2/4 A Crash course in topos theory : the big picture
Formal Metadata
Title 
2/4 A Crash course in topos theory : the big picture

Title of Series  
Part Number 
8

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. 
Identifiers 

Publisher 

Release Date 
2015

Language 
English

Content Metadata
Subject Area  
Abstract 
I will sketch an overall picture of topos theory and of the theory of locales. It includes the notion of sheaf on a site, the notion of forcing topology, of geometric morphism and Giraud's theorem. A useful principle is that a topos is a commutative ringlike object. Every topos is a quotient of a free topos, like every commutative ring is a quotient of a polynomial ring.

00:00
Area
Addition
Observational study
Closed set
Parameter (computer programming)
Mereology
Amalgam (chemistry)
Event horizon
Element (mathematics)
Category of being
Computer animation
Meeting/Interview
Lecture/Conference
Topostheorie
Right angle
Maß <Mathematik>
Cloning
02:59
Point (geometry)
Functional (mathematics)
Presentation of a group
Statistics
Link (knot theory)
Confidence interval
Multiplication sign
Maxima and minima
Power (physics)
Group representation
Thermodynamisches System
Meeting/Interview
Natural number
Diagram
Set theory
Descriptive statistics
Focus (optics)
Matching (graph theory)
Generating set of a group
Closed set
Physical law
Coordinate system
Price index
Limit (category theory)
Category of being
Musical ensemble
Object (grammar)
Maß <Mathematik>
11:27
Group action
Presentation of a group
Confidence interval
Multiplication sign
Equaliser (mathematics)
Direction (geometry)
1 (number)
Explosion
Group representation
Mathematics
Spherical cap
Positional notation
Manysorted logic
Different (Kate Ryan album)
Analogy
Diagram
Arrow of time
Series (mathematics)
Position operator
Process (computing)
Closed set
Moment (mathematics)
Physicalism
Coordinate system
Price index
Complete metric space
Category of being
Radical (chemistry)
Uniformer Raum
Duality (mathematics)
Order (biology)
Triangle
Right angle
Computer programming
Finitismus
Free group
Divisor
Similarity (geometry)
Element (mathematics)
Product (business)
Morphismus
Lecture/Conference
Wellformed formula
Complex number
Stochastic kernel estimation
Set theory
Condition number
Focus (optics)
Uniqueness quantification
Projective plane
Physical law
Commutator
Algebraic structure
Limit (category theory)
Numerical analysis
Network topology
Object (grammar)
Musical ensemble
Family
Maß <Mathematik>
28:00
Presentation of a group
Group action
Multiplication sign
Decision theory
Direction (geometry)
Rule of inference
Product (business)
Morphismus
Lecture/Conference
Analogy
Square number
Diagram
Condition number
Chisquared distribution
Process (computing)
Weight
Physical law
Projective plane
Limit (category theory)
Equivalence relation
Category of being
Arithmetic mean
Isomorphieklasse
Order (biology)
Right angle
Object (grammar)
Resultant
34:46
Group action
Functional (mathematics)
Direction (geometry)
Multiplication sign
Decision theory
1 (number)
Maxima and minima
Product (business)
Prime ideal
Morphismus
Thermodynamisches System
Lecture/Conference
Meeting/Interview
Term (mathematics)
Wellformed formula
Diagram
Extension (kinesiology)
Curve fitting
Condition number
Area
Focus (optics)
Physicalism
Variable (mathematics)
Flow separation
Proof theory
Category of being
Arithmetic mean
Funktor
Ring (mathematics)
Order (biology)
Universe (mathematics)
Right angle
Resultant
Local ring
Maß <Mathematik>
00:02
room however I'm among long known the CSS here saying the whole of the the
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day huh after the close of the stock so the it should the
00:30
finishing I hope it would be very
00:36
abstract or part of the world and supposedly is attempting to remain also according to the study we will hear an argument we could go on and on I would say the book is sold In addition to the right to protest closer and then in 2 % you you can talk about for example esteem on the meat between area a lot of the Supreme spinoff costs you have also been fuming as well as the sacred joined school this is called me this is just a few hundred the at on Monday at 118 when it exists between 2 elements in opposes this ,comma Scotland me just a technology but this is called a joint now if Europe is possible by categories then you have also this editing here this historical event before going around so you have a background tickets corporate periodinstrument boldness of the and here you could take the top priority of all the better if indeed the Amalgamated prompt unit commuted the defendant ,comma missing on the same said that
03:00
to the category is I'm the lettuces ,comma so let us it's simple that his music so credits is
03:18
