 everybody to the conference entirely devoted to the concept of topos here at Yahshes which is the place where topos were originally introduced by Grotendik in his seminar the the geometry algebraic du buamari. So the aim of this conference is to illustrate the wide-ranging impact of the notion of topos across different mathematical areas and encourage further developments in this direction. Grotendik in fact was himself fully aware of the importance and the unifying nature of the concept of topos. In his text Rekolte Semai he compares the notion of topos to a double bed realizing an harmonious marriage between the continuous and the discrete or also to a large and peaceful very deep river in which all the kings horses could drink together. This explains our apparently eccentric poster. So this is what Grotendik thought about the toposis. He looked at the toposis as a unifying space and the more precise here is what he wrote. For those of you who are not familiar with French here is a translation. It is the theme of toposis which is this bed or this deep river in which come to be married geometry and algebra topology and arithmetic mathematical logic and category theory the word of the continuous and that of discontinues or discrete structures. It is what I have conceived of most broad to perceive with finance by the same language rich of geometric resonances in essence which is common to situations most distant from each other. So we hope that this conference will stimulate more people across the different areas to study and apply toposis in their work as Grotendik instilled and deserved and hoped for. So I would like to thank all of you for coming here so numerous and in particular the speakers for accepting to give talks that promise to be very interesting. And of course I would also like to thank the other organizers for joining me in this adventure and of course the IHS for hosting us and also for the financial support which was also given by the Foundation L'Oréal d'Unesco pour les familles la science and this is basically the funding that made it possible to organize this meeting. So thanks again everybody for being here we hope that the conference is going to be enjoyable and fruitful. So now I guess that it's time for me to leave the world to the first speaker Pierre Cartier who will give an historic introduction to the concept of topos. Thank you again. So I will give a historical account of the theory of sheaths. Sheaths came before Topoi and Topoi is just a synthesis of all different kinds of sheaths. So I will start I mean here are two Bibles the book by Godement Théoride Pesou and the Carton Seminar two Bibles and I will refer to them later on. So the first part will be prehistory of Riemann versus Weierstrass. So the problem which was tackled by both Riemann and Weierstrass in many in very different ways is a problem of multi-valued holomorphic functions. First of all, holomorphic functions can be defined as follows which is Cauchy, Riemann and I would like to add Diderot no no no equations. What we call the Riemann Cauchy equation which can be written as df over dz bar equals zero. We are written explicitly by Cauchy and then by Riemann but about 50 years before by D'Alembert in his study of the wind motion wind motion and so the Cauchy Riemann equation of an interpretation in full dynamics and D'Alembert was studying the motion of wind at a global scale of the earth and discovered that he could use complex variables. He could use complex variables. Of course the theory of complex variables were just in the preliminary form but he could use and so he could use for example sign of x plus or y y etc as other facts. So but of course it has been more or less forgotten for a long time and it's only recently that good connection with the score. So Cauchy Riemann equation so it's a kind, I will not write it more explicitly, it's a kind of differential equation first order system of first order differential equation if you replace f by e3 a part and imaginary part and the idea is that in slightly modern form you consider an open set u in the complex field and you look for a function which is defined on u f from u to c and which satisfies which is a solution of this differential equation and as in modern slang I mean the collection of all these for all open set forms a sheaf in a modern form and more generally if you can replace this by an arbitrary linear differential equation or even non-linear differential equation you can have the same situation but the point of view of course the challenge is to study the multi-valued function. The most simple example of multi-valued function is log z of course what is known is that by definition d log z is d z over z which means that if you consider z as a real positive variable and you make an initial condition log of 1 is 0 then you get a unique solution over the real axis but soon it became important to study log for complex number the first and it was done by Euler and Euler has a question if people repeatedly what is the value of log minus 1 of course if minus 1 to the square is plus 1 and if you use a functional equation log of ad is log of b you conclude that log of minus 1 is 0 but if you want to do to be brave enough to try to consider log of z as a complex or z as a complex variable you soon realize that and Euler knew it perfectly that you when you approach so if you remove if you remove the negative line minus infinity to zero the rest is simply connected so c minus which is to make a cut this is simply connected and the differential from the general principle of differential equation this differential equation d log z is d z over z as a unique solution provided the normalization log 1 is 0 which is defined in z but when you approach the two lips of this cut you'd get different different values why if you approach minus 1 from the from the from above you get pi i from below minus pi and there is a difference so that was a challenge and what is the true value of log z if you don't make a cut if you don't make a similar problems occur when you want to solve not