New variations on the notion of topos
8 views
Formal Metadata
Title 
New variations on the notion of topos

Title of Series  
Part Number 
19

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 
The notion topos is a prominent member of a family of notions which includes that of abelian category, of locally presentable category and of higher topos. We propose two new members: the notion of locus and that of paratopos. The category of pointed spaces and the category of spectra are examples of loci. The category of small (weak) ncategories is an example paratopos for every n\geq 1.

00:00
Addition
Euler angles
Chemical equation
Multiplication sign
Sheaf (mathematics)
Insertion loss
Complete metric space
Local Group
Prime ideal
Formal power series
Arithmetic mean
Computer animation
Funktor
Manysorted logic
Lecture/Conference
Topostheorie
Analogy
Right angle
Arrow of time
Stochastic kernel estimation
Bounded variation
Abelian category
06:24
Area
Complex (psychology)
Chain complex
Observational study
Diagonal
State of matter
Tournament (medieval)
Embedding
Set (mathematics)
Injektives Objekt
Prime ideal
Field extension
Category of being
Triangulation (psychology)
Universe (mathematics)
Quotient
Square number
Module (mathematics)
Triangle
Right angle
Object (grammar)
Units of measurement
Abelian category
Position operator
12:40
Pulse (signal processing)
Presentation of a group
Euler angles
Tournament (medieval)
Embedding
Kontraktion <Mathematik>
Complete metric space
Proper map
Theory
Number
Morphismus
Frequency
Envelope (mathematics)
Lecture/Conference
Average
Analogy
Gamma function
Analytic continuation
Abelian category
Units of measurement
Position operator
Condition number
Addition
Theory of relativity
Commutative property
Product (category theory)
Uniqueness quantification
Physical law
Median
Variable (mathematics)
Functional (mathematics)
Connected space
Field extension
Summation
Commutator
Funktor
Network topology
Universe (mathematics)
Module (mathematics)
Right angle
Object (grammar)
Resultant
Local ring
Freezing
26:34
Point (geometry)
Covering space
Computer programming
Group action
Presentation of a group
Multiplication sign
Survival analysis
Ring (mathematics)
Wave packet
Funktor
Lecture/Conference
Network topology
Duality (mathematics)
Right angle
Monoidal category
Object (grammar)
Identical particles
Units of measurement
Abelian category
32:25
Geometry
Group action
Fundamentalgruppe
Presentation of a group
Differential (mechanical device)
Modal logic
Mereology
Theory
Morphismus
Goodness of fit
Spherical cap
Manysorted logic
Forest
Arrow of time
Symmetric matrix
Abelian category
Condition number
Social class
Chisquared distribution
Area
Product (category theory)
Mass flow rate
Closed set
Grand Unified Theory
Set (mathematics)
Local Group
Maxima and minima
Radical (chemistry)
Funktor
Logic
Right angle
Object (grammar)
Family
Local ring
Fundamental theorem of algebra
Directed graph
43:23
Group action
Polar coordinate system
Divisor
Direction (geometry)
Uniqueness quantification
Moment (mathematics)
1 (number)
Mathematical model
Equivalence relation
Maxima and minima
Explosion
Morphismus
Category of being
Funktor
Spherical cap
Meeting/Interview
Lecture/Conference
Factory (trading post)
Energy level
Object (grammar)
Units of measurement
Abelian category
48:30
Complex (psychology)
Group action
Musical ensemble
Multiplication sign
Sheaf (mathematics)
Embedding
Complete metric space
Parameter (computer programming)
Perspective (visual)
Grothendieck topology
Spherical cap
Positional notation
Arrow of time
Fiber (mathematics)
Descriptive statistics
Stability theory
Spacetime
Product (category theory)
Theory of relativity
Basis (linear algebra)
Infinity
Price index
Sequence
Automorphism
Radical (chemistry)
Category of being
Funktor
Right angle
Identical particles
Thermal conductivity
Resultant
Point (geometry)
Computer programming
Existence
Number
Force
Frequency
Morphismus
Goodness of fit
Lecture/Conference
Operator (mathematics)
Associative property
Abelian category
Posterior probability
Units of measurement
Focus (optics)
PoissonKlammer
Physical law
Total S.A.