concerned with the environment and the recent remark by solicitations of his supporters not to the emotions of course concrete category there is a category follows on the political so proud of you complete delivery is very much like to a super lettuce if you like in the world of carries around in and is solutions of a complete category this close on their limits I want you to know that everything is a small and it's something that you have a small diagrams and put it in the lock on yeah for financial and it's like to call that kept the I don't like this are allowable under the law and I can say that a complete you have here who ruled over the whole truth about where it is at my own living there is leading the Union all we had was that it's like they're saying it opposes closing of Supreme it was a way to get the idea that war is that it's an abuse of language right here and yet but the interacting can make it's strictly up to here with the of you have all but hasn't had the idea that when you take time limit for background you take it indicating of proceeds and if the firm so sold it always exists in the category of procedures and and if this limit actually representable then you say that the economists close on the limit so if we add up the point of view of of quick can use the word OK so it's not completely wrong not because of the nature of the year OK button so what is a presentable category presentable category isn't it a bit of a technical notion that roughly speaking it's so cool completed category with generators this is a small set of generators it's not too why is that it is generated in some nice so instead is a small accordingly inside and so I could take some time to explain to you what is a presentable category but I'm afraid that if they use a lot of my time so I will supposedly notion that is known about 2 writing some low turnover and make it to give the definitions and animals so presentable the actual examples of presentable category is that you take a small category then you take a procedure called a was to add that this 17 confidence in front of from data sets which are of course also same :colon informed from a up to set this is an example of of a presentable continues in this case the generators citizens to buy a so that you have to unit Franklin and the continued to receive and deserve every object ever appreciate physical limit of a presentable so that stares you somehow that the community of perceived as just by but not only and debate but it is freely convertible it is for the United that's importance for the category of receives 1 of the people of the world exactly exactly so what what this means is that if you have a I hear from into account political complete getting in there and this is essentially a unique way to extend it into a coal consumers hunkered down low enough for that to see what local consumers hunkered is the frontrunner is good consciousness is Pacific preserve continents is constantly it preserve them it's OK so given a photo from a to B. it is essentially a unique focus the Wallstreet which is go continues and which extended if this function that can be described easily found the history of the left extensions of FLYder statistical but it is a concrete description of what the reason is that if you take a perceived then they perceive as a coordinator for the representative of the mechanical way which means that it's exaggerated :colon Over the diagram they were exists wouldn't be that the Russian understanding from recently links but powerful of flurry of I'm sorry but don't reflect something that has a tenancy when it was 1st observed at any price if it is a canonical conservative representable this index Is it a category of filaments Amin and using the unit at the match to identify the remains of information from the presentable
11:29
Hawaii it's as the news of solicitation is a from a to something so I a right arrow from a x where a is an object of the category as of noon because on the nite of the Commission's celebrated like this and I know that this is the same as an element in its and so a category independently of filaments of affixed but the objects are then a where a presence the morphism our whole commuted the triangle of this kind of thing should say a isn't in the continuing so this is a indication for this category which includes his accuser like the category of animals suffer from terminal there is no reason by GM it could element fix known as a slice is played over X is the candidate of representable objects mapping 2 weeks this is the same thing so you have no understanding of filaments of appreciation the plan on December action from the Arab world 6 2 to pay and you can compose with a uniform to receive money and so this is a projection and you get a diagram of a representable presentable index made its original offer comments of a OK so that any evidence of a here fixed in the midst of a series that's to be the object is like that and the value of this factor at a given element is just 1 of 8 few to forget this structure the small areas in new anyway so every PC visit technical coordinator for presentable it's a nickel I mean I do that mathematics works with a certain number of medical care in the sense that it happens that something becomes very very simple and we don't really know why it is so but this is it's it's it's like the complex numbers knows it a simple but what is it that I don't know but too but uh but without the simple facts that Maddox would be very hard or impossible and it's 1 that we the unit of and this is the fact that a very pretty physical limit of Forbes interval in economical way makes life very few easy for us mathematician and because if you want to it extended continuously local continously folder use this formula here because if I have the fund which I want to extend so I assume