 Zdaj, najbolj všeč sem odpočiti organizacijer za to, da so počke na to, da ga so vse moj zašli. Počeš, da vse sem tudi se zelo vse vse občasovati in olačiti olivi Caramello in Lorena LaFarga. Tako, musim počutiti, da je loran i olivija. Zato sem organizačovalo v počutku. Prvno, da sem tukaj dobro tezidrat. To je tukaj dobro. Zato, da sem tezidrat. Tukaj sem naprati, da je to vseh zelo, kaj je tukaj, začelj, tukaj je vseh, tukaj je tukaj, tukaj je tega kategorija, tukaj je vseh, tukaj je tukaj, tukaj je vseh, tukaj je tukaj, ... in nekaj nekaj drugih, da je, kaj je, nekaj taj taj, taj modivov. ... in nekaj nekaj, da je taj modivkomplex. Protočno, različno različno na timi motivnih, kaj sem videl, timi motivnih je teoretika. Tako, kaj sem videl, teoretika je fizika. Prejsteš, da sem videl teoretika motivnih. Teoretika je zelo, da je vse teori. To je ideja. Ideja v teoriom motivu, kaj je vsega ideja, da je vsega vsega vsega vsega vsega, kaj je vsega vsega invarijača vsega, vzgovoril po vrognih otkatnem vse, ne, kaj lahko, zvrč, nekaj, se zbizljala, da je skupil da je ideja. Vsega je, kaj je vsoj, da je to bo, je odpočina, čao motiv, motiv, kajce, nab strongly vsega vsega, kaj je izgleda stilne korrespondencije. Zdaj je to kontynučnja vzgleda vzgleda. Vzgleda je tanačnja vzgleda. Vzgleda je tanačnja vzgleda, ali je tanačnja vzgleda vzgleda vzgleda vzgleda. Vzgleda je tanačnja vzgleda. Vzgleda je tanačnja vzgleda, šta je vzgleda iz dete, tega vzgleda, a to je več vzgleda. Vzgleda po vzgledo s počja. popreč je,here. Steljamo,ko je toto, to je motiviktega,kaj je se nasplot, nekaj nekaj nekaj, Ono je, da so tukaj, komoloji, omo ležite. Stajem, da se vsi dogodil na vse kategorije. Zelo, da je menej ideje na vsej vetišči. Zato, da zame vsej vetišči, sem zame vas na vsej vetišči, zelo vsej vetišči. To je zelo ambigos, nesel unike. To je zelo, da je teori, to je zelo komologija teori. Zelo, da je to, da je to, da se nekaj počutim, nekaj ne bo, nekaj ne bo. Zelo komologija teori v logici ne, da zelo, da se nekaj ne bo. ... vse odlišel. Nekaj ne bo se o te, ... z občasnjem z vse, ... in zelo, da je to, ... spremnjen pravič, ... z točenim vse. Zelo, da se se konjuncti, ... in da sem boš nekaj, ... da se se zelo, ... da se se vse, ... in da se se s Bahami, in vsega vsega vsega. In je početno, da je početno. Početnje je model. V ovom različenju jaz sem vsega motivija, kaj je vsega vsega vsega vsega. Zelo, da je objevo, da se pričo se potrebno nekaj zpravimo, da so vsega komologija teorišica s potravim časom. Tak. Počutno. Pobrej, da se teoriška, hrana, komoložija, kaj je v kategoriju skim vsega in kaj je vsega njega se in vesega kategorija, kako se zelo za kategorične spasje. Tako, zelo sem vzelo, da je to zelo vzelo. Tako, kaj je taj motiv? Tako, taj to zelo vzelo vzelo vzelo. Taj motiv. Mi je nekaj effektiv. Taj je nekaj stabilizacij, da ne vzelo. Tako, taj nekaj effektiv. kako je biljon obježene obježene. Vse je to, da je tudi obježene in obježene, kako se prišličuje, kako je vse vsleda. Zdaj je, da imaš 1 milijon. Zdaj, da vidiš, da imaš 1 milijon v parenteziji, ker, včasno, imaš nekomutativne temotivnosti. Zdaj, da sem... Zdaj, da imaš 1 milijon grub in stopov. vse je tvoje tvoje modifje. Vse je vse. In tvoje modifje je konstruktible. Vse je tvoje modifje, da je to vse. Ne znam, da je to vse. Zato, da boš vse na tvoj modifje, da boš tvoje modifje. If you drop up and you put groups, you might have an uncomotative situation. So what do you think, pi1? Yeah, yeah, yeah, this in this case, okay, you're right. I mean it's not, I don't mention here, so yeah. But not committed because you want to include pi1. Yeah, also, you can include pi1, but what I want to say that in the abelian or commutative context, this mode should be a grotendic abelian category, like this. And this should be just in the abelian. Here you don't have, as you are talking about this parallel between abelian categories. Topos is, so you have a non-commutative analog that could be just exact in the sense of bar. Okay, so you have that in general abelian is bar exact plus additive. And if you drop this, and you add maybe some other, like you might get semi-abelian or other kind of categories, but you drop the additivity, you have this notion, which is already quite good. And in fact, this category of groups inside the topos is bar exact. So, okay, so this is not, it's not much. What you really want is a realization function, so this is really the idea, grotendic idea of motives. So, from the point of view of topos, realization functors, which are now associated to t-models in suitable categories, like, as I said, bar exact or abelian or topos, and maybe other. Okay, so that means that if you have this theory and you take a model, which is t-model of this inside this e, this helps realization, and this e should be what I said. For example, if you are in a situation where this is a grotendic abelian category, you want to realize in abelian and grotendic here. So, this is what, or exact, or of course, for constructible, you might just drop the condition of to be grotendic, just abelian category. Okay? Okay, so