Incidence algebra
Set (mathematics)
Calculus
Affine space
Equivalence relation
Local Group
Field extension
Voting
Network topology
Object (grammar)
Fiber bundle
Table (information)
Family
Matching (graph theory)
Spectrum (functional analysis)
Local ring
00:02
the and and if you nothing and yes
00:15
maybe it was a bit too cool diversion will hold little to come along and do it in the air on careful on Air Force One on careful to choose that because I didn't have the time to really prepare it before August conference and I had this course to do so I could not accept the things that I wanted to prove the soul of and so what that yes workinprogress and that I would be happy to reduce it to have your European about maybe some of the statements or conjectures that I make maybe you already I know how to prove that yeah sold early this morning the last all crew by my press 2 there was a a Preston developed an analogy between them Top Asturian abelian categories so this is this analogy between a began categories and proposed to me is very much so on the mind of the of many peoples and uh I would like to see a few words about this not just because I have much to say actually I am and that presently doing some some work with to get custody on that and there I I'm bit intrigued because I thought that and you knew the subject than and the more I think about it that realize that too there are many things that you don't know what this but may be a bible for Abidjan categories is to you that this is sort of Pierre de videos in the category of billions on yes yes Prime is Celia Bible because here if it had a big impact on so I don't pretend to understand everything which is in this Oh yes of course to Wilkie was before Cocu 1957 or something and like that in the White published a half 55 58 it is isn't that yet probably was also written that before a publications of formal publication of the 60's what 61 you and I am not in their domination of the interesting things so that I have never observed before 1 and 1 of them is the construction of the perfect category From a narrative category and let me give you the construction sociable that pay Clinton additives category by this immunity you do have backs songs right of course 0 1 and finalize the better the from songs otherwise I would call that a player attitude could agree on and on and on In then there is a construction of or maybe it could be called a well I'm going to call at like this Lex of the left exact completion of phone and the army In an additive category there are many ways to do that left exact completions of balanced exact mean that the additive category now has Colonel you do you don't ask for an abelian category but just left exact category I mean the unedited category where every map has a kernel of the otherwise right exact edited category would be the 1 in which every day map has a core kernel of and because section there is is very simple because what you do is that you take the category of arrows the author of a and if you don't mind out by an ideal the losses of ideas makes sense in any of the and additive category or even credited category it's simply for each arm Everybody would you need to have a there are some groups and now which is close on the compositions of on the left and the right so it's a case of a man it says is a twosided it's a twosided idea of yes right that's a good way to save the right yes the Armstrong's there clearly is addictive and now we know what is In additives some Franca for functor Jay's justice of some Of the owner
06:26
so it is a bit like saying that the twosided ideal His by module of the fish yes OK so what is this construction but you can describe it like this are not yet so you could you have 0 81 but these are an object of the category of furrows so a map indicating referrals is premier the square like this so I'm describing ,comma maybe or maybe combatant was this is the set off all commuted a square in the city of 0 the F 1 you would become "quotation mark J J is a set of 4 I'm after what SAA have 0 F 1 is ingenious means no injuries in the area then there existed a man page from anyone to be 0 such that F 1 is equal the so what you're asking is for a diagonal but not exactly about was weaker you just want to make this a triangle to be communities a and B. and Gabriela study this category and actually on the explains where it is coming from and it is actually coming from a constructions of captain which is uh uh constructions of the derive category of did agree that the city's is the beginning of the end of a fourday solutions 1 would be of Chen complexes so Kathy Watt described to derive category of billion category using a chain complexes of injected objects and Baker were to be glasses of maps between injective objects and for the amount of utilization I'll buy got right you were that's correct the heeded the call to the suggests that the U.S. uh then they should truncated this construction you get that you get that's about this construction there seems to be due to Gabrielle about these was inspired you get Gabrielle states explicitly in his paper that it is inspired by In this constructions of kept saying I was surprised because this is what Pierce told me also beat told you it that the idea may be offered derive category maybe once at the beginning of the 250 was you have written a political positions this is the 1st book the price of triangulated category right here with new here you sold the nice thing about about this there is that this and in the additive category that's innovative category you take the quotient of the definition is that you take care of the the caution by J. the this twosided idea and then and now I know this thing has has as :colon it is relative category in the tournament and its solutions of the universal problem in other words if you have that is an embedding into this is CollettWhite because of its like the unit embedding so wide of 80 it is this 0 9 whole you embed the 8 into elects the new plan and this cementing as a universe of property which is there to 1st of all that easy it is entered and left exact which means that it has Colonel than therefore editor from 2 years and well that exists but extensions let's see if Prime which is left exact an essentially unique in the world in the call yes right fragile mind efficient are mean changing yes OK
