 So, the title is Motives from a commutative point of view. However, as a reminder, kind of take away from the last two talks, is that it's a reminder. I have introduced, so you have some fix some, some base commutative ring K. We have A, which is K algebra, associative unit will not necessarily commutative M is an A by module. And then I define the invariant, which for lack of a better word, I called with K theory of A M, which is roughly speaking, I take the tensor algebra of M generated by M over A, I take its K theory, I complete it with respect to the natural iteration by degree, and I take out the constant charge. And this is a very nice invariant. First of all, it's the structure of a trace theory, which I discussed in my last talk, which allows you to do lots of computations with it. In particular, the study of this for general A and M reduces to the situation where A is just K, just by general trace theory formula. And then if specifically K is just a perfect field of characteristic P, then the answer for this is very surprisingly easy, namely that this K theory of K and degree I actually vanishes unless I is not one, so it's only exists in one degree, which allows you to construct all sorts of explicit models so this becomes generally very accessible invariant. So today I want to discuss cyclotonic structure on this guy, but before I do that, there is something else I want to explain. So this is a very useful invariant, but this is kind of not the most fundamental. So there is another invariant, which gives rise to this guy and some other invariants, which was kind of the original idea behind whole business, rather than step. And this invariant is devoted to the notion of a cyclic node. So I had this for two categories already when I introduced this theory, but just for ordinary categories. So let me remind you of this category lambda, comma, if we consider the fiber category over the object of one, so this is objects and equipped to the map to one. And it's actually equivalent to delta, the category of finite and empty ordinals. And this is clear geometrically, so objects and lambda are both categories of quivers, right? I mean, there are different definitions, but the one I gave you works like this, but there are some quivers. And when you have a map to one, this means that one edge of the quiver becomes distinguished. There's one edge, which is now distinguished, this one. And you can just erase it. And if you erase it, you end up with basically, you know, so it's a quiver now, which is not a wheel, but just a string. And so the corresponding category, just a partial or totally other set of some elements, as this gives you the equivalence. So now if we have some small category I, and we have some functor from I opposite, that is I to sets, then we can define a simple set, which is known as the, no, I'm sorry, so definition, cyclic nerve. So I had this already last time, but without coefficients. Now I need two coefficients, so cyclic nerve of I with coefficients in F. It's a simple set, just a functor from delta opposite to sets. And this interpret as now opposite to this fiber category. So let me describe you what the values are. So if I have some object here, which is, yeah, I think of as my wheel quiver, then for the usual cyclic nerve, the values would be the set of configurations like this. So I have an object, each vertex, one, two and so on, I, N, N, and then there are maps, F0, F1, and so on, and then, so F, L is actually a map from L to I, L plus one, for L from zero, and then the last map. So for the usual cyclic nerve, it's again just a map of the category, but here I'm modified. Here FN actually is a map from, it's actually an element in F applied to I, N. I see, all right. So this does not have a cyclic symmetry, I broke in the cyclic symmetry by choosing one edge, but model of that, it's okay, so it gives me a superficial set. So there's a question, how does a map from two, one gives a distinguished error? Okay. Or is this a map from one? No, it's a map from one that would give me a distinguished vertex, just for this vertex. The point is that the category lambda is actually self-dual. Kind of the fanciest way to see it in terms of this functional description which I had, I mean, I described maps as functions between some categories. Well, those functions have adjoins, and you just take the adjoin. But also, if you think geometrically, you think of realization of a quiver as some kind of, you know, cellular decomposition of the circle, so there's a map from a circle to a circle, but then if you take an edge in the target and you take its pre-image, then it lies inside a single edge upstairs. So there is some kind of sort of a factor which associates to my object. The set of edges of the decomposition is actually a contravariant, and this gives me the distinguished edge. Perhaps I should draw a picture. It's not so easy to draw in this zoom thing, right? So the target is just one. Here I have some points, and then if you look at what the map actually does, you see that it has to contract all the edges, except for one to this point. So all this must be contracted. And there is exactly a single