how to construct the free posts so if we start with a small category C we can add finite limits to our to see and take appreciative of this commanded Ali the press chief over any small category is already at Opel's it's not free but it's OK as as the particular case of appreciation I have if the because U.S. bases it's to the post the category of goal and spaces so this is a tumult appeal and this is just a space with an action of GE L I can do the usual thing with the area medical theory and it will have an enveloping infinity doubles let me give you an example of very silly theory
15:50
so so I can consider the freedom posts on 1 viable so this is the the topless classifying objects so is the theory associated to the To this theory was 1 by and that's it 4 an action yes I can consider the slice the postMilosevic older X so I will call it like this and this 1 is classifies objects with the global section so I'm going to college .period hello finally let me mention of a source for example and accessible localization which is left exact although posts is again at the so this is this wasn't expected so as an example of such thing and I hope that all the other time to develop it further but no mention that In the work of God willing political reconstruct left exact accommodation of the stoploss so this time was completely it's fun from finite pointed spaces L and the theory of good is essentially an the studio but are wary of left exact localization of this top boasts the call and excessive organizations so that you know the embedding of these In 2 SOX is noticeable and excessive from talk we see on the the the you from the problem from this this yes I'm seeing this as a subcategory of of the full subcategory of this the law you usually was strong None of this is this is at a high polls yeah this it is the carrier's bases here note this 1 is pointed this 1 is not .period yes fact this is the promised to hold yeah you exactly a finite way exactly OK I that ended was for his part of the settlement a at that the it's good to know so let me say a word about the difference between the Infiniti doubles as in 1 of the busses so that as many differences I'm just going to have to mention 3 so the first one is 1 that I always mentioned that the subject classifier it is financed into an object as fair I'm not going to collaborate on this 1 but they just say that this is useful for example to rights In those or I'm going to use something like limit amenable language this is useful to write songs like this now this makes sense now this is an object of the notables model the size issues OK so now the 2nd point which is probably the main the main difference is that left exact localization are no longer controlled by goaltender topologies national goaltender topology still makes sense that's not every last executives nation is associated to that so let me give you an example if X is the political space it's possible to build an Infiniti doubles of she's on access and these are fertile from their bonds of satellites into space is an analysis by some condition and the Commission is the usual Commission for coverings of anew opponents so this is the list of you should use their worry is thinking about
23:35
causes the apple farming in vintage and now there's another
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doubles by can construct associated to 2 x it is the socalled hybrid completion of this is that I am but I impose this sense not only on coverings but on Haifa coverings and it's possible to prove that the 2nd 1 is left exact organizational the first one but it's not the kind of this not an organization which is given by Tenochtitlan so this is the most difficult technical thing about Hyatt opposes that all the technology of public apology is not enough to control of exact position so every time we have a proof of federal Seibel fury that uses a government apology will not it may not generalize is the other a statement for invented intervals so this is not the usual number as well the loses some of whom descendants is of that who is currently so yet another is yeah this that there's a difference between the 2 some conditioning it this is the latest about the movement is no a yes yes faltering catered objects only achieves with value integrated spaces then the 2 motions on so it's only because we look at shoes of a possibly and that her home to be types that that the 2 things are different OK let let me mention the .period it that in 1988 since we can impose some fun equations like something like this so this is an equation that has known until solution in an essay but it has many many models of Infiniti's appeals they seemed a yes or a signal suspension and so unassuming axes appointed object and this is the suspension of the rules space selective sorry who could you were going to have survived the destruction is convinced of the OK so little space is that is defined as the the final product of diagram homotopy phytoplankton designer L the suspension of and suspension of an object is defined as the push out the home of the people shouts of these diagrams so if I computes the suspension of the above today to see that as a canonical beers and we ask that this market is immigrants so this is when we come wait for example can consider left exact localization what that would be denied to by and I don't think you'll be generated by Botanic apology this is also a chronic condition that it does not seem Michael tunic topology that then someone the the idea tunic is just that a government apology or inventing monomorphisms only and the question is can we control any marked by by inventing a bunch of monomorphism and the answer is yes and when the categories truncated by the new things together its monetary buckets on the French elaborate soaring again half my time so let me ask a few questions about this analogy with we then committed hearings so so if anything it opposes committed rings then way we can ask a service the following question the day compared to all the Marines 2 other kind of categorical rings Is there a corresponding algebra if I'm thinking my rings as functions there such