12:36
there are other ways to construct could Alexei and
12:41
I Mike Preston this morning was giving another construction some of the of the dual maybe the wishes of her right exactly but if you want to and go Colonel you can't take what you can press would call the MoD the lot a well so you could take good as the procedures on 18 of finite presentations so that's what I mean so a billion proceeds of finite presentation what would you tell them all that would be to do all so but book too but it's very much so that's another way of describing uh this construction OK no there is a lot then there is a little miracle which is that suppose that you stop With the names of category which is which has Corcoran suppose a as Cochran it has "quotation mark then you do this construction not knowing it as Corcoran you're treated as an additive category of soul and and so you have this completion you add something you had colonel to something having a the coconut well then let him medical is that 1st that this has keep having covert and those in Siberian it's a so that's the miracle because to be abating and means that our relation between Colonel Oracle Colonel exempt from any you'll be hearing just do the less completion not taking care of the fact that Coconino not and not imposing a billionaire but you do get something which is a big so in other words this is a billion and a number adapt to and universal in the sense that now if you take easy and that billion category and you take him out of right exact from .period because you you need to take care of that right exact then d left exact the extension would be exactly so it would be an exact Franca between abelian category show as a consequence of that it is as if you want to concept may have been an envelope of definitive category so is attitude on so you want to construct the Albanian envelop you 1st no let's say they're right exact conclusions which is just the do all of this thing OK and then you don't know left exact conditions of the right of the completion and this is a beginning and as a solution of the universal problems I'm so that you know they're staying in 1 way or another right I think it was carrying your talk you right the idea that this is the objects CSU you call them imaginary objects were OK yeah but yeah but because they are not modules over a year but you call them imaginary sold so here you started a function which is an inclusive know exactly just as his and here is the use of the default ever doubled prime is exactly isn't exact contract between a bit included Of course you could do the ability the of the construction is the other way I made you could take and if you do it this way you go to a Rex or you could do with daily legs next wrecks and you get the same thing because of the use of some the same thing in world is there isn't another morphism between the 2 things still not for that is a commutation operator between that and ,comma completely on the left and on the rights essentially ordered similar to the law whatever it is that there is something commutes no what about the DAB 5 so if there's is an analogy between 9 doubles to reenact the category so 85 at In the financial quality find 1 billion category this was OK I will 4 for both of "quotation mark now that if you will and you work with the goal of the cabin categories and the average like pulses
19:37
and in this case the use of the results of the candy soften proved or maybe there's a version which if you start with a left exact the right that exact category and you take care the unit I'm betting Chloe preseason on it as the yearend embedding the other yet that's the edited versions of the unit I'm betting that now we know that this category's IBM because there because Abbas billion pocket but of course in addition it is also at the end of the course of bruising the comedian category but is the solutions of a universal problem so In other words if he isn't authentic the median category and if you have they left exact from list exact then you will have essentially unique core consumers extension which is exact and the coconspirators means that could preserve all local products all sums right so if you look at this picture it what is left exact spot and the whole thing is the it's it's it's uh uh negative category which kernels on something and you consider that a defunct which reserve colonel you don't ask it to be a the local and then you exit you can extend it into and the exact Franca which which is cool continuous running right Seoul because of that you can easily construct the freeze Wilson did get category generated by narrative category and and others at this tournament of going beyond and proper school right to which which is going to say that any a growth of the Cambrian category is a left the exact localization of that and of course the ref example positions are doing bye some analysts said subcategory and assess subcategory to play roles of the topology India being in context and everything just repeat the goal to of after topless theory is completely very very very almost statement by statement you can find the analog statement people she wrote a book ,comma there should be a handful where you'll make this parallel very very strongly because you OK now but what else there is about a billion categories well the heading into categories all right to the additive category can be cancer the citizen notions of perfunctory but added remains viable so when you work and creates the