edge which is not contracted, and that's the one that I choose. I hope that's clear. Yeah. Okay, thanks. Great. So there is the cyclic nerve, and then you can consider this geometric realization with some kind of topological space, spaces. And now specifically for A, so you have P of A, which for me was the category of finally generated projective A models. And then the bi-module M defines a function of P of M from P of A opposite that's P of A to, well, to carry out the space, but at least for the sets, right? It's just a very naive one. So because P dash times P goes to home from P dash, and then M times A. And so I can consider the cyclic nerve of this category P of A of the efficiency of this function. And this can be called cyclic theory space. Perhaps I shouldn't, oh, it's a space, right? So dot would make the most of the drops. So this realization of the cyclic nerve P of A. And historically this I think is what gave rise to whole. So this was considering this guy was suggested by Goodwill, you know, somewhat famous, but unpublished letter. Waldhausen, Waldhausen, I'm about 80, 80. The point is, I mean, this is just a space, but it also has a new loop space structure if you just take direct sum of everything. So it's actually spectrum. And it's somehow better than say with K theory, which I considered because it's reasonably small. So for example, if K is a finite field, then this is just countable, right? So it should be something which is computable, but in fact it's not. So a problem, at least it's very hard to compute. There is not enough structure. For example, we don't know as far as I understand, even now we don't know what it gives for just, you know, the point, even the case of finite field. So in order to get some handle on this, we need to do something to use it to construct some other invariance, which are more amenable to computation. And in practice, what you do, you need some kind of completion. And this can be done in different ways, in fact. So for example, there is a theorem, which I believe is due to Lyndon Stowes and McCarthy, which says that if you take this, so this is a factor of two variables, you treat it as a factor of respect to variable M, this additive variable. So you can do what is known as goodwill completion, completion in the sense of goodwill calculus. Well, for people who have more algebraic like me, this just means something like proper polynomial completion with respect to M. So some factors are additive. If functions are not additive, you can write down the correction to this cross effect. This is a factor of two variables that can again be edited for not in each variable and so on. So there is a whole theory of polynomial functions developed originally by both for WDU. And you can sort of try to compute the best approximation polynomial expression to a function. And if you do it for cyclic theory, you get something which appears also in the theory of the cyclotonic phase. It's a spectrum called TR of A, efficient in M. And already this behaves nicely. So already for TR, again, if you just plug in client fields, so TR of K. Again, I can consider any M, but if I consider, if I write just TR of A, this means diagonal by model. So this again behaves as my with K theory. So this is zero unless I zero if K is finite, say perfect, positive characteristic. So this already can compute. And of course, if you look in the... There is a question, Dimitri, there is a question, the good willy completion is just the limit of a good willy tower. Yes, yes, yes. I mean, personally, I always think about the polynomial completion because it's easier to visualize algebraically. All right, and by the way, the topological Hohschild's homology appears as the first term of the towers, actually just the additive, if you want the de-tivization of this cyclic. It's basically the linear part, linear respect to M. Another thing which you can do, you can look at these guys, if you're all fine and filled, you can try to look at them as an algebraic varieties. I mean, the cyclic nerve is a simplicial set, but you can put some kind of algebraic structure on that and look at it as some kind of simplicial affine scheme and use that structure. This would also give you some completion, which is more or less the same. This has not been written down, I believe, but because, I mean, we can do it more easily by good willy completion, but it can be done, gives you the same answer. And then also there is something which should be Ethereum, but I think there is no direct reference in the literature. So I put quotes here, is that my with K theory is actually also the same. So this polynomial completion also gives me why with K theory, but up to a shift. So let me think about not using groups, but say spaces, spectrum, this will be, so I need to shift by one. So there is this loop thing, all this WK, K. This is not an literature, unfortunately, we discussed it with Thomas Nichols at some point. So I mean, you can cook up a proof if you just assemble known results, but it's probably better to do it in some kind of conceptual way, which as yet is not done. So there's no radio reference, but this is nevertheless. So kind