that the fund to preserve coordinates so I will have the formula that if AstraZeneca according it would be the quantity of half of what with a 2 this and other times I mean it is annexed ethical questions since it will have it will have to be there but FFY in his native F this is so proceeded to quality so we have a formula for extending not narrative from going into a local hospital from which she there that is unique and is a unique way to do that so the definitive received is the Filippo completion of on conditions on the requirements of a category and I would like to think of this as the position of the of the category because I want to think of a way of Superman as songs and I wanted him to qualifying limits that there's oversold what I'm looking here is that he had visions of category it is very much like taking a set and embedded in it they to the he sessions of intruder the Aberdeen Group generated by this is a summary of the report because of riot )right parenthesis it's going to be here with you OK OK so we have this unionization process not do we have to be other process which is adding mitigation missing members that it is made to step from Sept it takes community president remains we are we all so we need to understand what is the 1st as a 1st step office building industry category which finance limits to how do we build up the 3 category with courage it's From I'm from a different category so in this case it would be like cats cap and woodfired units and then he actually revolt was in trouble with so we have I'm going to tell you how to construct a Phillies deposed generated by a category so at least we still don't know what I propose is that defined but will be able to construct the the troubles in the so how do we had found the commits to a category where there is no easier to and from limits because we know how to add Paul continents is the unit of getting this all units sold in I have found that a limits you may just need to take this subcategory of financially presented proceeds some procedures that are finally put it suffered a sensible so that and appreciation is furniture presented essentially if it can be described by opponents of kernel of a finite corporate confidence intervals costs look at the so you
19:56
take a fanatical product affordable product that you take the poor equalizer and that's what you have you that there is a fabulously receive a final presentation know the candidate preachers of finite presentation is actually a close on the phone and call it's it's not difficult to check company and the union of and used something here which could call a unit of again wetlands into a small category and this is that the whole competition on the front foot and it's of a which mean that now I knew that if he is if he is financially call from plead for completing the final coproduct Glencore coorganizer of the fountain here you have the ability to extend your front In the essentially unique way unique opportunity as amorphous and to have a triangle which commuter opportunity Canada has an office and so we know how to the "quotation mark complete and and small category for complete on the phone equipment but what about pleasing owner from units 1 of the marvelous things about categories mentioned is that the category has an opposite trend it's a marvelous I mean before Edinburgh McLaren we we we had made a few examples of duality you see that this is the category has an opposite we can always changing and it's just not amazing thing still there but it's a very important process so that the article or complete on the family the limits are needed so you take of that the opposite end of the 1st take the opposite of and that I think it's a finite call completion of and then I think the opposite and come from the United Front or you will get the here which you could call the again but this isn't exactly the dance and this category has finite limits Disney free conclusions on different units OK so maybe we should give it a notation on May on on the other hand positions such as OK all so I would pretend that it this like to completing alone joined this season joins are finite and the man who could have known completing the need to focus on growth will aid the moral order could be Maine had come from big completions look this is that than it would appear in focus that has no pain no financial None aspiring work all OK them under the was direction and it is natural for me yeah but I would put him on his election the for fashion completion of the rocket thank you letter the government of people that the problem of it's very yes a agree that a year annotations are extremely difficult and terminology all cylindrical music was very good to find the right conditions in indirect terminology and and I have a lot of communications you know there is still OK now what will be the fatal post generated by this small category well the federal posts I did and did not even tell you what is a troublesome what is the category but if it opposes rejected by a will be there the procedures for the column completion the financial conclusions all in all these all procedures followed procedure yet is not apology because it's free topology comes when you want to make a caution that remembered as in the case of frames we caution friends were indicted