differential equations very simple primitive but primitive but let's say if you want to calculate what to find what is the solution of y square is x 1 minus x of course if you if you consider the segment from 0 to 1 real sigma to 1 you have two solutions we differ by your side choose one let's say choose that y is square root of x 1 minus x for x between 0 and 1 okay now the problem the problem the problem is try to extend this for a complex number how can you calculate the square root of a complex number from a modern point of view of course if again if you make two cuts if you remove now the real line from minus infinity to 0 and 1 to plus infinity the rest is simply connected and it's a general principle that if you have a continuous function on a simply connected space you can take the you can take the square root provide but there are two branches plus or minus so but if you want to have agreement over the real over the real part 0 to 1 you have only one solution so you have a solution you have again a complex complex solution of this equation so when x is real and complex provided it not on the cut y is well defined but you have two branches plus or minus one but then again when you have the same problem if you approach if you approach from from the from the bottom from the top or from the bottom the cuts then you get different regions if you start with one determination let's say and you approach and you approach it from the top then you meet the other real is the other branch minus one which comes from the from below so that was a that was a challenge and and similar similar things occur for differential equation the most conspicuous example is about the uh about hyper geometric function f2 2f alpha beta gamma z i will not repeat it's given by your certain power series in z which converges with ranges of convergence is is one so you have a function which is well defined on the which is well defined on the unit disk but then you want to extend it and you have a certain differential equation no need to write it in detail but now as oiler discovered and kumar exploited this equates this function as an integral representation and the integral representation is for the type 0 to 1 t alpha 1 alpha prime 1 minus t beta prime 1 minus tz gamma prime dt don't have well there are there are some relations between alpha beta gamma and alpha but it's not important but the point is that if you consider this if you consider this integral and you consider that you went uh yes then you can you can you can see it's well defined because it's a real integral integral over z1 but of course the difficulty is about this t alpha t alpha prime okay and then if you choose z which is not on the cut from 1 to infinity then you can choose a proper branch of 1 minus t z power gamma prime and integrate and again you have you have a solution which is defined outside of this but it's a second order differential equation and then you have two solutions to independently independent solution the other solution will not give the precise formula the other solution appears to be uh holomorphic except on the cut minus infinity to zero so in order to come to have all the solution you make these two cuts again you have a if you make the two cuts you have a simply connected domain and according to general result a second order linear differential equation has two linearly independent solution and so but the same problem which was solved by kumar if you if you take one branch that means one solution one solution and if you approach a cut from below or from above because the differential equation remains valid one solution will approach a linear combination of two one solution from the top two certain linear combination from the bottom which means that when if you have a if you have a vector space v of s of solution which is two-dimensional then when you approach from the top of the top that means if you make a full turn around zero you have a linear transformation which is exactly what we call monodromes but since there are two singular points for this differential equation there are two monodromes one around one one around zero and they generate a certain group of linear transformation which was defined given in explicit form by kumar so that's that's and now what do we do what do we do Riemann motivated by this example introduced the idea of putting putting a sheet and gluing them together for example for the square root you have two sheets on one sheet which is a copy which is a copy of this simply connected domain you have the you take the the branch of the function which is positive on this interval 0 to 1 and then the other and then you put minus one on the other sheet so you have two sheets and in order to to to describe the the connection the connection rules i mean the boundary conduction which brings one solution to the other one when you approach the cut then you make a you make a geometrical picture by taking two sheets of paper cutting and pasting of course this is highly intuitive and which was advocated by by Riemann but the criticism came from Bayerstrass and it's also a political a political fight the two major centers of mathematics in germany at the time were berlin and heidelberg and guttingen and there was a bit of fight between the people in and in the guttingen mostly around around felix ryan and heidelberg and in berlin mostly around bayerstrass there was recently in berlin a celebration of the 200th birthday of an anniversary birthday of bayerstrass and many good explanations were given at that occasion and i was there so now this is riemann which is quite intuitive and it's not by it's not by chance that one of the books by heidelberg is called unshowlish geometry which is translated that geometry in the imagination is not a good translation it's more geometry as you see it unshowlish geometry unshowlish in philosophy has a very strict sense which is not the imagination okay so then but on the other end bayerstrass came with a different solution his solution was the following if you consider if you consider a function holomorphic function