of the take away is that the fundamental thing is the cyclic nerve, which generates all the other environments. Cyclic nerve itself is hard to compute, but then you apply also various sorts of approximation to it, and this gives you computable invariance, which can then compare. Okay, so this is what I wanted to say about cyclic Ethereum. Now let me discuss cyclic Ethereum structures. So I need to discuss first equivalent aspect and so on. So let me start with the finite group. Now, first unstable. So as I mentioned last time, actually when you think about J-covariant spaces, there are two types of homotopy category you can consider. So one is the naive homotopy category, only the crude homotopy category or G-spaces. So object space is the action of G, and then we just look at maps, which are the covariant maps, which happen to be with equivalences, homotopy equivalences without regard to G, and invert them. And this gives you, so in modern language, so if I denote by point with the index G, just you know the group point, which is the quotient of the point by G. So, you know, one object with group bottom of this is G. Then this is the homotopy category of functions from this guy to spaces. That's the naive type. But then we can do more refined thing. You can again consider J-covariant spaces, and you can see the maps between them and classes of those maps up to J-covariant homotopies. And it's important to realize that this gives you different action, much larger category, which can also be defined like this. So, sort of genuine G space. This would be again a homotopy category, but now what you do, you consider the category of G orbits, or G is the category. So, a G orbit is a finite G set where G act randomly. So, it's a quotient by some subgroup, and morphisms are mapped between those, which are G-covariant. So, it's a category with, you know, some objects. I mean, since G is finite, there is a finite number of objects that correspond to subgroup. Well, isomorphism classes correspond to conjugacy class of subgroups. But still it's a category of non-verbal maps, so it's an interesting thing. There is, of course, a forgetful factor, because if you look at just the orbit where G acts on itself, sort of, so quotient by the trivial subgroup, then, automorphism, so this is just G itself. This gives you an embedding from this point model of G to orbit. So, there is a restriction of function. If you have a genuine G space, you can forget the rest of the structure, and then consider just naive, but there is something else. So, this category on the left is bigger. Alternatively, if you don't like, so orbit is something where G acts transitively. You can actually drop that condition. And we consider the category of all finite G sets. So, sets where G acts in some way, not necessarily transitively, maps are J-quarant maps. And then say that the function on gamma G to spaces is additive if it sends disjoint unions to products. So, there is always a map, let's say, contravariant. Let's hope it takes me. The fact, I'm sorry, here it's also supposed to be contravariant, yeah. Because what you associate to an orbit, if you have a space with G action, is just the space of fixed points with respect to the respect of some group. So, it's contravariant. Then for every function, there is a map like this, and I want this to be a homo-tube equivalence. So, this is my notion of additive. And then, this genuine category of G spaces, from orbit to logical spaces, can be also interpreted as, I consider factors from gamma G now, logical space and vertical equivalence, but I consider only those guys which are additive. There's a full subcategory spent by it. Okay, this was like space level unstable. Now, the slogan is that for spectra, what you do, you add, you do just one thing, you add transfer maps. And the way to do it, I mean, one way to do it, but the one which I think is the world's best is by modifying the category gamma G. So, you can see there a category, which I will denote Q gamma G, it has the same objects. So, Q gamma G, same objects, sorry, just same. Objects are again, just find it, just sets. But morphisms are now correspondences of zero as one, and morphisms are correspondences like this. Now, you could just take the isomorphism classes of those, but this would not be the good thing to do because it's too crude that destroys symmetry. So, you really should think of this Q gamma G, not as a category, but as a two category. So, here are objects, morphisms are like this, and two morphisms are isomorphism between these guys. So, for every two sets, the category of morphisms is actually the group of diagrams like this. So, this is what you do. And then, if you want now to define JQ variant spectra, then this is just a category, you notice like this, which is spectra, this is just a category, so it's a category of this Q gamma G. So, I need to consider stable things. So, this is a function from Q gamma G to spectra, not spaces, and they have to be edited. In fact, if you do it like this, you don't even need to consider, you can also put spaces here, and see the functions to spaces. And this would automatically give you a spectrum