a thing as a distribution various measures there for the more than the as formula for use good the was the new rules were at work in which 3 0 sorry as I I'm assuming that some people attended the talk of all actually so that problem with the rest of the world OK it will get together and get let me make a remarkable there you're sorry I should have should have said that as so it's convenient to introduce to categorize so I'm going to introduce a category that I will call she and another category at the local topples so this will be the algebraic side and this will be a dramatic signs and this category would be opposite 1 another so here in the objects they all the top posts on that's a it's convenient to have 2 names will fall when we look at the topples dramatically and underregulated like locales we have a thousand frames and so I'm going to call them categories of sheep and I was a kid the name doubles phylogenetic side of the cold In the the Olivier friends and I an like it's so that the morphism Solar Circle algebraic morphism that only defined so between 2 categories of shares and looking at the front office that Apple continues and left exact and here the move is another genetic morphism which illdefined just by defining its category is the opposite of this the I won't talk about it to say that note that other lifetime Olympic title at the head of the movie you can ask again the question at the end OK so less than are connected to my to my questions
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all we think that we have yes so so the comprising of 2 divorces without an accommodative rings is on the edge right side so there that in the economy it's also a sense and the finite panicking it's as soon as some kind of politics and so the the Mofaz and sold them the front of preserving sums and products In OK last question but if this is indeed the will the refrains and if the emotion of polynomials and their notion of differential calculus OK so I want time to answer all the questions and actually I don't know how to answer all the questions but under going to sketch pieces of insects so let me say brief word about the 1st .period it's I'm going to use this poetry so in my head I had the follows the following an energy that is helping me so yes it's a good thing that assurance the Sunderland drawing table so where I thought was symptoms the space and associated to that Will the notion of locales than we have set an Associated too that we have won top post and then we have spaces and associated pursuant to this we have initiated a post and the parameters aligned services and this year there will be no more than that he said he thought that I'm going to compare lows of this specification without classification and all that I committed a bit rough and another thing to to be more precise but essentially can do this so going to the writer can explain it it's possible to think this way so separatist bases its functions with coefficients In 0 and 1 so it's like polynomial was coefficient in 0 1 In in doubles will look function was valued sets and so by using the cardinality we can think of them as function with valuing integers a foot that functions with value spaces we can think of excess something it the Celebi mention 1 and essentially the way to go from 1 the left to the right is taking some of the characteristics and then we see that at post is a kind of the algebra and then is a divorce would be as Umax algebra so this is all denied apologies so nite apologies to the poorer for the possibility of Watson and wage material was to is to adulation intimacy or over more complex objects they said they 1 it's it's and I'm thinking that acts it is an internal without wondering if there is a problem no experts 1 would be the circle the 6 who has no no no it's not that it is so this is all the areas around this is all the knowledge about Sociedad rating it's an honor to committed agreed if you want I can ascend space here to its characteristic series that's an X is justifiable for them look certainly on something which is not a top post but it's useful to have in mind the 2 minus category of spectrum an early Kadeer's Uzbek trying languages correspond to his friend so we have invented explains that's there's no Xpress 1 is the circle and we invent Exelis much so it's it's so that on the circle will be X minus 1 plus it's a it's a 0 cell and no 1 minds Swanson uh 1 of the 20 minutes and so this comprising uh so the freedom plus compares to the to the free rein only explained that's earned and at the poll that the post without points is descending as the rain with no rational L L I the 1st of theory or a sites this is just not enough to generate a ringside to presentation unfortunately we hope that there should be another the With the disease I have no idea if they have a new worlds of OK OK but the question is what is another break equation the 1st love them and care when you have a system that is once this is world that this would have to be an example of an algebraic equation yet of only 4 you go get kind of admit we're going to give that after you have to have been the last few days .period with the the decision was made in an interview no it's just an analogy just analogy I don't pretend that this is the this allowed to and it's easy to think that the ordeal of the evening this is the duty of the world .period and we think the last filing was the scenery yesterday there were more than 1 who and amateur amateur understood then that the questions maybe we can discuss that later on OK OK so that was but I don't know what is the descendant of the I you he and a very good question I don't know if that happens that I guess this goes without saying that was differential calculus what is an ideal is there such a thing as an ideal it's not it's not obvious because this is a highly nonlinear and we where we have a kind of somewhat what would honor the section so it's not clear what and I don't think that this reflected his belief that on the the leader of the group this isn't the 1st is the ogre Mary Ann so where was I prominence so let me say a word about early in algebra so