what did you say a solution of the coche riemann solution equation are the given point z zero then you know that you can expand it as a power series sigma a and z minus z zero power n for n zero are positive suppose that you have two points z prime one z prime zero and z double prime zero and similar expenses are put in and then coefficient in n a double prime n okay you have two you have two solution two two power series expansion for instance they represent solution of algebraic or differential equation how can you say that it is the same solution how can do which is a problem of analytic continuation of course analytic continuation as a sound solution in the complex domain but people now try to do holomorphic function in a period is much more complicated and it would require you had three hours to explain how do you do it in the period case okay so now you have this solution and then the proposal of the proposal of of various tasks and that you trace a path from z prime zero to z double prime zero and you follow by continuity this power this power series this power series well more precisely inside a circle of convergence if you have another point then you can transform this this power series expansion into another one which is not centered at zero but another point and then you follow along the path and you get so you get a transformation which is analytic continuation if you give me a path and you start from you start with the power series mentioned around the origin of the path can you go until well of course it may be obstaculate you can assume that there are no obstaculate and if you have a differential equation or algebraic equation only the singular point can be can be can prevent you from following so now the proposal in slightly modern form slightly modernized form is the following by various tasks you consider the complex plane and your running point in is called z and then you consider you consider the following you consider and you consider some power series some a and t and and positive we call it phi phi of t phi of t so a series correspond to a collection of coefficient with some aboundness condition with some gross condition which which will give you that radius of convergence is not zero so now what you have if you call if you call s the set of all this series you are really dealing with c times s c times s you give a point and a series now analytic continuation can be as follows if i give you a point z zero and a phi zero and a phi zero and that's what i explained you consider the radius of convergence of this series and if you try and if you have a power series which is centered at z zero by transformation which are made completely explicit by habel you can generate a family of of power series so for this and so it means that given a point in this space you can have a neighborhood and you take this you take this as a neighbor so in the product c as the ordinary topology and s has no topology at the moment and then you can you can have this you can lift if you can if you start from this you lift this a small disk to this and of course it says that in the collection c times s you have a you have a collection of neighborhood and it generates a topology it generates a topology which is the one showcase use that in this you often in this lecture on analysis this presentation which is just a small variation but then if you now you have a topology on c times s and now if you take this this this topology is disconnected there are many components and the Riemann surface a Riemann surface is just a connected component of this so it's a connected Riemann surface called it r is a connected component but it has obviously a projection over c because it's part of this product space and it has also yes c which is z but also you can have you can have another projection you have another projection well not a projection this is projection which i call it z and then you have f of z which is the function which is a function will give corresponding to the Riemann surface so a function or homomorphic function in this setup is not a function over an open set in the complex domain you are first to consider a covering which is of course et al covering as we saw we need we know now and then the function is not defined on c but it's defined on this that's the idea of Riemann I put a Riemann surface okay so now it can be well it can be it can make the connection between the two approaches and in modern terms it corresponds to consider a shift by its section over open set or by its et al space corresponding et al space so but this is the pre-story and already all the ideas are there and they were quite explicit by Hermann Weid in 1911 in his book the idea of Riemann surface the idea of the Riemann shunt flesher that is more more philosophical okay so that's the pre-history of shapes now what come next second second is Leuré Leuré from 1941 to basically 1950 well as it is well known Leuré was he's made a work prisoner and he spent five years in a camp which was called an off-large officer lager which is camp for officer in German and which was rather my rather my conditions rather my condition and there was a kind of home rule in such camps that means the French the French officer elected their leaders elected their leaders and their leaders had to negotiate with the German leader with the German German commander of the camp a little kind of home rule and Leuré was able to organize to organize a so-called in the in university within the camp and which was a way to well officer were not forced to work as ordinary soldier were sometimes forced to work in feats or in factories but they had to they had to find something to do okay Leuré organized that I suppose to have been the French leader in such a situation would have been sometimes difficult situation I can imagine it is known I know from family witnesses that the Jews were not usually chased in such camps except if some Frenchman pretended that they were Jews and they don't send them to the open okay that's another story so Leuré who was an expert in fluid dynamics was approached