because of some version of the single machine. So, all J spectra, which are connective and appropriate sense actually correspond to spaces. So, there is a fully faithful embedding. None of this is in literature unfortunately, as far as I know, although it has been around for, I don't know, 15 years, everybody knows this, but there are no ready references, and that's part of the problem why we can't really prove some comparison theorems, which we would like to publish that. And part of the reason for that is technical because first of all, this is a two category. So, this has to be really done some kind of a categorical setup to make it work. And that's, I mean, that requires some writing, right? So, people are kind of too lazy to do it. But nevertheless, it's all true. So, this is how things are. And in particular, there is a version of the single machine for the G spaces, which was introduced by Shimaka a long time ago, which boxes Shannon Watson used. But that unfortunately is not strong enough. So, it doesn't give you enough control, it doesn't give you what I write here. So, one actually needs to strengthen that, and that's something which is still a bit of a loss. Dimitri, there was a question by Shan Tilson. So, he says he asks, so where aren't stable splitings equivalently? Stable what? Splitings? Splitings. Where? Not sure. Shan, can you be more precise in the inclusion from G spaces to G spectra? No, no, no. This is a fully faithful embedding. And, I mean, downstairs it's not G spaces anymore, because I added Q, right? It's a function from, I mean, spaces would be if I just put gamma here. Okay. And this is Q. So, if you want, so there is a function from G spaces to G spectra, which is the suspension spectrum, of course. But this corresponds to some kind of induction from gamma G to Q gamma G. But if you have a function from Q gamma G, but to spaces, then this is already fully faithful, so you don't need to. In fact, if you don't have G, you just take Q gamma, then this very close, I mean, this contains this category of pointed finite sets. And that is exactly what goes into the single machine. Okay. Yeah. All right. So, this is how it goes. Okay. So, how do you see all those fixed point functions in this language? So, for G spectra, there are two types of fixed points. There's the stocks, for example. So, there is the geometric fixed points and there is something which didn't have a name and then they called it categorical fixed points and now they probably call it genuine fixed points. So, it changes. So, one of them is very easy. One is just a valuation. So, those genuine fixed points, how does it go? So, let me denote this by psi. So, for any subgroup H, you have a fixed point function, psi H. And in terms of this function from Q gamma, this is just a valuation at the general page in fancy. And for geometric fixed points, so what you do, I first define something called inflation. The term inflation comes from the theory of makey functions. And by the way, the makey function is the same theory except target of your function are not spectra, but complexes. And then get the notion of derived makey function. So, for makey function, I actually wrote the paper about this like 10 years ago. It was not related to you. But anyway, so inflation. So, if you have some subgroup instead of G, you can consider it's normalizer, N of H. So, it's normalizer, and then you consider the quotient. It's somewhat similar to while group and group period. And then you observe that for any H, or for any H, we have a factor which does the following. So, we take, I notice by small phi H. So, we take Q gamma, and this goes to Q, U gamma G, and this goes to Q gamma W of H. So, you take a set, and you just send it to the fixed points. So, if you look at the set of fixed points, then the normalizer acts on this set. But H actually acts trivial, so the action actually factors through an action of W of H. So, as H is naturally W of H set, so that's this factor. And then you prove a lemma. So, you have the factor, and then you can consider pullback. Factor commutes with disjoint unions. So, it's an additive guys to additive guys, and then, so you get pullback from, well, W of H spectra to J spectra. This is called inflation. And then lemma is that this is actually fully faithful. This is something you have to check. I can't really comment on this. This isn't, I mean, this is not that difficult once you realize it's true, but it's kind of surprising. If you have to prove it, it's not obvious, but they won't do it. And once you do that, so once you do that, you can consider it's adjoint. So, it has an adjoint, they left adjoint, I place a hat from G W H spectra, and then the geometric fixed point function is exactly this composite, just forgetful. This evaluation at W of H, it just goes to spectra. So, geometry fixed points of some X are obtained by applying this guy, and then evaluating at the trivial orbit. I mean, not trivial, but the biggest orbit. All right, so this is how we see all those guys. And the general picture is as follows. So this category of G spectra, it's actually a triangulated category, well, stable category if you want. It