 Right so let me start with a brief reminder of what I mean I wouldn't remind everything with that well done yesterday but there is one thing that you need and so the setup is as follows. I have some key, in most cases it will be perfect to fill the first characteristic but and some key something that can be just commutative measuring that you measuring you have some a so last time it was community but in fact it doesn't even have to be that say associative unital what k algebra of course if we're over a field it's automatically flat and now I define this environment actually I didn't introduce a name for this let me do it now so I consider algebraic k theory of power series in one form of variable t of coefficients in a I completed so this is actually the inverse limit of k theories of truncating guys and then I notice that it splits as k theory of a plus something else which is of interest for me so let me call this something else with k theory it's not standard name but I mean I need something right so why not this double a k of a and then there was this observation this is what really curtain assessed for the lecture so if k is p local this guy carries those endomorphisms silent n which are almost idempotent if you if k is p local and n is a divisible and n is prime to p then you can invert n and then you get an honest idempotent and then the whole thing splits into a product of copies was smaller guy called so this is kind of big with k theory and this is typical with k theory this is just the kernel of all of all those other and this what is our interest to me this is what I want to study so I said that uh in situation one case perfect field and ace community can smooth this gives you the there are weak forms and particularly it has a differential so uh whereas the differential differential comes from a circle action this was the end of last so how does this uh go let me explain this now so I mean there are various ways to do it uh the one I prefer actually involves a little bit of category theory but I think this is in the end the most conceptual clear one so this is what I'm going to present uh partially it already appeared in talk by Tina on Monday in the form of a so-called cyclic object but let me be slightly more precise about once a reminder because most people heard about it but still uh so uh just a simple thing take some n and let me denote by n brackets and long the nose there's uh it's a category but it's a very small category so I take uh a wheel we were with n vertices and this is a wheel we were and this is the path category of this we were so there are n objects and morphisms are just you know paths it's of course I mean there is a orientation and the only way the only invariant for path is actually its length you just go around the quiver for as long as you want so basically alternatively you can think that objects are just residues mod n and then maps from a to a prime are just integers such that a plus l so l is this length it's least zero and l is a map from a to a prime such that a plus l is a prime model of n of course for every guy we have an endomorphism which which is a path which goes around the the loop around the wheel exactly once so there is the stole a from a a response to f so it's a path which goes around the loop exactly now there I mean it's a it's a category but of course category means something huge and this one is very small but still the same kind of there are objects morphisms so okay now if you have two guys like this and you have some funk from m you observe the following you take some a object you have this uh endomorphism to a and then you so a function sends a map to a map right so this will be some endomorphism some endomorphism of f of a so it has to be some power of and that guy actually generates the story generates the endomorphism monoid so this guy raised to some power which is a non-negative integer i don't know by degree of f and in these observations that this degree does not depend on a so it's the same for all these does not depend it's an another one it can be zero for example if we just send everything to a single object and all the maps to the identity then this will be zero and I wanted to be actually non-zero and for now I wanted to be exactly one so definition there's a small category denoted by lambda it's the category so objects are just you know numbered by non-negative integers we denote a tradition like this can and brackets so there's some discrepancy in literature where they start numbering from zero or from one and there are valid reasons for both unfortunately so i'm going to start with one so this is just the number of vertices in the keyword and then uh those are the objects and then maps morphisms are just you know functors from some n some n prime of degree one this is a very very well known gadget it was invented by lmcon like almost 40 years ago and it has different definitions but for me this one is now if you look at the category you can look at uh it's nerve it's an official set you can take its classifying space and then this classifying space which I denote like this is actually so it's not uh it's simply connected but not contractible it's a cp infinity or equivalent to the classifying space of the circle where circle is the group consider the group the usual group structure or it won't be you want the same thing right and uh this means the following so if you have so if you have some function from lambda to some category e which is locally constant in the sense that it inverts all maps so every map and lambda goes to something in the target category and then since the thing is simply connected locally constant means constant however they can do something slightly uh more refined you can now take as our target the category of spaces in whatever sense you want so it can be there are everything which models homotopy types it could be biological spaces or completion sets it really doesn't matter then if you you know i'll denote by co of lambda the homotopy category of functions like this so you take functions and you invert point wise uh weak equivalences you need some technology for that but that's standard and then if x is locally constant in the sense that uh this means that any map goes to something which is not directly an isomorphism but a weak equivalence a weak equivalence is a weak equivalence for all f so you have a full subcategory here spanned by these guides i denote this by lc that's for locally constant then this uh locally constant then is equivalent to the category of spaces uh with s1 action we'll see the biological spaces for to be precise and then so you take s1 space equip the continuous s1 action and you take the homotopy category so Tina mentioned this and so this is a precise one of the ways you can make the precise statement that cyclic objects correspond to things within the circle action here is a fine point which i want to mention explicitly so this thing here can actually mean two different things and in my experience to topologists it usually means one thing and two people from all algebraic representation means the other so you consider spaces with s1 action you consider the homotopy category but the question is what kind of homotopies do you allow do you insist that homotopies are also a covariant or do you or do you just invert all maps which are homotopy equivalents without regards to s1 so the latter is actually a much smaller category in the form because if you also allow homotopies which are s1 aquarium or only those homotopies then for example the space of fixed points with respect to some subgroup becomes a homotopy invariant node so this kind of refined homotopy category contains a lot of information whereas the latter thing is actually very very stupid well i mean it's a much smaller category for example the classifying space of a point would be something contractual in s1 action would be just homotopy equivalent to the point in the latter category but not in the form so this here the small one so small one so invert homotopy is not necessarily s1 