this is one point, but to be motives, you have, you need motive functors. So, the last point here is that the motive functor in this context should be something that associates the object of your category, motive complex, which is a chain complex in t-motives, whatever they are, okay? First of all, t-motivic complexes, which are just objects of a chain of t-motives. Yeah, yeah, sure. This is a very important point. This should be independent from the realization. This is something which sends an object to this chain complexes. So, this is a t-motivic functor, exactly. So, then you have, because this, you are, I didn't say, but of course, your realization should be exact. In fact, they derive naturally, but I just use the chain complexes here, okay? They just, so be, and you will see in a second why I'm just using chain complexes, okay? So, chain of e, this is again h, and that's essentially all in a sense that if you have this, you have quite a lot, because, in fact, has been noticed recently, as I know by Dagger, that we have a universal homotopy category associated to any category, okay? So, this is a simple shell pre-shifts on c, with a model structure, okay? Which is universal with respect to any other functor to model structure. So, this is a functor, which you see is just induced by Yoneda. And here I have constructed another functor, which is in chain complexes, but chain complexes do have a natural model structure. So, you immediately get a quill and pear, like this. So, then there are variations on this, you can take other kind of pre-shifts. So, enrich the category of pre-shifts. Yes, it's a quill and pear. So, it's a pear of a joint, which is compatible with the model structure. I don't want to, in fact, I'm not using at all this part, but what I want to say is that if you put a c equal to schemes, this is a way to present the model, model Vevovsky structure, which is just a quotient of this. In some sense, these categories, these universal or motopy categories are the free model category. And then when you have a model category, you have a kind of presentations, as you say. So, in some sense, there is a notion of presentation. And for schemes, you get a model Vevovsky structure just by contracting the line and imposing some conditional shift conditions. So, gluing conditions. My avietaris. Sorry? My avietaris. My avietaris, if you want, but this is just shift conditions. Okay, so at the level of, this is a stable motopy category, it's not, okay. So, in some sense, the idea is that these play the role of this one. And then, in some sense, you see that there should be a way to think this chain of theoretical motifs, so this motivic complex. And in a way that present, there are categories of motifs. Because then you can also contract some interval object that you have in c. And impose a condition that you want. So, if you have this factor, you can impose conditions. So, this is the idea. And I will show you that, in fact, for schemes, I can describe this fully. In fact, for schemes, this chain of three motifs maps to both Vevovsky construction and Nordic construction. So, in some sense, is, so, let's say this, but I want to go ahead. So, this is just schemes, get chain of T motifs. So, here of, nori means derived category of the in category of nori. It's not, it's a little bit, but I just put nori and Vevovsky. Just, I'm sloppy, sorry. But I will make precise how to get this factor. No, it depends on the choice of a realization. Sure, sure. Yes, this is why I have this factor. In fact, as I'm showing a key point of the picture, is the proof of Olivia of the construction of nori motifs. Is a really a key point, so I will show you in the next. It's not coming from nowhere, but everything is very simple. It's not, okay, so let's go ahead and start to see what are nori motifs. Okay, so. Before we raise the blackboard, I will just talk to make the remarks. So here, of course, you are requesting this possible motivictopost to be connected and locally connected, but in fact, we can wonder which type of property. Yeah, maybe, yeah, of course, yes, you are right. So, maybe I should say that Olivia is looking for something much stronger, which is, to put, it could be too very atomic, too valued, atomic. So, which would provide, so if it existed, of course we don't know, if it existed, it would immediately yield the fact, which is expected that all classical topology factors have, give the same dimensions for topology spaces. Yeah, yeah. So, this could be a characteristic lifting of this factor, which, of course, is expected, but we absolutely don't know how to prove. So, this is a good point to make because, in fact, you see, this, the construction that I made is independent, is independent, as you will see, it's completely independent from grot in the standard conjecture, which are related to what Loran is saying, because property of realizations, like, I mean, It is not exactly the same thing. It is, for example, a case of that is independence of Ella, but it is not exactly the same thing as... No, no, no. But, yes, if you want to show that