has some kind of semi-orthogonal decomposition. Maybe I should write this, because this I think important to keep in mind. In fact, it's already in the original paper of May Lewis and other people where it was invented, but the longest error was different. So it's not stated cleanly in these terms. So it took me some time to extract it from there. So let me share this knowledge with you. So in general, what happens is that this guy has a semi-orthogonal decomposition. It has a filtration decomposition. It's glued out of pieces. Pieces are numbered by conjugacy classes of subgroups, or if you want isomorphism class of objects in O of G. And the pieces are just very naive things. So these are spectra with the action of W of H. It's like, if you want, let me note like this, the homotopic category of factors from some guy to spectra, as T here means stable. So this is the most naive version of the covariance table category which you can consider just spectra with action of a group. And the genuine thing is glued out of those guys. And the gluing factors, gluing data, are some sort of generalized state, homology, respect the families of some generalized state. So that's the basic picture. The category is huge, but it has those factors corresponding to subgroups which are easy. And then there's some gluing, but the gluing is not that. I mean, for example, if you restrict your attention to things which have fine homological dimension in some sense, the whole state of homology would disappear and the pieces would just be earthen. Okay, so this is the general setup for a covariance table homotopic. Now, this was finite groups. And traditionally the level of generality in topological sources, I mean, people who invented it, would be for more generally comparative groups. And one can do the version for compact league groups. Of course, now those categories have to be topologized. So the home sets would actually be some kind of homotopic types also. But the point is that we don't need it for our applications. This is maybe strange, because you would think that the circle would come into play, right? But in fact, if you look closely, nowhere in this psychotomic business, you have a genuinely a covariance spectrum with respect to the circle. Because, as I said, the G equivalent category has those sectors corresponding to subgroups. So for the circle, the subgroups are cyclic subgroups and then the circle itself. But the latter never enters the picture. So people only consider the part which corresponds to cyclic subgroups. So, effectively, it's only genuinely equivalent with respect to all those cyclic subgroups on the circle. That's all you need. You don't need the subgroups. Conversely, as I will argue in a moment, what you do need is to arrange the same cyclic subgroups in the other direction. Instead of taking kind of the embeddings from one to the other, you should take the projections. And not the kind of inverse limit, which is the group of cross-review, but conversely the projective limit, which is the profine completion of Z. So instead, let us... So instead, what we need to use is actually a version for finely generated, but profine it. Profine it. For example, Z. And then the option is that it works exactly the same as before with one modification. So where am I? So now if G is a profine, then you can... It's absolutely the same, but you slightly change the notion of finite set. So instead of finite G set, we now consider admissible G set. And this is two conditions. So first of all, for any point, the stabilizer of the point is finite index or finite. So it's a union of finite orbits, but maybe infinite union. But however, for any fixed co-finance subgroup, the fixed point set is finite. So it's a union of orbits or finite orbits, but each finite orbit can appear on a finite number of times. So they can grow, but then the size of the orbit has to move. And so this is the only modification. So this gives you some kind of category gamma and G hat. And the rest of the story is exactly the same. So they give you some kind of notion. I would respect them now for a profine. Okay. Now, what has it got to do with the next part? And now I can answer the question which I was asked actually at the very end of the last talk. So I said that it deserves a good answer. And this is the answer. So the question is about my definition of the category lump, which I defined categorically. Maps were some functors, but I only allowed one, which had some degree, but they only allowed functors of degree one. The question is, what if I allow all functors of degree, which is not one. And the answer is that this actually fits very nicely with this perspective point of view. So definition. So Dimitri, there are questions about the finite index. So by Remy Ponderben-De Bruin, he asks, when you say finite index, do you mean open? You are very continuity condition and objection. Yes. I mean, I'm working with profine groups with the profine topology finitely generated. And then open and profine is probably the same. Finitely generated, profine group, I mean profine completely. So it's profine topology, so open means profine. I mean, the example I