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sensitive the thing goes Fontanini ruffled opposes a is the theory of whether categories so this is the category of presentable category laminar views for the fullfrontal and the model size issue that I want to enter into this to discuss now there's a the underconstruction Of all the free oppose form interval category it said that an analog of taking the significant about all of this was going to this is executive sodium is it will work here and we can even denial as a national supposed by removing the patentability assumptions OK so far the category of preventable at you is a very nice getting going into mind the semantic close category part of that is due to the this is yeah yeah well that's that's a good question it OK so let me describe the tensile politicking In the particular case of 2 preachers category the principal actors just appreciative of doctors and product of the categories and and then the general of product is computed from this phenomenon by composing was organizations on the left and on the side helps I don't have time to give the order but the precise formula for the about it it's exactly the same as the the 1 computing tensile Pollitt the vector spaces would take something pretty generated by parents and then we impose a tonight on 1 side and United News side and and this gives the former Soviet ineligible for book yours was vector spaces is perfect it works all formulas and all the results of that no 1 can expect when such the the fastest and most the boat there yes yes shortly I II and III this year it will be here the world it neither is it is so that this is done for example is this is and a newly hired about on OK where was I didn't want to lose but would be significantly more presentable getting the guy who was in this category this a systematic nominal preventable categories Bloomberg said Scotland said this was a closer more prone to violence is well as Williams this in the category of preventable categories None other that that there will be preventable categories with all Semitic Memorial Institute researchers 1st of all this in a way which it wishes which will be closed because it will have to be OK as I didn't even mention what can a morphism I want here I won't go continues more will follow all this is the at the local continuous morphism between patentable go continues morphism go continues rentals we have been sorry I couldn't hear you Everything is infinity category but works the same as classical in the classical setting the location of the wearer as the example of an example of the money's here are so so as is the unit following because your spaces the unit for the distance apart if I look at the category of truncated spaces there is a and today but important assistant thing for spectrum to several of the states all of this OK I'm assuming focus so sorry targeted spaces a space was home to people of Spanish and at after all In this case so I fight and saw the Cordero spectral itself again spectrum and this is related to this this formula here if I sold this over as Eurex I get itself and then there is the removal release of the Universal of all the of the vessel yes yes it's the universal object that classifies a beaker continued suffered or around OK so these distinguished the more this guy will be presentable and categories helped the module spectrum will be the stable multiple categories for income 1 foreign visitors were the OPS yes so categories would no higher morphisms degree higher than and uh located on late Monday trade the it is also more about the handling let me say a word about the measure theory so if it a is a committed rain I can define a distribution on a as a so I can executive same fault sorry all feel that I have not apology on my ring but it's it's very committed differences all In the accommodative setting so if it is a top boss I can define a distribution as a cook until start off from 80 to spaces actually there's not a lot of examples of such things the point of view will be examples but it's difficult to produce some although examples so that sideshows useful to generalize a bit this definition by saying that instead of looking at this additional value in S we're going to look at his mission with value in any category so I'm going to have to take to the case where the category the spectra so such such rentals go continues for their Kolko shoes also so we can
51:08
define a kind of deal of light looking at go continues frontal from into Spektr and we can also defined by a base changed the purpose of stable she's so let me that this is the category at 12 she was all spectra so the distance of politicize very useful to produce this is not the usual definition but it's a very nice this same the raw yes that this is the that of all other political party was designed to prove 1 to was To do that part of the the yes it's a it's a pheromone he what she is on on the the so there is actually that say that these categories assenting as a category of complex of coal from also from the old to sp other continues you that's a general
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formula for that and support to the citizens of the year should should have said it's this has nothing to do with the fact that post commission demands 9 from group a OK so let it let me try to enter into this so I like to explain all this is elated to review In so that it lesser candidates to sitting of operations on the category of she's without you intend complexes or inspect and um as a nice results of animal who came up with this idea to it's about it this is a there's a statements in 9 and a high of rebuke of fury and that's not immunity should be seen as a Gennady between category and its deal in the sense of presentable getting so but duty with Mary is to set up the exit is the Compaq Dell still space then I can define the category of Infinity shares on next and I can take it shoes a spectrum they can also computer the deal In the sense after and the statement of filets that's embedded it again understood as an equivalence of categories between those 2 things so this is this can be