in the particular case of what we are going to see now, it is independent from the choice of the theory, you are absolutely right. But, if you want to prove that you take the right category of this kind of tea motives, or even the Voski motives, the realization is conservative or faithful, then this is equivalent by Baylinson to the standard conjecture of grot. So, property of realization, which is very... And here, I'm far from this, okay? So, I'm not saying that this implies this... But what is the guess is that, you will see at the end, that this, the right category of tea motives provide some tea structure. This is my guess, okay? So, which is something that we are looking for on the category of Voski. So, this is... Ah, sorry. For... Let me go a little faster. Okay, so, but I want to prove this... The proof of this is important. So, first, I want to give this. So, this is a theorem, let's say, Nori Karamello, which is the following. We take a diagram and a representation of this diagram. For simplicity, I take a billion groups, but you can make variation on this target. So, this representation is just... This is not a category, it's just a graph, oriented graph. So, associate an object to an object and arrow to an arrow. So, this is a proof of graph, okay? Then, we then have a faithful exact factorization, which I write through a billion, a billion category, CT, universal among the factorization. So, it's actually initial. What I mean? So, that... That is, we have... Sorry. Put here the CT in amber, till the Ft, then you have A, a billion, and S and F, and then there is this one that I want to call this RS, RS, which is making everything commutative, and this means strictly commutative. So, this is really... Even the statement was a little weaker even for a billion group. In fact, there is no statement for a billion group in the situation of Nori. You have a fanatically generated a billion groups. And here we have all a billion groups. Provider F is exact and faithful. Yeah, sure. Thank you. So, what... This is exact, thank you. Exact, faithful. Also, sure, exact and faithful. So, how is the proof going? Yeah, you don't see logic in the statement. This is, I think, really beautiful that you really can prove this in a very elementary, putting together elementary things which are coming from the theory of categorical logic framework. So, proof. So, what do you do? You construct what is called the theory of the model. If you just construct or consider the theory of the model which is... I want to... So, let's denote this t, t or the theory of t. If you want, this is... You have the theory of the model with the signature given by sorts d for each element. So, element of object of d. So, this is just a symbol. It's not... And function symbols d, d1 for arrows plus function symbols for binary operation, unary operation and zero array. In order to formalize the structure of a group, a billion group. And this is for any object. So, this is what you have. Then the theory is just simply is the regular, is enough, theory given by all sequence which are valid in t. Yes, we have a model. So, this... We want to have a theory whose models are like t. So, in fact, we put the axioms which are exactly... So, for example, I don't say that we put axioms like a billion groups because they are a billion groups and these are regular sequences. So, they are true in the model. So, are in the axiom of the theory. We don't need to say that are homomorphism because to be homomorphism is a regular sequence which is true in the model and so is true in the theory. This is the idea. So, now what is CT? CT is the... We have seen variance of this but this is the exact completion of the syntactic category category CT reg. Regular syntactic category. Put reg here. OK, so if you want. Of course, you can take several syntactic categories associated even to irregular theories as Olivia was explaining. Sorry, yes, sure. Thank you. We need these. Yeah. We get rid of this t at least in a moment. OK, so... OK. And the key fact here is the following one. See if I can get this blackboard. Maybe I can use also this. OK. Yes. So, the key fact is the following lem. Is that because you internalize this operation of a billion groups and so on you have that actually t reg is additive and you can easily see what is the zero object which is true in this which is the final object as well which is sentence true. So, this is given by the objects are formulas and so you have a formula true which is the zero object of this category and then this propagate by construction since anyway I don't want to spend time to in this, but this is a key fact. OK. So, after you have this you complete the category and you get an abelian category because what I was saying. So, as a consequence c t is abelian. OK. So, now we have that the t t models of e are by the universal property of the completion are exact factors from c t to e. This is a key fact. So, this is for e exact. I drop bar exact I mean exact. OK. So, this means that if you put e equal to c t we get as usual a model in c t which is our t tilde in the