actually need is on the Z-hat, but I think the theory works for all finite indexes. Okay, so definition. Cyclotomic category lump R. Objects are still the same as lambda. So these are just n. But now maps are another condition is that the degree is not zero. So there are some functions which just end everything to an object and all the maps to the identity, which are not interesting, sort of degenerate. One thing which are not degenerate in this sense. Then I allow all the degree. Now, an example of something, so if degree is one, then this is our lambda. Let's say that the map is horizontal. It is degree is one and vertical. So what's an example of a map of degree which is not one. The basic one is like one. We take some nL and this has, so this is a circle, so this has an anatomorphism of order L. And we just take the quotient. You can take the quotient and the level of the quiver. In fact, even for categories will be the quotient. So there is a quotient map. And I want to say that the function is vertical if it is isomorphic to this kind of quotient map. There is also a categorical notion of discrete cofibration. If you know what that is, then this is the quotient. And then this horizontal and vertical maps form what is known as a factorization system of lambda r in the sense of both fields, which means some things in particular. It means that every map in the cyclotomic category factors uniquely. So every map factors unique isomorphism as, you know, there is first of all some kind of horizontal map and then some kind of vertical map. Horizontal this is vertical. And conversely, if you have a diagram like this, we have a vertical map and a horizontal map. Then in this category you can form actually a Cartesian square. So there is a pullback square. This H m will be horizontal and this will be vertical. Okay. Now, if you look at this lambda r and you only consider horizontal maps and this I will denote the index of lambda r on the horizontal map and this is of course just lambda. But if you look at the vertical map and this the category of orbits for the group Z which is the same thing. I mean orbits are finite by definition. So this is the same as orbits for its profiling completion. And so lambda r combines both lambda and orbits for this group. And so in order to define the cyclotomic things and this would be kind of the clean definition of the cyclotomic spectra which is I think the most conceptually clean one. You just take this lambda r and you repeat the procedure you did for the orbits. So you first of all you need to add this joint unions. Somebody should mute it I think. Somebody mute somebody. Okay, thanks. Now add this joint unions and in fact finite but also infinite but sort of admissible in the sense which I had before. So it has formal unions you know. It has to be done. It has to Vanya. Vanya, microphone, switch it please. Sorry. Thanks. You know how to do this, right? You go to participant list and then mute somebody. You can do it. So you can see the things like that. Admissible means that for every n i number of n here. So this gives me some category which I need some notation for so let me call it what I don't know. What's in my nose lambda r gamma lambda r gamma. And then you do this Q thing. Well, for orbits you can do just correspondences and when I said correspondences I didn't tell you how the compositions are defined. Compositions are of course defined by just taking pullbacks and that's why it's not a good idea to restrict yourself to orbits because if you take two orbits and take the product it will be a g set but it can now split into several orbits. So you get the joint union sort of way in the system if you want to have compositions. And so in this case this way I had to add these joint unions here too and then what you now define now consider Q lambda r gamma, right? So objects are as in lambda r gamma and so maps are correspondences of the following types. So there's some kind of generalized L. I mean this N is actually a disjoint union of some kind, right? There is some M dot and then there is a diagram like this dot. So the only difference is that here I can allow on the right I can allow any maps whereas on the left I only allow vertical maps. And formally I need to do this because my pullbacks only exist for vertical maps and in lambda in this category lambda r gamma once I added the disjoint union vertical maps admit all pullbacks with respect to both horizontal and vertical maps. So this will define category, well two category maybe yeah. Now the definition definition cyclotomic. I put quotes here because it's not the definition of the literature and the fact that it's the same is not there is no published it's what I want to be a definition an additive this guy and why I think that this is a good definition because you get everything basically for free now once you have set this up in this way. First of all this is obviously cyclotomic spectrum this definition I mean I explained why this is a cyclic object just by considering if you have a quiver you consider a tensor algebra replace the tensor algebra with the path algebra of the quiver consider k theory for that and this is