Variation on sur information theory: categories, cohomology, entropy
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Variation on sur information theory: categories, cohomology, entropy
<abstract language="fre"> L'entropie d'une variable aléatoire discrète, introduit par Shannon et généralisé par Kolmogorov, Sinai et autres, satisfait l'identité 0=H(YX)H(XY)+H(X). On verra que, si on considère une catégorie S de “observables” et un topos de Grothendieck associé à ce catégorie (les préfaisceaux sur S), l'égalité cidessus s'interprète au niveau cohomologique comme une condition de cocycle. Sous certaines hypothèses, l'entropie apparaît comme le générateur du premier groupe de “cohomologie de l'information”, introduit par BenenquinBaudot l'année dernière. On parlera de cette approximation catégorique aux probabilités classiques et quantiques; quelques constructions de Gromov seront aussi mentionnés. </abstract>

Alternative Title 
Variations sur l'information : catégories, cohomologie, entropie

Title of Series  
Part Number 
13

Number of Parts 
17

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 
2016

Language 
English

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00:00
Geometry
Spacetime
Free group
Link (knot theory)
Logarithm
Multiplication sign
1 (number)
Survival analysis
Function (mathematics)
Content (media)
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Selectivity (electronic)
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03:51
Commutative property
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Group action
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Insertion loss
Mereology
Variable (mathematics)
Variance
Connected space
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Logical constant
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Moment (mathematics)
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Partition (number theory)
Category of being
Quantum mechanics
Simplex algorithm
Arrow of time
Relation <Mathematik>
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09:25
Geometry
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Real number
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Infinity
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13:32
Area
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14:13
Spacetime
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00:02
at the time of the of year and the and on and on and on and I salute everybody else to contribute to these links with geometry and like the quantum gravity in whatever language and talking about flagcarrier has some links to get area and the Gomel in series and we probably tea and particularly with the information so I will introduce you to Herr so this is the last will likely presented the characters like Winscombe only with this information then well the tube but for you because there was talking with observables communities whatever and then I would take to explain how all this is related to to enter the cell air in the German to like this thing called let because the ILO's has to distinguish the shape the shape of things so you can distinguish his fairness here from a Taurus because there's a hole in the side bad there is another mathematical Whalen unleaded break way of saying that there is a holder and basically is the fact that these cycles a and b the drone that do another boundary of and sold they are like cycles that I know about November's so they are like a nontrivial cycles I that so the notion of shape and the enjoyment is related to the interference and that there is stable on the deformation so we can change the structure we can do know transformed this year into and out of sight basically in terms of college we have so now we'll try to define a Moloney free information series of incidents I also have to do this also where the information theory and that videos selection and introducing the 48 this way of measuring measuring how much information is seen this random servers so you think that the like of the resources thinks that can like tells mint and different messages the innocent and then you went to measure how much information is there so you can like right to this expression you know like the song between all the criminal case in a lot of the privileges and that's a measure of information was because here is if you know in advance the output there is no information and maybe few communication you know the output so you don't have to transmit anything to get his hard to the message stimulating there's nothing to talk but on the other side is if you're like equally undecided to know all the possible options but that's so you have to decide your communication device to respond equally well to every hour so they can be and is willing to let formation is how undecided you are between the outputs and the and that's related also with the optimal ways of calling so they did now is likely to try to put the the Our problem the that assets like some underage approved a the idea here is like will consider a lack of funding of possible obstacles like you can think that these other possible sources of information that may be enough ecosystem this unlike the possible things that become measures and then we will put the narrow between 2 observables you like they have several peaks it's a refinement of survival like wine in the sense that the odds are anything but it's it's like finer than the other ones and we will certainly
03:52
that because that making like this kind of grass in a supposedly there you have like a consistent run the model and then you have this random violence that like this can distinguish 1 .period Minnesota To Hear My Big Swan and his denoting Nevada will go to the states like Santa mind you on this point blank and certain other vital while on the on the compliment but then I think that this value defines ignited around then I'd just like to distinguish between 2 variables that find the same thing my solo in the fight against this partitions so here I had the petitions the retention of distinguish 1 from the petition undistinguished to British and 3 but then if you will immediately when you have the yellow is the competition by reduces very poor example with its wonders I and also I could remove like 1 of several for example may not be able to do observation exist and you can take like a physical example for example is quantum mechanics so you Beglitis said momentum this observables are related to this moment and then you cannot measure like to all of those at the same time because we don't commute by have this 4th observable and that is compatible with the at the expense of wines sold at the end you can organize like about there are different refined and relations between the observed it was and I will put their name to that so the name of these will be information structures on information structure will be with the and getting worried that again I solicited soul you having their as objects of these categories are as the Burtis of the graph observable and then as ROC you have these refinements relations and then we will assume that each time that you have a bottle if the the semifinal of for the 2 models you can make this construction that gold enjoyed by sewerage entirely when everybody knows that the media is at the level of of petitions this is their costs a petition that is finer than the petition of why the petitioners said so on the KSE you have is structure and we will see this as a brother but fact is the categorical brought up on so those are the assumptions that we make on this suspected to have this nicer the 2nd year in a is that you have to adhere compositions of Boro's also identities with essential Our I'm like remember like what it's like the entropy .period Anteby each is a certain like functions while the broader duties on the petition it's so the naturalist space word these leaves it is a member functions that goes from providing on you so the real numbers so essential services implicit in information theory that we we want fakes like 1 providing like that we want to play with families of that possible priorities that we could define this petition so I would call this finding is surprisingly school so on each cheek you harnessing arrest makes and you can like consider certain family groupings that we assume that part of a tried in some way and then each time you have a refinement you assume that there is addictive map that goes from creeks too Kuwait so frivolity example is like you can take the the thing we considering before so only a 1 2 3 the 3rd petitions that discriminate against 1 blind is that the meat petition so all the possible loss on the stomach petition to concede is said that the 2 but just like triples of numbers that someone and then if you have but if you have a problem on the on petition you can always transform that to appropriate in this petition is like this summer component