factorization and if we put e equal to abelian groups we get because model of this theorem in a category are from this to this. So, since we have t is a model of this theory particular. OK. So, now just note that I use this now any factor sorry any representation d to a which is a model yields OK. So, this is because of this property. OK. So, a is any abelian category but in fact this also for also for exact I think a abelian category because I want to prove the theorem which is factorization for abelian but this is true even for exact. So, if d s now we want to have a model s out of this condition that I put in the diagram. So, you have s a with f to ab which is a factorization of t but the key point is that f is exact plus faithful faithfulness is very important and then it reflects the validity of all regular sequence so, since you have this composition f is equal to t s so, this implies that s itself is a t model in a and then everything works OK. So, this is so simple and so beautiful and so deep I would say because it gives a pattern OK. What is coming? So, let's apply this to the case of schemes OK. So, there is a step in between. OK. So, let's apply to schemes so, which is the case I am mostly interested but, of course I am sure that applies to many other situations so now let's see what are non-remotives and I denote non-remotives by e h m where e is effective h is homological and m is motifs and I mean, this is notation of non-remotives and these are actually mixed motifs, not pure motifs so, the abelian category that you have is not just semi-simple is really an abelian category with extensions you want to distinguish between the two pure motifs and the mixed motifs Right. The main difference is that you have extensions in mixed motifs OK. So, you want to have abelian categories with extensions, not just semi-simple abelian categories and if you, yes OK. We turn on this point later because you might wonder how do I get pure motifs but there is a way OK. So, let's take c to be schemes and now put the condition to have a subfield of c and of course, as I said, the schemes are are of funny type OK. If you want, you can reduce to a fine scheme. There is no problem to extend everything from a fine scheme OK. So, d is the diagram of nori she is objects are x, y n with y inside x closed and n an integer and you have arrows which are of two types same n whenever you have y to x x prime x prime here commutative and associated to this diagram z, y x y here that I call boundary there is something which is going from the other way which is n to y, z, n minus 1 OK. Responding to the excision exact sequence not the excision but the triple Yes you formalize OK. So, this is the diagram and then what you take as a representation you take in this case you have a very natural homology which is a single homology of spaces because you can, if you have a algebraic variety which is defined over k, you can pull back to c and you can take c points and this is as a usual topology and so you can get the diagram which is t from d to ab x, y, n associate to hn x, c y, c, n and sorry? Relative homology of yes, homology of a pair this is a singular homology of a pair and oops look at you about this here is also sorry yes, thank you I'm trying to go a little faster but let's see so and then of course you see you get c, y, c to hn minus 1, y, c z of c which is the boundary map let's say call it delta n which is associated to the longest sequence of the triple so e h m is c t by definition and here you have t is the if you want regular theory by definition again regular theory of singular homology of course they might maybe finita number of axioms but no problem ok so now this is my stopos which is the following one so of course you know because there is a regular theory you have topology which is regular topology given by regular ap and this stopos e of t is actually equivalent to the sheves of hm this is the abelian category given by this where the topology where here sheves are for the descent topology in fact is the effective descents regular ap, ap are all the same in this covers covers is a regular ap for what are the covers sheves which contains one ap one color one arrow whose image is the identity so sorry regular ap are subjective maps because the factorization in a regular category the image are given is a cover for the abelian it is the same so you take the descent topology in fact I think is not an effective descent even on the regular category but ok let me go back the first one is an abelian category e hm and then the topos and this is just all sheves for this topology which is actually the same of the topology given so this is t t j if you want a reg by definition and this is an equivalence ok we could call it the enveloping topos of the apelian category ok good and inside that we have ab as I said of t t ok and here you have a nice object which is object sorry is a nice sub category which is ind e hm which is also containing of course e hm as you see you have t motifs t motifs and these are constructible t motifs and in fact you of course it's yes these are