also written has infinitely coaches with the new Inspectorate on X and so we have an agreement between the category and it still has what the space combatants OK maybe maybe you right maybe a locally compact should be enough the content of this of you what Over here it's that its precious adherents coaches coaches all this is shares with value in spectrum so that you have the floor is devoted to polish With value inspectors who yes he's yet exactly so so uh this is a very intimate in those categories of from 2 OK so but that was we are working on the statements about analyze this and that expanded its what is going on former title poses so let me explain what we could do so the question is can they can replace explainerto Paulson what kind of iPods doing a little such as such a statement is his true at all sorry let me mention that the way to to transform Shifrin took a shift is by taking section with common compatible Florida the centered the discharge of producer
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Koshy from shift and vise versa so we need someone to look up capacity condition so question What is it look at contacts doubles the protected on the list several possible answers to 2 questions and here is 1 that is satisfying but locally compact space is the same thing as an exponential object in the category of so this is the the well on construction of the old compact opened apology on the space of French and so we can use this to define the Kenekham octopuses as exponential objects in the category of toppers so to go back to as an exponential troubles so sorry to hear that the view of the regular physical clashes with the but yes yes it's at its Poland in the Book of Peter Johnson for example instance spaces and actually any expenditure will at is based With the Little Rock is voters in the local form of programs there's less lawfully use as long as you know it's it's more locally quasi compact list this this factorization is actually a local equal I compact spaces this question the no not hostile the working minus 1 and 2 and a half years Selig and try to write a statement so here is the 1st result at the police's exponential if and only if it's category of shares is what is a continuous categories and underwrite the definition immunity is fact of a category of an object why From dolls filtered couldn't so this is exactly the same statement as in a bear profile violent incidents the thing is the proof is different because they use continent apology to prove the state and here we cannot use continent apologist when in in another approach and now let me finish my statement which is not fully written so I'm going to call it 1 theorem the only that's what we have here in the Stations of the Cross the group and you have to live half yes I should maybe I cannot draw the Infiniti because I'm used to forget about that stuff Ferguson and here OK so let me finish with the kind of India and the so if he is looking to Compaq winning exponential and where mechanical bag diagonal so which is a kind of separation assumptions then there exists in front of it will play the role of a section was conducted fault I am this franchise the trace of a full menu structure on the category of able so this means that have a frontrow for section was compatible with civil spectra would value spectra L this induces the perfect pairing between James and so it just an equivalents between the category and it's still so now it will finish with 1 last remark so this noted it is In an nice because it explains the nature of the stick frontal uh we have she's and go she's and both categories are not only from Doyle was picked to eat so here we have from it is like this between chairs here we have the but we go like this between coaches and using equivalence with companies this fraternities into shares and this is building this from and around now because it's a full menu structure the committee was rock structure here the semantic nominal structure here produces a semantic go roster churn on this site so it appears that the the new position of attending against be completed because we have a call mystification also take care what we think of the Commission's possible the results of this verification provisions for this is that the knife not badge right Fellini so so it's like the switch of Mexican demands the whole world it is the end of the Nunavut that as significant a product of Hughes and and so but but by duality because this is how this is compliance so it produces a caustic traumas what do you think procurement minister Peter you the political implications always the 1 I ask a question the use of force and not America top of the list b we have some of this visibility in the the I thought that was the real thing I at at her home again OK OK so let me have it 1 film and so this to convention fall to cells in the category of 2 post the SGA Convention and the Convention of Lydia and so the usual convention is to take care to look at the airport and invest nature and to take note of information between investment and this is not the SGA convention this is a convention that is nice for application in logic because with this convention there the category of points Of the fatal blows this is the category no the on the domestic side the it's so by definition of officers from a F is another right morphism from after it and and the natural it to sell between F and J will be to sell between now and the corresponding frontal and Simon getting the older also fall to cells so at this convention the point of the and the deposits as of banks becomes as so both conventions on all useful this corresponds to the convention where the point of a space although by specialization this is the convention whether although by generalization so I guess it's upon of allegiance is to the justices that the provisions of this program performance but models here and I like to call them at a later :colon models there is a lot result as a category of models in the category of points under the category of appointed the opposite of the category of mobile and you it's convenient thank you might have had the Heinrich to you