08:26
Our and I would if I like the last ingredient that is dysfunctional models so you have Europe that thievery with Claribel then to each model can associate a certain set of proteins and now I said that if I want to go see the light all the possible will function that they probably reducing give me real numbers so sold in particular like the entropy the entropy of pigs and you and that the soldiers we find out here the there is a natural action of commissioning the new can be here so you have a certain nerve function as that is the final the priorities its and can commissioned this wedding like this was into vision 2 on live data on Of this function 1 example
09:28
closing Our they're 1 of the features of all these can be extended like quite naturally to the quantum keys so you considers that of only I take like a buying dimensional complex space our amendment of cereals after break the hours and then there's also like a kind of notion of refinement because each of several defiance of the composition of the space and directs some of some spaces so you have abated the composition that refines the other with the narrative so the guys like the constructors very universe and well that the information part of the Prodigy and that we want to light go to this algebra and geometry a delay likable more theory related to this problem I'm well I want into this very clearly like the construction we did not I will today will do knowledge this city so the use for example is the Runkle monitored in 1 of the suits some political sneered at day care to function the final honest open said of the plane and even mother air if there is another function they can be as such that the the gradient of that function gives you this so it is it is You Are there is a necessary condition know because in the functioning of the a smooth you know that the the Crossville across derivatives like I have to be the St. on the history gives this necessary condition and then you can ask business conditions sufficient so not always is not always sufficient about you Yukon proven well this is a suitable interference that the restart shaped writing comics throughout year is like you have the is a sufficient condition you will find that the but the it you have whole thing the plane with the whole team so the answer is no and then you can consider a simple example you think these 2 functions and then when you assume that there is such as your rights contradiction because well in the 3 competitions so in fact the answer depends of the chain of so if moving is like as a whole for example we know the answer is no exploit this fact the diamond contracts and others
12:00
so what they now directors will write writers like differently so would you say that we had the continuous functions that term a continuous functions we can apply this operator that is called the gradient so so this year In this victory like that is that because the victim of an amendment to a vector we can apply this this quarrel operator that not too To this function and we know that if we if think that girl of the gradient and that's so we know that the image of the gradient is in the care of the group that is not always and equality economic and we just saw them so you consider this this that we call like an excessive and then you look at these groups this is the 1st call at the Group of I'm so they can of the accrual delighted that the image of the great so we just started this thing is especially this is everything that is like the Court of someone then the written as the gradient of 7 everything that has proved put to 0 can of the region of someone but if you remove various things so Netherlanders here we have likened example in fact you can prove that these equal to a is isomorphic to the real numbers power and where n is the number of schools so I hear you you've got to rely
13:34
very very good the shape of the answer depends on the tip of well soon well you
13:42
can arrive instructions are here displayed was leaders the black books if you want but asserted that the wants skull dollars globally but that's not the thing is you have you owe you can't it here in that the area of the information you provide needed that set on the this functional model that we define as well the advantages of construction for a so at the end like a
14:16
sequence like his nose sequence that this sequence we had before is that the signals but continues so you can define Gowanda groups as the 4th disfigured this mad divided by agreement to this 1 even so what happened in the singles observations this observation said so I will likely take again like the than the simple example so you it's only a 1 2 3 these petitions slope of the dummy petitioned here and if you follow this like a dinner of construction there you come the recision along cycle now will be refined military functions so that makes wanted to and that satisfy these equations on all those flying but the others on the board and and and these equations are appealed on Tuesday for example here we quarrel people to see so we wouldn't want to be chronicled 0 1 do something more complicated people to 0 but still slated this approach equations on these equations are functional equation of any I said that these things are functions I but you can like well some people from which is especially in India's so hot and they have been the doing that 30 years so that no like so set of solutions so add the that if you find an equation like Beasley disagrees with this solutions To this thing I'm
15:48
not multiples of the and so she has selected a very simple example is just like the entropy concentrated in 2 elements but but well you you can eating dinner with if you analyze the patients in a was the moral of the story effect is a further than were mentally fit the information come already this year thing that we can define from damage where I had a number of situations generalities onedimensional from onedimensional but space and fact this like uh composed but all this real multiples of the ship entry so as to conclude this wellmeaning the whole of the but I the media while they did you we can use this maybe general approach categorical approach to right NOC probably due to different for example way and this is an idea of grumbled we weakened we could take like the observables as the primary thing and is like this only as a kind of model like the selecting of the base for a victory space that it already has military nothing like really ignorant to produce because in fact and that the situation is likely to serve the needs of from the matters of whenever you change for me on the way so so fact you could think that you have just observables under several sigh of finance sets with the possible upward and then well this out but do have laid altered throughout with their like needing another observable there so you you you you make a category of observables and then to try to black the only other match all this observables but the construction not unique and you could study different constructions like that said the button well maybe the perspective of what this is like trying to understand well with the geometry and allow how can we like what like something is useful for priority to carry out like remaining guests is to obtain like hiding information functions like a ways of seeing like common information is shared between 3 forward like 100 variables and try to tool light find that thing the fire :colon would you groups but we can define the state approach so well preserved that the papers like the bravery of the Devinney account with display like the introduction of the concept and the paper of Gromov that is this categorical approach to biology to plant thank you I will send you growth for in the 1st place this question yeah no only it's not let's stand improbability and and