by especially Hasse to work on different projects for the German for instance to be editor of central blood or maybe to work in the field in fluid dynamics but then in order not to be able he stayed which was very brave he stayed in his camp he stayed in his camp and developed topology it's not my chance because there was already the fixed point of theorem of Schauder and Leuré which was already topological in spirit but which was used to find have to find prove uniqueness or existence in uniqueness solution of non-linear differential equation especially in fluid dynamics so what Leuré invented so Leuré invented these sheets and his main motivation was the following suppose that you have two two topological spaces and a map from the space x to the space y okay now you want to calculate the homology of x and at that time homology means check homology the homology of x in terms of the homology of y and of the mapping so basically the idea of Leuré was the following if I take a point y in this small y then I consider the fiber the pre-image fiber f minus one y I consider the the the homology group of this fiber and I study how they change when y moves okay in under reasonable condition the reasonable condition for instance if it's a locally trivial bundle then of course it's locally constant because in a small neighborhood around why you can identify the fiber with it's also and the common with it's also but what's happened when you move out of course the idea of a local system was already known by I think it's with Whitney will introduce this I still know the Whitney I'm not sure but at least it was known to both so the idea of local system and all the topologies that you have a system of coefficients which vary from one point to another one okay now you want to reconstruct the new data of that you have for each fiber it's homology group and you want to reconstruct the homology of the total space out of this information and that's where Leuré invented the so-called spectral sequence which sometimes is called the Leuré-Cosul spectral sequence for the reason that Leuré invented it but in a specific situation and Cosul used it in another situation he used it for the homology and common of Li algebras but then if Cosul fed necessary to give a general formulation of what is a spectral sequence and then there are sequence and he gave the solution that it's a collection of it's a collection of complexes each one being the common of the previous one which is a very very good very good and sometimes misleading very good but sometimes so this and then this spectral sequence is something which enabled you to reconcile the common of X from the common of Y and the common of the fiber and Leuré and those many other people I mean develop this idea and especially Borel, use it to consider the common g of groups because if you have a group and a subgroup if your d group h and g then you have the projection of g over gh homogenous space and it connects the common g of g with the common g of the subgroup and the common g of homogenous space and Borel used this with great anesthesia with great benefit okay so that's that was already known and but Leuré did not well because of course of the conditions he published his final account only only in 1950 so it will not enter into the I think he was motivated by his study in functional okay so that's what we had then the final account was published into only 1950 and then in between so the Armand Borel used immediately Armand Borel who was at this time in Zurich and the Ecole Polytechnique ETH in Armand Borel used that for common g of groups and that's the main one is of his best contributions okay then in between then other people joined and Carton Henri Carton studied that already starting from 46 47 and Carton was invited in the Harvard in the in the beginning of 47 48 so beginning of 48 and there is a story that his plane was delayed and he could not miss a very well okay you can write a novel with that okay so and the the daughter of Rodrigo has written a novel about okay so Henri Carton and then Henri Carton had another approach another approach at this occasion he developed the idea of a greeting or which is in French carapace and I will come back to this idea later gating so it was another approach but then Carton and you have also to mention André Vey who entered into the game and entered into the game and André Vey was obsessed by the theorem of of Durham, Durham theorems and he repeatedly said that the core of geometry the Durham theorem is a core of geometry so and André Vey published a version of a proof of the of the theorem of Durham which is by far the best possible the best the best and which influence which influence Carton of course André Vey would pretend I don't know what is a sheath it's just a stenography would say what that's what's part of his way of of thing but is when you look at his proof it's clear that he perfectly understood what is a sheath okay now that was so there was a period of hesitation a period of hesitation and Carton when coming back from from Harvard in 47 48 and 40 not 48 49 in his first seminar in his official seminar which is available in this collection of books collection of books this was his collection then Carton explained both in his lectures in the in the Harvard and his seminar a way to define sheath grating etc but he was not happy with that he was not happy with this and this part of his seminar has been dropped from the published version of the seminar and it started again in 1950 1951 in a long seminar which was a collaboration with Samuel Ailenberg and it's my the first time I attended a mathematics seminar I was a first-year student that he called normal at the time and then it started again it started again to develop this in the last part of his one this recording in this volume then he started with a completely new point of view completely new point of view and then he gave a general definition of sheaths he gave a general definition of sheaths but mostly under the influence of Michel Nazard who is credited for that he developed this