Mandela category of offered presentable edited category if you want and there's a nice so there's a product in their soul where should I write this mission so it's it presentable the categories presentable really just mean that you have connections from generator essentially are and that there is a protester props between presentable additive categories that's a it is the solution offered the exhibiting no perfunctory witches come a presently this is the scholar Tatum or whatever you want a functor which is additive in these variables I and that's it I mean that defines you what's the solution and examples of a functor edited in these variable is precisely the homfunctor for example that it's plenty of examples of from additive in these variables OK there is a home loan a bean is just the category of where a call continuous Coke onto a sedative Franca and "quotation mark conscience and mean you want to put this to be preserved because we working with presentable category where essentially everything is generated by calling it's OK but there is also a subcategory of that I in which consist of a Iberian category and then it turns out that there is a and then there hasn't been tense as an additive category is obvious I'm sold in this way you can construct on the Net owes Monday dealt to category where you will consider cocounsel sometime which are also exact OK and when you have a lot of money category like
26:35
this emitted when I die category it's very tempting to look at a do Olazabal object so why when is it true that as that because Dick I category it has a dual but in a sense that you have a caring which we will go into added unit we Arab 2 satisfying the usual identity 4 the duality and examples of that on our given by bye the 3 1 2 3 3 over and additive getting so any of these and if a is small the social life of a it's small and then if you take day on the take the median appreciated procedure they all but on the way a brilliantly then and you can show it is serious not difficult to that do you will know that there is that that thing has a dual which is a billion 1 day all so we have plenty of examples of their use had been getting these that can be allies doesn't Canadian categories that it can be to allies but the question is but what I have been injured mean so after hearing my press this morning I'm ready to make a conjecture is that and the idea that there would be an if you take care a uh smile at the Indian category this time he had a was a sedative and the you put on this day the topology that would say that every team offers him as a cover so that's going to be like looking for functors the which from payout training and With which shower left exact had been a take any smaller Abyan category and look at left exact from tourists from a up to 2 up but the soul of a conjecture met a problem I don't know you know it's a bit like a guest but too so it could be wrong don't take it too it's a problem for me is that these are exactly the I do Izabel the a little women I billion category and and the duo is what you take this would be it just the same description about with a instead of a you have to be a lot of for this is to be the bullishness yes which without action this .period trees will be utilized all that would be realizable and except from all K program of so it's an open problem OK now I want to go to another the 1st is situation the issue date top posts but you can generalize is them in many ways maybe if you could put what could be called to lanes so these are that Mandela category would answered which is cocounsel variables and with the unit the objects sold here are some right said that the unit object to a miracle conscious functor from set to Is this it's it's enough to know it on the point of survival and the point so that's just an object of the so but these are the same monitored in the category of presentable categories In a Semitic money don't seem to have no overwhelmed not for rings they just hoping but
32:25
they the interesting thing would be that the tool symmetric 1 significance so in this case the evidence product is symmetric and then you can Spitzer a bit further sending I have been telling it to the paratroopers because they are close to topple OK In this case the tense product is other captives in so you have to that kept isn't products well you will have a chasm product in these categories still but there is nothing that guarantees that they can indicate product will be will have a right and afforestation differentiation here do exist but it's which is packaged as a product so that the area the condition is that distinction mediation preserved "quotation mark at in Israel so in 1 enough because the product is committed or symmetric so and so it has a patent a small couldn't the literacy yet this is in the future not the focus of but not why why is this interesting well at 1st that I remark is that because of this condition here that in the core consumers frontiers between presentable categories as always has a right so the user of the categories market hasn't closed so in other words if he is about to the signed by its ability it is cut isn't and so sold there is a In forestation which is right 1 2 taking profits on no chapters close now this is it's not difficult To show director a presentable category it a trap that Opel's if and only if it is currently this kind of to become wasn't close is the necessary I'm sorry it's not what I wanted to see because what I just said is to because had the 1 I want to say is son yet this make it clear to them that this has to be in difficult to prove that OK on OK the presentable category it is about the polls the additional lives all this show me :colon disappears interval cutting if only flow is the only organization laughter the chief category the theory of the set In said see opposite the localization of oppressive category where the localization preserve final product it the organization the fund so we have done I see all can say that and we have the localization functor witches on Western jaunt to work to the intrusion on and goes to and you as to the organization from cooperative from abroad so there
