There are categories of ‘spaces' that are not categories of locales
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Formal Metadata
Title 
There are categories of ‘spaces' that are not categories of locales

Title of Series  
Part Number 
22

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. 
DOI  
Publisher 
Institut des Hautes Études Scientifiques (IHÉS)

Release Date 
2015

Language 
English

Content Metadata
Subject Area  
Abstract 
We described a short list of categorical axioms that make a category behave like the category of locales. In summary the axioms assert that the category has an object that behaves like the Sierpnski space and this object is double exponentiable. A number of the usual results of locale theory can be derived using the axioms: the (weakly) closed subgroup theorem proved, closed and proper surjection are of eective descent, parallel theories of discrete and compact Hausdor spaces emerge. An example is given of a category that satises the axioms but which is not the category of locales for any topos. We show how to embed the category of elementary toposes into the category whose objects are categories that satisfy the axioms.

00:00
Point (geometry)
Group action
Exponential smoothing
Elementary arithmetic
Transformation (genetics)
Diagonal
Logarithm
Multiplication sign
Maschinenbau Kiel
Propositional formula
Open set
Power (physics)
Positional notation
Manysorted logic
Lecture/Conference
Term (mathematics)
Natural number
Topostheorie
Subtraction
Stability theory
Condition number
Product (category theory)
Theory of relativity
Spacetime
Topologischer Raum
Prisoner's dilemma
Set (mathematics)
Axiom
Local Group
Maxima and minima
Category of being
Proof theory
Arithmetic mean
Computer animation
Network topology
Object (grammar)
Resultant
Local ring
08:24
Point (geometry)
Standard error
Exponential smoothing
Group action
Transformation (genetics)
State of matter
Multiplication sign
Correspondence (mathematics)
Equaliser (mathematics)
1 (number)
Morphismus
Manysorted logic
Lecture/Conference
Term (mathematics)
Alpha (investment)
Condition number
Area
Moment (mathematics)
Exponentiation
Expression
Bilinear form
Proof theory
Category of being
Quotient
Homomorphismus
Right angle
Object (grammar)
Thomas Bayes
14:22
Point (geometry)
Group action
Free group
Transformation (genetics)
Equaliser (mathematics)
Multiplication sign
Propositional formula
Complete metric space
Junction (traffic)
Open set
Power (physics)
Meeting/Interview
Lecture/Conference
Natural number
Term (mathematics)
Arrow of time
Data transmission
Stability theory
Condition number
Process (computing)
Theory of relativity
Product (category theory)
Prisoner's dilemma
Equivalence relation
Local Group
Category of being
Funktor
Topostheorie
Object (grammar)
Identical particles
Local ring
Resultant
Surjective function
Gradient descent
21:11
Point (geometry)
Group action
Exponential smoothing
Finitismus
Elementary arithmetic
Logarithm
Multiplication sign
Complete metric space
Proper map
Theory
Junction (traffic)
Power (physics)
Number
Manysorted logic
Principal bundle
Condition number
Stability theory
Series (mathematics)
Topologische Gruppe
Spacetime
Constraint (mathematics)
Product (category theory)
Prisoner's dilemma
Exponentiation
Bilinear form
Local Group
Equivalence relation
Connected space
Category of being
Computer animation
Commutator
Doubling the cube
Lattice (order)
Logic
Network topology
Topostheorie
Object (grammar)
Local ring
Resultant
Gradient descent
00:02
as if the what I can do it alone on ,comma I love and now for something
00:16
completely different categories of spaces that and also categories the categories of spaces that on all the categories of lowcost this time we're kind of attack is what is going to recall some of the properties of the category of locales notably that there is a government double exponential object in that category living look at bats double exponential object in in any category and prove stability results for that double exponential object and and that will actually give me up costs of categories that has a stove likes manageability 2 minutes and then the sort of point of the talk is to show that that costs 1 of an example of a can that form with the double exponential military cannot be a category of locales and then if I got time Noppadol little bit of 70 roundabout in and so justify saying it actually a category that does have this double exponential ability is like a category of spaces so you can do a lot of topological space a local theory within within my cat within that terms that within a category that satisfies as axioms the answer when it'll be about lock category locales I'm gonna write this for BSA Penske locale set the skillet ,comma main property that we have reason for all this for all locals X set Penske the statements came to the X exists and the to X doesn't always exists because not everything is likely compact so what 1 mean by that what we do have appreciate Fountain said Pinsky to the X came from lock Paul the Senate to the the opposite the category locals set and that sends any y lot White whiteowned context ,comma statements so you can define that and that is the exponential In the preachers category so when I say that this exists what I mean is that you need a recipient's to the power sitins GTX not once not preachy the exists In the preachy category local ,comma sets and it is represented when we have that we write the effects the double exponential it is a property of a captive locales this double exponential exists the key universal characterization the points of PX on winding the double exponential that they correspond to natural transformations more prisons in the appreciate category pacific and To the X statements to the y and once you have that you have the defines a strong it is also interesting tonight but once you think about this appreciate category you can do actually quite a bit of topology so for example you can prove that a local ,comma FSU next y it is open if an earlier there exists a left a joint recipients into the air so there's a natural transformations in in appreciate category such are the usual from being reciprocity condition holds and is also well known and that we can once we can say World open map is relative to any any category I guess with the final minutes of the final product and we can say what the discrete objects are said we say that excess In any contacts