idea starting not from the sheaths collection of section of open set but the corresponding et al space and it's it's it's clear that it is it's dominant it's dominant it is exposition and so it comes at the variance from the modern exposition well of course I will come back to that later to the to the various views about sheath but then he started again as that's Henri Carton and but he was influenced by Samuel Ailenberg who gave in the first part of the seminar an account of the homology and homology of group in the axiomatic spirit of course Samuel Ailenberg and McLean invented the homogyre and other people invented the homogyre groups of also but for the first time Samuel Ailenberg presented it into an axiomatic way and he gave a series of axiom for the homogyre group which could be more more less formally well more or less immediately translated as axiom for the homogyre of spaces but the point of view is changing the point of view is changing because so far when you where people develop the homogyre of spaces where there was Czech Spaniard there was Durham they choose coefficients a space and coefficients they kept the coefficient and let the space vary for it so you take the homogyre with real coefficient and associated to the various spaces the point of view of space is a dramatic change of point of view now you have one space but you take the collection you take the coefficient as variables that means you consider one space you consider the category of old sheaths or coefficient whatever whatever over that space and the homogyre is defined as depending on the coefficient the space is filled by the coefficient right which is a totally new point of view of course and then we have also to remember that about the same time to pose a historical purpose to make that Ailenberg and Stienrod published in 1951 foundations of algebraic topology where they gave an axiomatic description of the homology group of a space with with a coefficient which are fixed not to call not to shift and and then they prove that for reasonable category of space comprising all the standard spaces the homogy is uniquely defined by this axiom but one crucial axiom is that the homogy of a port is trivial h0 exists it's a coefficient of the homology h1 h2 it's r0 and this small but later on later on it was this condition was removed and the so-called extraordinary commodity were invented were invented mostly by Krillin and other people can't Krillin and so on to get account of this so but it's not it's not the point so the change of emphasis is the following the space is there but it's just the place to put all the shifts okay now about the development so Henri Carton in his seminar and I spent yesterday evening I'm reading again this seminar and so give an axiomatic description prove a uniqueness existence and uniqueness theorem but with some restriction on the space is used well Henri Carton was obsessed by the idea of treating commodity with compact support and commodity with ordinary support at the same time so he introduced a certain collection he has a space x but he introduced a suitable collection of close subsets close subset of the space and he considers this the commodity with support in five five maybe the set of compact subset of a localized compact space or all the open or closed subset of a compact space compact spaces are a slight extension of locally spaces which was invented by Geudonné in 42 or 43 for the purpose of decomposing of a continuous function in two variables as a sum of product of one variable so that makes the exposition slightly slightly disturbing but the point is that there is a restriction this restriction on five forces you if you want to have general commodity with a close support that the space be by compact what by compact as I say so slight extension of locally compact but of important reason is that any metering metric space is by compact okay and that was enough for applications for various applications but there was a restriction there was a restriction and soon after that sir came as a dog in the garden twice first of all in his thesis which was defended in 1951 he applied this idea to to to homotopy problem so and the main object was the loops the loop space or the loop surpass space of if you have a space x then you consider the space of all paths originating in a given point zero and you consider this with the proper topology and you study the this is certainly not a locally compact space and you study the commodity of that in order to get information on the homotopy of the given space that is thesis that is thesis and with great success because you had a great success with a homotopy group of spheres with the hope that we could calculate all the homotopy group of sphere but we are still far from that unfortunately okay so sir came and he had to but he had to so he needed both commodity for these new spaces which were not locally compact not by a compact and also he needed yes and he needed these spaces and he needed the spectral sequence because the specter you can consider the state of pass is the vibration over x over space x what is called now a cell vibration which is characterized by the lifting possibility of lifting of polyhedral then he had but in order to calculate p the point is that p is a contractible space also commodity of p is nothing except for one is the same commodity as one point but you want to have information about the homotopy and homotopy then you can consider that the loop space is a fiber of this of this projection so the strategy is to combine the commodity of a space and the commodity of the fiber to get the commodity of the space that's what it is doing for instance in the in the in Borat but here the commodity of this p is no one it's a contract but so you get an interconnection between x and omega and the point which gives you information about the homotopy group but for unfortunately both the both the few of shifts and the commodity of shifts and the notion of fiber did not work so he had to invent he had to invent and he explained to me recently that he listed this he made