left exact factors on the abelian group abelian group in the side of the topos are the sheves of abelian groups for this topology they are left exact factors and in the no but you can have non additive the abelian sure sure sure but you can have even if they are abelian they might not be additive so what you are saying is something is a little more fine but you are right if you add the additivity are equal so at the end which is the left exact factor put in this way pre sheves of e hm are here and then you have ab so let me do that abelian pre sheves here you have additive additive abelian pre sheves over e hm and and then and then I think that the intersection of these two is this one this is what you are saying which is the left exact yes so these are left exact I will use this e hm op on ab this is the group of the cardigan category and blow exactly so you have ok so this is the pattern you can remove the conditions on being so you can so yes I don't have so much time but I can try to explain how to remove t the index t everywhere sorry sorry so so you just start so two so t motis but I mean only abelian for the moment so you start with c and m ok so m should be sub category of arrows of c which contains iso ok so for example you can take mono if you like in fact it applies also even to c abelian what I am saying every even delta I get in this way as models of this theory so I consider c square the category that you can imagine is given by objects are squares where these are in m so sorry objects are these and morphism are squares ok and we shall just denote x, y to x prime, y prime such a map so this is the map and also as is there yes I have also this for in this case are my the cross sets are this m and now you take sigma is the same signature essentially where you have signature r h I can denote this which are if you want x, y, n just this is just a symbol ok and the same signature as the same as above of course as above where d is this diagram so I am a bit fast now what are the axioms so the axioms now I have to put because I don't want to use a model I just want to put axioms so I put the minimal axioms and we know at least in algebra geometry I think in algebra homological algebra are good axioms so h, n is a group if you want a billion group 2 that h, n x, y to h, n x prime are homomorphism so the one which is given by h by the boundary h, n minus 1 y, z and these are for n in z I just take all n so I index the index is z so if you want these are just point of a diagram not don't think this is a take situation as Olivia would say so now I also want I don't know how good I still have some time so I so three so of course you want x, y, r on c square and h, n x, y in h, n minus 1 y, z is a natural transformation so these are all these all are equational conditions so up to now we are in Cartesian theory and also the next is of course that you all way for the boundary you can factorize as this diagram this is delta ok so you have I have in the signature these maps h, n, z h, n ah sorry this is y, z inside this this is equal and so here I have h, y and this h, n minus 1 of y, z I have all these maps in the signature and now I can put the condition that this is exact which means that the composition is zero I have zero which is going to zero then there exist some elements so that the only so if you want in this arrow so the only axioms which has been considered by Olivia as well is that if you have consecutive arrows and v, y is zero then there exist an x such that v, x is equal to so this involves the quantifier existential quantifier and so if you put this you get something which is no more equation ok so so now you get this theory which and you do the same pattern exactly the same pattern as above and you get an abelian category which is the abelian category constructible t motifs ok so I don't have to explain how to get the factor but this has less axioms than above and the same proof gives that this is additive so implies that so e t is abelian so what is the model for this theory is just a family of factors on c which satisfy these axioms so you can see that if you start for example for c to be an abelian category and you take monos you have delta factors as models but if you consider the pattern of singular homology I give you some models in abelian category so homology of course is a model but I have a slightly refined model which is suzljimbevosti singular model ok so this is you take c equal to schemes ok and you take a to be preships over correspondences and here I have to put additive preships sometimes this is preships with transfer and this is abelian even a growth in the category ok and you have for x a scheme a representable preship which is just I don't have these correspondences are just multifanctions and are given on smooth schemes so I don't have time to explain what they are you can define this ok so and you have s of x which is c star of l of x which is suzljimbevosti singular chain complex right so this is a functor and in fact is an exact functor on this category so you have if you have a close s y inside this so y close