with the lot some objects at Penske axes discreet if and only if the finalized diagonals are there and all of this works all of this works relative to arbitary elementary top so you can have a discrete objects relative to lot well it's at any age let's just he again and beds in the cashiers locales relative to a venue elementary troubles set the fair and I want to prove police outlined the priests about double exponential currency it's essentially saying the public's legibility is closed under the formation of G objects the proposition so and if see just as to have finally products that's why any assumption so I hopefully if nobody's interests only the local theory little bat spaces they'll see this is an interesting result about the likes manageability so and we have an object a such that a and EX exist In see for all object facts we have G which is a good thing which is a group in the day's top products so we can't about internal growth then the results says that way the point 2 face points text ,comma exists for all g objects thanks ,comma adjust the clarity so I'm talking relatives now To this category pursues appreciate nation notation for it but it has as objects and they have to yield as objects so that a typical example is X and Due to inspect their steps
08:24
and of course the point to use just a with the with the trivial action on it the move victims show this will here so that data pointed to a with the trivial action they supply to air the trivial action on it X X ,comma a for an arbitrary geologic this double exponential exists so you just said geisha the proof of that police In outline the uneaten I need to construct a subject relative to GE objects therapy a G X ,comma it's going to be have its objects the double exponential respect their acts observers is G actions something has defined through the strength so the strength of the moment that once you have double exponentiation give a strong man and so other have a strange morphism here magazine that it so that shouldn't surprise you that if he had a this is how we action on here that's is defined set a complete the predators we need to show that In that right we need to show that the collection of G homomorphisms the state that the proof we need to say that for all other Geel morphism say why ,comma ability but that the collection of all morphisms from I want to see the effects that are actually G. Holmes the homomorphisms that these are isomorphic to the natural transformations From now makes the point too to the end X :colon 8 . 2 why ,comma Bay like a lady because that's exactly the universal characterization the double exponentiation that I give you earlier the said in summary released this lefthand side will be corresponds to natural transformations so they correspond Alpha that goes through Szubinski to the next Sibierski to the wife and then unraveling the definition of G home In appreciate category gives you In particular splash so it's completely times act that's a Penske to the judo talents why it's almost a sort of warehouse debate a 2nd into to the baby 2nd stadium sorry I realize I firms delimitation will error citing ill health said you have to bear with me that might Szubinski is equal to the area I White whereby made
12:07
an error on just replacing him with the form that ACX unraveling the lefthand side in terms of the construction that point of enough transformation you have that that GE home condition you so that has to be that you need to unravel the the definition of strength the on the righthand side here I know apply if the following coequal like this so the following equalizer In the appreciate category for this actually in bed said 10 equalizer diagrammed here right 11 but is say what it is said times why the the free free um uh uh the Freiji algebra Y G 2 and G Todd's wife again the free ones and the reason why this is a the equalizer in appreciate category is effectively you can apply the exponential to those the well known in and it is the rapid well expression of the an eligible why Kobe as a quotient so there is In the category all Geest G objects we always have this question and it's a splits we always have this question in the M in the
14:24
category of uh and catcher objects and if you apply G 2 that question Tenneco equalizer again equalizer arrived here because you have an equalizer over here these natural transformations impaired into and transformation make ,comma points to To the Exco take a look alike to me now it's times comments on sensed something it at the identity of the free the dead the free on the air that 10 decided then the the free action on white so you just called and
15:14
wanted and so this embeds in that because of that equalizer and the following here the final step in the process is to observe this is what I meant by Fleet got this left to join in the together from there that said she magnetic possessor of their captive men as and minor enjoyed induced by that but is also satisfies for beanies reciprocity and because the service was Rabin is reciprocity this actually extends contravariant only to an injunction in the appreciate category um so that's ,comma blanket so unwinding that junction appreciate categories we see that these natural transformations actually can be related back to natural transformation from a to B X aides why because this is the this is the free G times what Jeffries on the left on left a joint is visit contravariant so that's downright joint misconduct and there's a little bit of checking to do that the condition here corresponds to equalizing with those 2 arrows says an additional check public recounted but that's how you should that if you have an object in an algae the note you guessed which just as finally products then double exponential ability is is G stable but Florida so knowledge to make the medium of promises little incentive of give you an example of a category that has this property but is not necessarily the cashmere locales so I'm going to prove some insight GE is saying opened let Kaliningrad and went through some of the consequences of an assumption then g of lock is equivalent the lock can give locales over some mn policy so is a managed troubles can develop positive red was going to happen if I do you make the captives that locals of the G accident action the same things that catch be calls professing to observe is that as a 70 so big opening remarks that we can embed the as the