he made a comment i have to reprove when he was cautious about not yet not not disturbing loray and i had to reprove some statement or make a new connection etc and he explained to me that it was quite delicate not to offend loray who was not a music character okay so loray did that and obviously one needed and then he invented ad hoc method ad hoc method he in order to fit with the vibration he introduced not a singular homology but not with more than which are simplices but tubes now we have much better much better term not better tools in the in the in simple it's implicit homology and we can deal but we're using by simplisher island there's it burns so on we can do this directly but he was forced to use chemical homology but he had to prove that chemical homology was the same as simplisher homology so a lot of discussion of course great great great progress but at the cost of costing the boundaries now sir did a second time did cost a boundary a second time where he used well first of all in between there was the work of carton and sir on complex variable many complex variable and then they use civc homology with a great success to deal with problem of many complex what holomorphic function of many complex variable where the topology is quite delicate a remand surface on one variable is rather easy to discard a remand surface in many complex variable is an awful object an awful object which is really not completely understood okay so but then carton and sir used they use civc homology they did not create a new homology they refer on the carton homology which was defined as I mentioned first in 46 47 and in new form in 50 51 okay but when after that they obtained the so-called theorem a and b which are famous theorems in the chronology of coherent shift of holomorphic functions but then carton sir developed application to projective spaces and using a certain trick which was given to him by someone in robot someone in the group and then then he used civc homology but he used for projective space common geofrojective space and then he realized that his method were basically algebraic basically algebraic but what was needed what was needed is the risk topology by the way the risk topology as we know it today is not what inventor was I just invented invented the variant of that topology on the net of all local of all valuation rings of a given fun or given feet which is not exactly the same but who was used sometime by other people's ice kid topology but the size kid topology the rumor is that this was invented by chauvalet in a seminar given by the risky of no real proof but it's it's quite full quite you it seems quite reasonable to assume that but the series kid topology was first used by on the way on the way in 1948 gave a course in gave a course in in Chicago which has not yet been published unfortunately I tried repeatedly to have it published but I did not succeed he gave a course about what are fiber spaces in algebraic geometry of course and then if you have a fiber space the main property is that it should be locally trivial but locally the human that you have a new need of topology and from various conditions it was clear that the good topology was the risky topology so it's really on the way who invented what were used for the first time seriously and also he was connected with so-called cousin problem which is to reconstruct a meromorphic function in many complex variables out of the set of the poles and divisor of the function and zeros and poles okay so Andre Bay was the first to insist on the risky topology as a tool to imitate topology in algebraic geometry and then sir knew that perfectly sir wrote a report for the Boba key seminar about this work of of Andre Bay and then and he immediately tried to imitate what he had done with the help of carton for a complex complex projective space viewed with common of shifts over projective complex positive space and he as he explained to me many times I mean it was so easy now it was so easy now so easy now compared to what he had done with carton in complex variables so it's sir who invented and his famous paper which was finally published I don't remember it well it was it was written 53 but published in 59 55 that's there and and then I remember my surprise when I presented to me part of his manuscript an algebraic variety is a topological space in algebra it was a surprise to many people okay then but he had to invent he had to to use shift homology but Zeissky topology is a terrible object compared to ordinary topology it's it's completely exotic compared to ordinary topology and there was no shift homology available for that and then he had to reinvent it to reinvent it in a slightly different way so sir twice cause the border cause the border showing the necessity of going beyond ordinary spaces and all the chiefs over ordinary spaces but this time you had to the challenge was to invent a homology of shift without any restriction on the space uh at the time got and it was back to Paris and there were many exchanges at the time with Grotonnik and it's Grotonnik who provides the first solution he provided in his typical way first well of course the main challenge was the following from homological algebra non-metal of homological algebra as the book of carton and island man was just printed at the time he appeared just at the time in order to define homology you need resolution well if you have if you have a module for instance that the modules is a quotient of a projective module this is easily a quotient of a free module so you can define a projective resolution quite easily but duly injective resolution injective resolution at first injective objects were invented just by duality because island man was obsessed by the duality and obsessed if you take a diagram and reverse all the arrow what happens if you were defined by diagrams of property of being a projective reversal the arrow you have the definition of injective but it was a quite exotic object an injective module and it's not easy to prove not easy to prove that the modules can be embedded in an