actions would do nuclear arms so if you want to make a cautioned you need the analog of the nuclear was in this context and this is what exactly what those at the Bellagio providing OK that now should give you some action year I have some sort of of similar actions Is this a moment here a year ago but it with a small problem because you started a small category and land was to be ones and so among others should be an undefined but about otherwise associated with that of the law that will be used to call for a new programs for the record yet not finished said I did not sure yeah yeah yeah yeah this is a member of amorphous and between frames are and will match that preserve supremo and and meet the need of finite and the muscle the dissenting for the proposals so I will define what it proposes spending during his election year you could use In general the definition include unwilling oppose is general characterized still pose listen was so even before Monday's action efficient on the cocoa presentable I think we need to his troubles the following the conditions the holes were
28:03
sold in the 1st when the pullback from tourists
28:17
"quotation mark issues "quotation mark the descendant action is that the following Square is a pullback or and B. so if you take the core product of 2 objects in the new look opened the intrusion on on and and you take a pullback to take the intersections of a and B including corporate then that the initial projects this 0 so this isn't to work on a treaty agreements suggestions are affected which means that if you have any politicization is is always the equivalence relation for offer map and there's always always obtain them pulling back diagonally it's like OK OK that's away a version of June election result exactly as you action but I want to make some remarks the 1st action here is distributed to missing the decision Tuesday after a battle order it so early because of its size is that if you have an opera and meaning and if you have a background he is not going to be sold is a diagram of fraud this small community to deal with so you have to defend the subjects of debate and then you can take care the economy the pullback on opponents and thinking a little bit concerned yeah yeah presentable category fortunately have exhibits presentable category have but if emissions goal its but they have also altered its own its history venerable of this Karen will result in injury but if you have a process which is assumed that its supreme then it must have on the Net mentioned it yes I mentioned this to this and the exact analog of that is to say that the presentable category which is actually on as a continuous clinic has minutes or so OK so this is why I can write to you here put that on there will be a canonical map from from this to a requirement of the CIA and the condition is that this kind of income that is an isomorphism initiated this is really a distributed denial because the example if the Bloomberg is physical product at the same time saying that this is the same thing as the core of the Office of the across the community that not around and the funding oppose this way Is that is equivalent to the physical motion that can be to right it's interim it's not obvious completely it's essentially 0 2 around but that's 1 way of defining what it cost is you can may have a modification because there are many ways to federal surpluses of the caucuses means preserving coordinates of the and within you know no nominally the forget that about the definitions of the regimented discipline that equivalent of all time that's equivalent I the idea that I cannot prove it there in this talk about this definition is equivalent to the testicle notions of that until I'm so we can use this 1 if you like them if you don't like it you can use another 1 but the reason I'm putting it here is that it's not a souls that this limit of financial and it's economists plays an essential role in the notions of a total OK now what is a morphism morphism I'm walking here in that in the category of troubles is not in the opposite category it's a bit like walking into category of friends ,comma indicating a flowchart there so I'm going to do it defines the as a right notions of a proposed rule of law and order directing cocaine so from office and an algebraic morphism as written off some of the draft proposes the answer from the duty prominent I welcome their it's a fountain which preserve economists and that's all or none of this weight as well as a black morphism OK so this this
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direction and that about offices of proposes a functor work from