inside this you have this so this implies that you can define the relative one and if you take homology of this gives h i y in this category a which are model of our theory what does it mean it means that we have a functor from I think I am done I just end up with this and then you might ask questions if we have time so what I want to say is that yes I just want to say that there is a functor to this one because you have a model in this category a which is actually in the complete so you have in in a and I don't have time to explain but a very simple lemma gives a condition I mean this property is not a pair but a triple so when you have I would say not x, y, z but x, y, z and w you have two boundaries and a small magic as you would say give the composition is zero so you construct a spectral sequence in the category 80 and then you can take total complexes of these spectral sequences and you get a natural very natural functor from schemes to in the category of 80 which lift the construction of nori so when I put also nori also nori is a model of this theory because nori you remember you have a t inside ab but you have a lift of this so also for nori you have this t tilde from c schemes over k2 eHm gives h tilde I x, c, y, c which are models are also models of t so in the sense of Olivia this t is the theory of t is bigger so I add axioms of course these are regular axioms so they are satisfied by the nori's singular model trivially satisfied and then you have many more axioms so you have this and this means that this one maps to this one and then you go to ind again and you get a factor on ind and in the construction of nori you really need ind to get a factor it's not just a joke that you go to ind you really need ind it's not that you cannot avoid this in and sorry for a little bit just about half of what I was saying I was planning to say we take few minutes for questions so here you have constructed this filter so do you expect this filter to be an equivalence or not you are sure it is not yeah I'm sure it's not an equivalence but I think the objects are the same because they are given by the formulas right so and when you put more axioms you put relations so what I expect this is a sir quotient in fact what I expect is that d of ind, the conjecture that I have if you want some expectation is that when you do this derived category of ind and you map to the derived category of ind of ahm what I expect is that this is the localization of this so where you add exactly a1 or motopin variance and my inventory is that you don't have level of axioms and similarly when you go to so you expect that it could be enough to localize this I strongly expect that this process stabilize as soon as you impose these two conditions this is of course because in some sense if you have the functor c to this then you have a model of your theory inside ind and you can iterate this because if you take homology here you get a new model inside ind and then you can iterate but you can iterate because I don't have any assumption on c as soon as soon as I have an interval object and shift conditions in some sense these are enough to put all relations which stabilize this iteration this is the idea but of course it's just you know but what is incredible for me is that we go the other way because in some sense we have a functor from non-remotives to we have a functor from DM geometric Bevowski motifs to the derived category of EHM and the big conjecture is that this functor is fully faithful and then you see because you have a faithful realization here you should have here and this gives a T structure on this triangulate category and what I have is something in here which is a presentation of both they share in some sense the same structure if I can embed both inside and it seems that the only thing that I miss is these two conditions in both cases and what we know for the last is why I strongly believe that something like this should be true is that because inside this one we have we understand very well what is the so called category of triangulate category of one motifs which is equivalent to the derived category of in the one motifs which in turns is equivalent to the derived category of EHM one ja so in some sense in here I am going up on the picture and on the bottom this is going to to be so this is just a variation this is really a variation of grotendic that pure motifs for curves are Jacobians it is just this so if you want you can see the category of semi-simple category of a billion varieties as a model of the theory of motifs for each one of the smooth projective curves are inside this and this is a mixed situation but still they behave nicely inside this so the real problem is that in some sense the T structure on this category at this level is well known I mean it is now well known and in fact there is some work here sorry I have to prove to prove this theorem so these are not just trivial theorems I have to add iub we have proven this with iub and kan ok so but at this level is is really something that you don't know what to put but this is a natural candidate ok