injective module buyer buyer gave a proof which is quite quite sophisticated then okay so it was clear that for shift you could you don't have projective shifts you don't have free shift projective keys whatever you don't have that so the need is you have to go to the other side and take the injective shifts but to to to to construct an injective shift were not obvious and no one knew how to make it okay then got and came from the sky I mean he said what in order to prove that any object can be embedded into an injective object I use a buyer construction the buyer construction and I look at what are the formal properties using this proof and the formal properties are basically or the basic the crucial idea is that if you have a module M and a connection of some module M alpha and another module it's a distributivity law which is intersection let's see intersection of M alpha plus M is something intersection of M alpha M alpha plus M something like that something like that I'm sorry ah okay okay okay okay you are right you are right well one basic ingredient in the proof was this so Golden Dick invented the idea come from the sky I will consider an abstract category and at this point invented the notion of of of a billion category which was about the same time invented by your books bomb under the name of exact category so that was more or less natural natural idea the new idea was to introduce this as an axiom somewhat corresponding abstract form of this action of this action and then repeating the buyer construction you can prove that any object can be embedded into an injective object in a given category so now the next step is to prove that for shifts it works well but it's a rather easy exercise it's rather easy exercise so in this way Golden Dick could prove absolute from abstract principle that a shift can be embedded in an injective shift and then you have injective resolution resolution of shifts okay but almost immediately Godamon came with a more explicit solution Godamon came with almost almost immediately with an explicit solution which happened which is the following take a shift f over space x now I consider the various the various fiber fx for all small small x into it and take the product and take the product of the fx which amounts to the same you have x space and you have x delta which is the same space with a discrete topology at the time you have a you have a natural map of course natural map we discontinue the right direction call it pi and then you play with the factor pi star and pi lower star of direct image and inverse image but for this map from x delta to x and then it's easy to show it's easy to when you have a discrete space what is a shift over discrete space which is injective each fiber is injective so it was easy so that's the that's the solution of that was provided by Godamon and the other variants of various form of these conditions so here we are here we are gotten dick invented well by his new category category called theory of a billion category is able to prove that you have injective can apply it to the shift you have injective see if you can repeat exactly word for word what was given in the carton rebe for modules and get in one in one or two pages all the necessary properties okay and then as I said Godamon in his textbook repeated that with a slight variable so now as we are well checked yesterday that Godamon book was published in 1958 now the famous paper of golden dick which is called talk because it was published in the talk mathematical journal was written in 55 and the copies were circulated and discussed among burbaki at the time but it was published only in 58 so the book of Godamon and the and and and and the good and the papers appear at the same time and they refer what Godamon refers to to Godamon okay so that's now what we have that's now what we have are at the moment you have a now a completely general theory of shifts and shift homology over an arbitrary space and you can cover both the point of view of sir i mean the sir vibration is and the path space is covered by this new new theory and also the application to a gemite to a gemite geometry so that's what it was around 58 now right no no no no no no no no no no no no no no no no no no no no no carton carton carton gives us in for certain spaces the common g is a check homology and already already learned at few rams comparing his chronology with check homology spanner singular and uh what was the last one spanner singular and uh doram so we are already many many many in with wait wait wait but already well loray already compared to take well loray loray come defined check homology for shifts and compare it with his common and understandable condition is the thing okay pardon see see see see loray invented the chief commodity no problem and then carton gave another version but it was well another presentation i would say but no no no no he doesn't define it as check homology no he defined it in general and then he proved that by by resolution by resolution i will come to that at that at that moment okay so that's exactly the idea of grading so okay so at that time since that what we had of course are what came what became very important and what precedes topos topos is that first of all there is a comparison between uh well the comparison between the two point of view it's either a trial space or it's a collection of section over the open in in carton seminar it takes a definition for a trial space gives one in a few sentences he said what we could also use it in this way we use it in this way but i think carton was uh dealing with uh complex variable and he was in full by the point of view of viostras okay so uh yes but uh the notion of pre-chief for instance which is so familiar today doesn't appear explicitly in both in eyes of case so of course there are some remarks if you start with what we call now a pre-chief you can generate etc but usually what was what was done is that you start when you had a pre-chief you associated to it an et al space and et al space was considered a shift but the problem of