38:19
are no words at battle posts it exactly like the horses except that you replaced the conditions of being left together left exact localization just by localization preserving family products and a apart from that the 2 war completely see that it did not want to get this kind of localization you need to exhibited to the tasks of morphism that you want to invert and what you ask is that this class should be close on the cap part not put back but just get in front and everything works the soul this emotions of geometric on algebraic morphism where the functor should preserve should be call consumers and preserve the final product instead of being left exact it but you just yet you all take cities in the hope that yes of course yes so that there's always a terminal object and then and determine logic would be preserved by the morphism right yet 1 9 a all the big difference of optical stealing just what for example you can the the composer juror that morphism fizzled known as the localization preserving for fun a top followed by something which is conservative you have call anything just repeats itself except that they be the cans of of of of category that you get in some sense much weaker so what is an example but there that and that's why I'm sort of interested in this subject yeah what is an example of a apparent posts the set this category of smaller categories uh you know that this is scattered and close the it's easy to check that this locally presentable so that an example of a palatable Kansas so you what it is the left exact localization of appreciate category Hankins exhibited it's because he you you look at cities says set the category of that so he these are this dictator of proceeds on the cathedral needed now evanescent has said has a fundamental category there is an intrusion here which is the nerve good and this nerve has left joint which is the fundamental category of assemblies says set and it is still around which is not difficult to prove that the fundamental categories of the Prophet is the product of the fundamental continued the problem is that people the don't want taiwanese the is yet that is the year I 1 by 1 as the fundamental group if a user group pointed instead of the said I would say that that would prove that the group is a profitable source of all right so that's another another example sold capped the example we reported Bush said missing Bush set is effects of subcategory of cat and then because of a battery of set the latter category and now there is an affection from good to have a category you can associate of Percent you know the kind of full use would you say that an object EIS more equal than object be if there is an arrow from a to B yes the he disease the father of falls and bless all a Yaklich was accomplices something quite different meanwhile was partially partially ordered sets passionate passionate about it said those little things that was we were a year by partially b old people set fire to the
43:24
opposite but please it says a reflexive subcategories of this disease where should gradually withdraw from the world of work for more than a year use this quarter all Yeah yeah yeah yeah yeah sure you have but I can partially so you want to but the audience said His strudel you could put the commands the category of equivalence and then there are almost In this list that can be extended in various directions because you have to look at this is the category of 2 categories the 3 got in and a double capped doubled categories this was OK I'm certain movies of his life there no known all this is to look at all yeah sure we are in 1 category the I'm not looking at 2 cells between 2 victories right now I'm just I'm taking to stick to caps and restricted captain and these categories that I made the AFP only understood to mean uh we understand what the topples is the property is unknown we would like to have her understanding of capped abstractly as a category 1 day actions that can be put them the 1st of course is that it cat close there are other actions that can be put but 1 of the exactness properties of I mean we know for example that in the post you Yukon factor of Vermont has owners nicest ejections and followed by accident we have all kind of thing and so it's a big questions and I don't know except that let me observe that too but in many many of the situation there is a factorization which is there are a couple of fraternization that are really interesting so let me just discussed 1 is that if you have the perfunctory between 2 categories then you can factory at for example as a functor the subjective come on objects it's easy following by a effect full earlier model morphism operating that's 1 possible for authorization look at that level and the intelligence of a full subcategory In a functor can be factors in this way are ending the unique unique way unity essentially what you do is that you you take the arrange of the year 1 of the objects of any and you take the full subcategory engendered by the image but it looks like a bad factorization because this map looks bad good I don't all but is something maybe interesting which was observed by the head of the Jewish people there is another factorization right there is this deadly factorization with where you ask for this map to be budgeted on object and this 1 to be just for the fitful and Europe models but I'm choosing this 1 because being on but where because it had that want to deserve famous resolved by a ban elections you have with with another reporter for that moment I'm sorry I apologize and would which unit which says that if you push out the push out of the offer of a fully fit for Franca the pushout intact so what if you have some fully faithful functor it could be more known object connected to it doesn't matter for 4 it is is also the fully fit for now