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Italy Prime is a function that takes on your yes and this category according a top and the inside the category of rings to a propulsion OK and you would think that might include that financial it's an arbitrary concerts and then there is the category of travel "quotation mark in view of the increase of the defendant is the opposite of that OK so this is like a year and then 1 missive from an opposition party cocaine is a very of accounts the world will do the same thing here is that university locales in our planet is kind of fiction with respect to France because it is just obtained by reversing things upside down will take well couples are also fiction they are just something that reversing debatable substance and these things are remains open and fair and what is the proof of that this is the fact that the construction site just gave rise to the top also works in other words it should take effect a category with financial it's so this is like hawk and then we know that this is the end of that article of the category with foreign affiliates in the sense that it has the this universal property well actually dissident and attend a unit of you have a greater community so started to know you take a couple of satisfying this condition take a couple's satisfying these actions OK fixed the couple's smoking now and was easier and I take it from there .period preserve furniture units preserve OK then I know that this is an extension which is that what we saw before it's the lefthand extension during our attention on the floor should be extended to unity just with the formula extended the new term this hunger is not only that it is meant to be considered it president OK 1 like this is some sense this should be could be the fundamental to was too but I don't want to see them because of this may be an engineer for questioning has rejected that so many things in this year for marine never read it systematically I may have missed this work we could be there but Wednesday's Is that this exhibited at all between Everything "quotation mark continents and from this distributed to all the motions of propose which is emerged that's essentially the idea and here and here is not I think I would try to get approval notes and but you may receive an altar in a couple of thoughts and have another incident that but that still around is kind of interesting to prove let me just there prove you'll and spend fit means that Minnesota this preserve final product in order that believe will he use it like the exact visited his misstated the phone to preserving can be said to be left exact so answered OK so understanding with I'm so we think it will be yes exactly exactly where we can see how good we for now it's amorphous and this type is another wreck morphism of troublesome sits front of the absence of OK OK so let me try to give you a quick proof of that but it would be have not completed when approved that preserved all tobacco just prove that it preserved product and approved is the the difficulties to show that preserve all back for products this an area we have to do something like that you want to answer the officer if expressway that it can have a lot to do with the politics Gross of flying this was amorphous and you want to show that this is another morphism from every day for a reason why so inedible that its environment presentable this look at the case where if someone doesn't so in this case you have of why a crisis and it goes into effect but often cross the company listed in this specific case but the product of 2 interval you know what on the of all of this is what I'm an IBM and finish so you to show that despite the focus on understanding the Canes whenever it is sensible but this is why I a presently because this is the category has fired him his soon as product and other Franca preserved so in fact you're looking at it's not like this and this is therefore of failed and this is from the because the floor freak expense so you saying this is ever of a Crosby going into of press of the with was supposing that effort is left exact not isn't as amorphous and in the case of represented it's not taking product fixing of 1 variable physical it's OK that's going it's about distributed so there is a diagram here and if you use the chemical clinical according Pentagon put it of widely used fact this another officer you from you fix 1 variable you play with the other than you you can't do it the same things you have to do it 1 at a time it's different because of 2 variables which is called in each but not several times because yesterday careful about them and that the term is not broken the difficulty is to prove their case this you have to size and there are some something that is more subtle than in the see I have to use the rhythm of the actions and what is it that a fundamental as a result would be finished with that is said that if you have a top posts in the sense that I had defined a with excellence Tennessee since it is a locally presentable category you can always find a subcategory a which is a large enough so that they can the corresponding functor just because local representable from this is a localization which the right and want is for the faithful that's just because it is a decision or the result for presentable telling you can always do that OK once you have done that what about the is well In this case there you could choose a date to be told the financial it's OK and then the intelligence has been fighters of the extension GM so this called on the intrusion the extension and style will preserve Charlotte it would be a dramatic amorphous and left a post where the parental joint with the right everyone is fully faithful well and this is exactly the other definitions of what locals say is it's last exact organizations for Topol says that proved the same time Jerusalem and to try to get step here this creature of Bangkok and between