shiftification starting from a pre-chief to get a shift was not really really i mean the present in this day it's really good and we introduce it in this sg for this point of for more general situation so uh another another problem is that about another problem is about a a continuous map so if you have if you have a if you have a shift as here you can define an inverse image x star g which is x star g which is here and you can define the direct image x star f there and then there is these two factors are adjoint to each other in a proper way and by the way when the the adjointness is not as mentioned in the both both carton and the one is not the right one they take a right adjoint instead of a left adjoint etc so but then it's uh this was not realized as such import such import but now of course the point is that if you want to define f the direct image is very easy using the definition of the cheapest collection of section of open set because a section of this u is simply f of f1 v but the the the inverse image is better using the entire space okay so it shows that you have to to do to do with both now the so there was this problem there was this problem and it was important that when you have a continuous map from x into y you have two factors for shift one which is the inverse image and other one directly made and they are adjoint to each other in the proper way and and the existence of this adjointness is considered as a crucial property by Rotary the the new idea of Rotary is to replace the space x by its collection of sheets by this category of sheets and now instead of having a continuous map from x to y you have you replace fx by its topos which is this category of sheets y by its topos so called x the topos these are now categories and then instead of having f you have to find the f lower star and f upper star which are adjoint to each other with suitable property so that was now that gave the definition of maps between what we call geometric factor between both topos then there was another important fact which is when you have when you have a pre-shift a pre-shift and as I said it a pre-shift is what it's just a factor already in modern terms you have a space x you have the collection you have the category let's call it's top tx topology of x which is a collection of open sets so that's the object of this category and the map are just simply the inclusion if you within v you have the injection of inclusion in u of v and that's only maps which exist so this call and now a factor called trivial factor from tx tool it says category of set category of things category of whatever category of cows if you want to have chief of cows and then then it's a it's a covariant so it's a factor tx op tool let's say sets if you want to all those things okay then it how can you generate how can you generate a shift it the idea is the following in godemort carton etc the idea is that first of first research a factor you associate a space e which is ital over x and then you take the section of this space to generate a shift but the point the point the disadvantage is that you have to use the point of your spaces and it's contrary it's opposite to the to the spirit of categories the spirit of category you should never mention rarely mention the set the point of a set or space but only what maps etc so this was not very satisfying and as a result it's not very mentioned it's good and it who really understood the importance of it and the point of the now the what is important is the shiftification which is so called pre-shift pre-shift pre-shift of pre-shift are just the I mean this is a collection of this this co-travagant factor with no with no relation and then you have shifts which are shifts shifts which are of course a special class of pre-shift now you have the inclusion map a pre-shift is a pre-shift and upon the shiftification the shiftification factor is what here you have the inject injection of this category into this category and there is an adjoint there is an adjoint which is the one called called hs and adjoint with suitable properties the point is that an adjoint of a factor satisfies automatically certain conditions with respect to limits and you put another limit so core limit general core limit sorry preserved by s but you want you have to assume a little more so now the point what what was really important for Grotendi is realizing the analogy between these two situations first of all you have if you replace the space by the top of all this shift then a map is correspond to a collection of two to a data of two adjoint factors and again here but it's completely different you have a you have a shift you have a category of pre-shift and you have a sub-category and again two adjoint factors and this is really really the crucial discovery of Grotendi in this case and then from starting from these two examples you can formulate the general definition of a of a shift which is the static sorry a topos well I elementary topos if you want but these are special these are special case Grotendi Grotendi topos which are a particular case of the of the of the of the elementary topos okay so I just finish that and that's a short historical account the last part should be much more developed because but then we are very close to Grotendi or you already got in Grotendi so I wanted to describe the pre well the pre-history the pre-history the pre-history and then well the history begins there and related it's a different story I would just wanted to show you that the notion of shift was natural natural after Riemann and Weierstrass the two point of view then it was developed in the 14 and 15 by especially Loré, Carton, Borel and Seyre that the people who should mention and then after that well after this big synthesis both by Carton and by by Godemard then what started but of course shifts are many many other applications which I did not mention for instance my colorful theory of shifts perverse shifts so I don't have time to to to develop all these the development of shifts I just wanted to tell the creation of shifts thank you very much