discrete objects relative to this category I can do exactly the same thing over here because I have a say Penske like object in here this just Penske with the federal action on it so I can define discreet and unopened and I can look at the discrete objects relative To this category and it's a dilemma but it's relatively easy to see by detaining power just described in terms of describing natural transformations relative 2 GTE's Back in terms of relative natural cultural and natural smell of natural transmissions relative to lock White unwinding the fair wages or proposition or just proved you can actually checked with the discrete objects relative to hear of the same thing as as BG so just the usual thing assets with the G action on it so the objects adjusted and just discrete locales so with that information playing we know that these 2 things as an equivalence and because equivalent that's not what we noticed troubles anyway but as a top also is the same as that topples therefore they don't have the same categories all locales but we also know from well my enjoyment Jenny result for any g there is actually a geometric morphism here which is an open suggestion on the sly synergy GE was opened in the beginning that's surjection and instead therefore of affected descent and is well known in the media statement effect being affected descent means that the calls over BG I've given by g objects relative to the group that you get by pulling back these 2 and lookalike geometric morphisms against each other that pulls back in then their captive also policy to be Angie hat which he had is the and Alamo dike as the Intel completion of due said this actually His isomorphic to the daily equivalent rather to medium g objects relative 2 The as Sergio Paez only it's how completion of G the walk up overwhelmed then missing a little bit of detail you could also show that these these at junctions are effectively as I ever lock 7
21:13
there's a technical aspect of that so using bad junction in the case lock you actually means equivalences commute with lock and then you can go OK with the Zirconia never lock they induce a ceremony on lock and therefore she is actually isomorphic did so it has to be a tall complete so since there are examples all groups the categories that are not itself complete there are examples of categories he is 1 of JI's not local complete but has this double exponential feature today and that was the main result but want to get across but I do I have a word with him had sought 3 minutes yet so to make it a bit more of railcars at a given time constraints have really been able to make that connection as precise as light as I'd like but OK that's very interesting but that doesn't mean a walls allowing you to collect thing Acosta's spaces they're on an ax medic approaches to locales you say I've got got in which category got finite products Monaco products coverups a pullback um stable above said Pinsky objects which is an internal distributive flattens it classifies open and close I've got a couple exponentiation um and got some other technical conditions around how this behaves interacting with equalizers and of reflecting on small office and things like that with those actions in play you have a category which you can prove that proper maps a pullback stable and this ejections are affected descent the year weekly close subgroup Fareham holds off from his Hoffman this loft Lofaro holds II could be topological lattice series is an awful lot of you can do which I would call local ferry but we've now shown you can do that local theory without actually being a M a catch so that's why I've them headlined it can't create spaces that are not categories of locals the top the list the amendment you question about the I'm of 1 of the nation's Communications in the sense that it will forget why you'd have to give me a bit more of a conflict a lot of questions the president of the pilot machinist of mission I think that after the Cup I've I mean just out of interest as it may be related to the double power there's anything like the express really comes from the lowpower local at lower power and author constructions and those were originally developed into main theory in in the rest computer science so you I actually in inaction ,comma decide the constructions era she come from various computers USB unlawfully with continuations of and cut becoming 1 more hit commented unless another question at the the other thing this is of even more interesting about it is that you can the 3 stop said you can you got your below Kowalski that works reached top the you could also frees geometrical forms of you can have in the junction between locales and that Junction care commutes with the double Power construction wherever it wherever I rented out and that she characterizes geometric Wolfensohn's so you can actually in bed the Academy of top bosses in this category of spaces and it's it's fully faithful and I guess what this is saying is that it's and you know there's more of them than there are of geometrical prisons and a dozen preliminary work and some of the properties of geometric morphism seemed to work in this In this logic category but 1 of you have hailed as 1 of the last day of the eye that is I don't think yeah that's a good question and and you know you could take the view a where you got it wrong you know you have captured locals are Armuchee taking the opposite view I think it's it's it's a more exciting interesting thing because and and another thing if countries that 1 of the key results a topless theory is that when got only category .period the points of the locality group would correspond to the principal bundles relative to the it's how completion but in this case once better in his logic and large category you get the principal bundles without having to go to the itself completion so my intuition is that it's actually in some sense that there was a lot of health among yeah this is as far as I'm aware there is no idea is known elementary characterize Asian and um 58 speed Thomas struck I think he presents as a fact she called an elementary characterization of dozens slightly awkward known elementary catheterization side never mailed to tie it down and I have been a number of years I guess this this sort of coming along and last year were earlier policies may be thing action to step outside looks like I think the candidates it was the 1st few as