obviously functional and exactly the same thing goes for cyclotomic structure. And if you have a vertical map then there is a certain function which has an adjoint and both induce maps on k theory this is really you get for free so one, two I also had this p-typical k theory and there is also a notion of a p-typical everything which starts with actually lambda r where I only allow maps such that the degree of the map is the power of fifth prime p and then I can repeat the procedure consider adjoint unions consider they take the quotient k, there is q category and so on so there is this category of cyclotomic spectrum we can denote this place and there is a category of p-typical cyclotomic spectrum but then what's funny is that there is you can construct some kind of inflation function and it becomes actually an equivalence once you invert p no, not p conversely, once you localize it p like this actually let me write this in words equivalence p locally so I had to split this with k theory into pieces to obtain the p-typical one but once you do the cyclotomic story then this becomes kind of automatic just the same k-type and then you can also describe this can be described in terms of as I said this filtration also can be described in terms of sectors in terms of semi orthogonal decomposition and gluing day in fact in this case there will be only one sector this will be just the circle so this would be spectral in action of a circle and as I said this generalized state which enters the picture often vanishes in this case in fact if you are p-local the relevant vanishes in all cases except for the psychic group of order p so p-square, p-cube and so on all these guys don't contribute so the only thing you get is actually order p and this gives you this Nikolaos short description this works like magic just because you know we are lucky and this semi orthogonal decomposition is almost orthogonal there is only one possible gluing and there are no conditions on the gluing either so this is some kind of free free algebra situation so this suggests that this is a very easy and direct description of psychotomic spectrum and this is exactly what Nikolaos and Schultz did for computations this is of course what you want to use and this is what people use nowadays with which success but for conceptual reasons I think the correct definition is the one that I gave and so these are some motivations and the kind of the main motivation for looking at this picture for me is that it also works with coefficients so kind of the whole gist of my lectures was probably that you should if you can you should generalize your things to non-commutative setting and allow coefficients because then at some point you can reduce the linear output so it's crucial to work with coefficients so with coefficients coefficients so WK of KM is not a psychotomic spectrum anymore we lose the cyclic symmetry but it's still a genuine spectrum aspect to particularly have those fifth points and then there's this KHH and this actually appears just as a geometric fixed point with respect to the whole group and as a reminder the guy itself should be a model of the written proof yet but it should be the same as TR how am I doing this time okay my time is up but let me maybe yeah, yeah practically done so to recapitulate so this W K as I said should be the same as TR historically TR was defined in terms of THH but actually the relationship is symmetric in fact so TR THH in fact TR is maybe more fundamental so this can be recovered as fixed points TR and then historically it's the other way around TR was obtained as inverse limit of categorical fixed points from THH but the difference is that what I have here you can do with coefficients and then it's much easier to prove because you reduce to A equal to K so without coefficients of course TR of A as we saw is just this inverse limit so without coefficients THH has this some kind of residual it has some kind of non-complete S1 equivalent spectrum structure meaning that it's a spectrum of S1 action equivalent to respect to all the circuit subgroups and then have this so this stands for maybe it's better to do it in my notation my notation was like this so in this sense the two things define each other but I think the picture with THH is kind of more relevant for I mean it's closer to both things on nature I think this is it, I think this is all I wanted to tell you and thank you for your attention in these three lectures and it was a pleasure talking to you so thanks a lot for the talk so are there questions about okay so maybe I have a first question so you said on WK AM with coefficient there's no cyclotomy structures so can you explain why? because it's a cyclic symmetry when you choose M okay but if is there a choice of M where you can see if you take the diagonal by and this gives you the the point is that even if you break the cyclic symmetry so you break the S1 action you still get the Z-hat structure which I think is important to use okay thanks so any other questions okay so I think there's no more questions so thanks again for your nice and rich series of talk Calais Dimitri and okay we meet in half an hour so thanks