48:31
what what it seems to say is that the cat the opposite of cash is actually In exact category this is inexact category you expect the pullback of 1st ejection this objection this is not true at all and cash but the opposite of camp as this month's property where the my suggestion is not played by fully fit for landing so I'm giving this as an example because I think that this kind of result should the 2 also In the higher categories in was there last year and should be to also if you were to take infinity categories where this for upon leaving much harder to prove so in 51 categories that should be to also sold my understanding of captives very poor I only that know that this is essentially covers and tolls that has this kind of property but it would be very nice if we could do extent of posterior To this kind of category possibly by adding a few actions but I think it's very much needed given the importance of cap of tle cap and the wing Russians and tributary of categories of 31 and also for 2 with his truck which should be able to extend the posterior through to these categories OK my own home ,comma stand on this is the way forward but that won't some of the classifiers all down a bit that seems to not know that that's not the title the Rangers voters both you yes I'm what did for um did too 10 minutes soon yeah I would like to discuss with you .period insects this is not the troubles but because the initial objects in terminal object coincide so the question is if it's not at all opposed to what what is it because especially with pointed spaces To this plays a big role in topology but it's spaces and so abstractly What is it so I would like to make the 1st an observation too there is a given a category any categories C or you can describe the the category of families of objects of C but the family the objects of is is what you think of this as a family of objects of and the match between 2 families it is also what you think it should be because what can you do to map this year and into this if you don't have any other information 1st you need to have a map from high to jail but In the end for each hour you need to have a map From AI To be called firepower for all what so the map between these 2 consist of there world and lonely before I don't know how to college to whom the bad notation on here are some whom it would it's a pair there may be to guy after all it's a pair consisting of 4 map from I to J. downstairs blast or maybe a family fight card of the match like this yeah OK so that's the aid given a Category C you can construct the category of families of objects of seeds and now this is the core product completions of C & C has product arbitrary core product and the IDD is actually that this family is somehow a former local program of the object AI you should think of it as the core product of the took a product of it with purely formal so they don't mean that exist in so I'm going to make sure that we know where we are so I'm going to write to brackets here just for the intrusion it's really like the unit that from the time that actually I'm looking at and this is actually a subcategory of appreciate category and I'm looking at core product prefer to intervals OK so and this is free this is an issue where 2 of maps for city into category with product you will be able to extend it along the unit operating too "quotation mark product preserving functor from the family of C 2 OK so observation if I data family offer this is set have pointed said the family of what it said what is a
55:25
family of what effect of this is the same this is this is a totals this is the same as that call money and founder ongoing on the category which I'm going to describe like this section retraction and the identity so you would take a category would 2 objects and we which I call 0 1 1 and S far it is the identity with 2 arose during arrows and you just ask this equation and that if you have a functor from this too said you would have to set big 0 Annex 1 you would have a retraction in this section and of course if you take the fiber of day after attraction at at a given point then you get to appointed said because this section is providing the base .period but the fact so you get appointed itself and in this way you get a family of 4 deceptive In the Congress can be done so that is an equivalence of category between the families of what it said and a band of sets equipped with 2 operators like this satisfying this a question and since this is the category of background and sets this is topples so that's the moral of the story is that of the soldiers who were there that who is the House you desire to old yes right you could or even have it with a with a projected yeah that's true yeah yeah sold definition I have a name that premature it is appointed those locally presentable the category such that where should take the family the focus of the says that if you take family the category is such that is the troubles I'm sorry for you want to reduce the of of life appointed category it means that the initial object and that the real object when what do you mean by way of getting exactly it means that the initiative for a presentable in is a locally presented as a presentable category for me to get a along with Oh well it will wish name would you like me to use a little point in his it's a category where that that . 2 notions right means that the 1st at the OMB sets out what it sets that's 1 way or you could ask for a better basis points to I'm for an object which is both initial and terminals and that is very close to the commitments which to right I'll be OK well uh in I would like to give you examples of locusts and this this notions becomes interesting and when we look at the incident it topples there's emotions of infinity which is just same except that families yeah by a family and uh an object here I remember you internal little normally allow you would describe the family as as as a function of free from From the discrete Category II too 2 figs OK but in the context of and fitted Opel's that I should not be discreet I should be in the would that is in a much appetite right this is for minor but it makes sense if you have an infinite the category Yukon consular families of the object could in it to index by an affinity group would actually buy any centers and set the example it's a it's a very very and up emotions In the highdensity category to so here may be assured that may underlined this because we are no longer for example I could be the given by a group of law then that that would be there is a category with 1 object but where the automorphism commentary yelled Let's give groups and that would produce like an object in the equipped with 2 and actions of that group's so OK then and then use if you take the category of spaces of the cases people I'm sorry the basically the actions this is for the anniversary and Finn 1 category credit and that's supposing that is top right so you know you could consider appointed spaces and families have of pointed spaces look families in this sense and this is again In the Simon only in relation he has as before sold a disservice relations shows that a family of pointed spaces family some extended sessions it is actually the same thing as a matter of space is equipped with a section "quotation mark OK yeah I will 1 of the I'm OK OK so what while they they the what's interesting here is that there is a notions of 1st victory of the spectrum the prospective apologies a sequence of spaces the league 0 x 1 it's run together with 2 map from the suspensions of and to extend plus 1 for every and there is a category of of 3 spectrum and in it's very easy to verify and that this is a locus well actually it's been appointed spaces yet that this is locus in other words you can have a bit of met this government this description With the base space available by space and this disk suspension over so you have the space with sections and on I'm taking this to the point where suspension and it it's a pretty spectrum over over B that you described and then come as the testicle Turi a place spectra is that the very suspect it is dead is a photo to spectra spectra but a spectrum of perspective is that it is at the spectrum if if the associated that went mad From the extent of Omega of eccentrics 1 is a that's the call sometime in the Omega spectrum more crude that's that's the idea for the Office of Inspector it me that you have a sequence of spaces With a sequence of equivalents nominee what you have is a prospective and then there there's an Associated the please and then associated spectrum that number None of competition now the Fonterra that takes a perspective to the Spectron there an explicit constructions and this is the left exact and you can use just construction is typical innit all the Toronto Blue space and it's it is only after it left exact but the I'm sorry site will to you have to know where we are in the IMF and the World yes OK but these 2 categories are pointed they are appointed and you can do this family construction USA Udall family construction on both sides so you consider families of spectrum and these are spectra over basis waterbased can be can vary OK and so you see that the category offenders of spectra is a left exact localizations of this category but this category Is it topples look at it it'll because of this kind of construction here because of this kind of uh dictionary Uganda a it's not difficult to show that the categories of families of prospective therefore the categories of families a spectrum of OK now this is something which has was observed by various people at 1st metaphor from the demand In 2007 and Charles Rex subservient to about the same time on and it was coming from a goodwill executives but the prove that I am doing making here is completely Elementary in some sense because it does not use good Goodwillie calculus it's completely understand during almost stand up construction and the fantastic thing about this a look at at this table category of spectra is very far from being a top post verified well maybe some people would say it's very close because like an abelian category looks like a proposed to but if you just take families of suspected of spectrum bundles of Spector I think there is a bundle bundles prospect aware amorphous enough our to you know what you think it should be OK morphism is a matter of consumers might between the base such that if you pull back the prime you there would be a matter over a pullback great as start Of the prime misses this category of parameterize spectra has been studied the thereby be there may for example there is even a book about this 1 but they have not observed that this is a topples well at that time still polls this itself was the existence really but balked at 8 seems to indicate that 1 good and bad that to me of stable categories In still posterior the new I don't know if this embedding is useful but I I would conjecture that if you take an infinity 1 of stable category I know that if you take family families exactness and constructions families of of of object and in this it has to be cocomplete so you need to have called product now I'm in the category of respect spectrum of stable category that you're looking at now should do this construction that you do get a topples off every time it including including the cases of Chen complexes I'm looking at the fact that Mexico could perhaps but that this may be a while the conductivity and lots of WiMax in Canada over the years just polite thank him for that so I had a I if I had time I would provide some arguments supporting it but I don't have time to focus so and some