and then so this s1 action but if your target is not just a space but something linear like a spectrum for example or you can also consider a homological version where your target is a complex of a chain complex or some ring then so if x goes from one to the spectra then you can actually split this s1 action into two parts so this one has homology in degree zero and in degree one so you can split this and then you get a natural map from b from value of x let's say object one it doesn't matter which object i take here because the thing is locally constant so they all do the same there is a map from this guy to its loop spectra homologically this would be a shift in degree and this is known as con sigan maybe right differential and this how the deram differential in deram width complex and also say the deram differential in the usual deram complex in the context of the corp of course there appears so this is the typical source the differential on this okay so what i want to do i want to consider my width k theory and endow it with the structure of a locally constant cyclical now how do i do this again i mean there are various ways to do it but i'm going to use a category theory again so one advantage of defining this category lambda in the way i did is the following so if i take any small category dmitry we have a question if you are taking the joint of the s1 action isn't there some disjoint base points it depends on what you mean by space but you probably want get pointed spaces but then i want disjoint base point which is s1 fixed and another question the question was about the differential on the previous slides should i show it yeah would be helpful yes so it was an action on a spectrum and then the smash product then is adjoined to the free loop space isn't it yes the point is that if it's a spectrum so we have the summation map you can split the free loop space into the product of x and the base loop space so free loop space splits into x times omega x so i have a map from x to lx just this action and then i take the component which lands in the moment and that's my differential okay thanks all right right so now it takes a small category and then i can define its psychic nerve this will be cyclic set the factor from lambda to sets just in a very naive very direct way so it sends some n you know just to the set of functions from n lambda to this completely parallel to the usual definition of a nerve or a small category except instead of kind of the category delta which parameterizes ordinals i consider this lambda which parameterizes kind of it's now a loop category or a quiver which is a wheel and not just a string but essentially the same construction obviously functorial respect with maps in lambda just from the wake of traffic lambda and so it says cyclic set uh i won't actually convert it to a category so there is something which is called growth and deconstruction you don't really need to know if you don't already you don't really need to know the full extent of it but what i want to do i want to consider the following category again so consider something which you'll denote lambda i it's some category which comes equipped with a function to lambda and its object appears of an object in lambda and some function so this projection here is forgetful functor which just forgets the second day forgets there's a functor and the fiber so if you have an object here then the fiber of this functor over some n it's just this discrete thing just the set of those i can do it for any functor to sets right if you do it for simple set this is usually called the category of simplices and it can cause the same day tomorrow less so you can recover again your functor from sets from given a category like this a projection which satisfies some conditions now but the reason i want to do it this way is the following assume now that what you have is not a category but what they call a two category so uh now take two categories they see i don't really need to know the precise definition you need to know it's something which has objects but then for any two objects you have not a set of maps from c to c but a set of morphology but a category and then there are compositions there are identity domorphisms and you can sometimes ask it to be strict so the compositions are strictly associated also there are some kind of constraints there and there is actually a way to package the whole thing rather a more convenient way using this growth and deconstruction i mean it's many places in literature for example i just recently had an opportunity to write up a survey of this so it's posted on archive pretty standard the precise details are not that important you can i mean somewhat technical but you can make it work but now what i want to do i want to consider the cyclic mirror for these two categories and and i want to do it right away in a way in the second way using this growth and deconstruction so then we have and this will now be again just a category it moves to the projection to plan so object appears again and gamma gamma is a function from n to c so this has to be made sense of but again this is a standard thing so objects go to objects morphisms in n go to morphisms in c but then whether it's a composable pair there is also some map which some isomorphism between the gamma of composition and composition of gums so these are objects and morphisms are the following so you have some let me do another board some morphisms you have some n gamma you have some prime gamma prime and the morphisms of is a pair f so y f is just a map and then y so i have this gamma which is a function from n to c i have gamma prime i can compose gamma prime with with f and then this phi is a map from gamma to f composed with gamma prime composed with f no the other way around i mean it's a similar two category of simplex is except there instead of this phi there was just a condition because there the things was just a set there were no maps between but now it's a two category so now there are maps so there is an extra structure there is this extra phi and this such a map technological thing so we should have another morphism is called Cartesian if this phi is actually inverted it doesn't have to be but if it is so again we have projection from lambda c to lambda which just forgets gamma and for example the phi bar over one would be the following so it's a category of pairs c which is an object into category and then f which is an endomorphism of the object c and then of course since c from c to c is a category now there are maps between those f's and this what makes this into a category okay and now a general definition trace theory c with values in some category e is a function of lambda c to e that inverts all Cartesian maps now this terminology is mine but the notions of I mean it surely was discovered sometimes in the 70 by the Australian school and then also Tina mentioned that there was a work by Kate Ponto about well several years ago five years maybe and she said about various trace structures in anthropology so she had a name I think her name was shadow if I remember correctly it's pretty close notion so the notion by itself is not that it's not unique and it's kind of axiomatizes some structures which is in nature so why do I call the trace theory so let's see what this is actually practice what what kind of data this guy consists of explicitly so first of all we have this function we can restrict it to the fiber or what so that's fun here which means that for any c and f I have some object okay but now turns out that the next next piece of data which this gadget provides so now we can consider the fiber over two this is a wilk we were with two vertices so consider and the fibers is what so there is an object here an object here and then a map f f right and so I have to associate something to this also but then since my e inverts Cartesian maps the scan they actually identified with e related to c with coefficients in the composition but on the other hand I can equally well identify it with e at c prime and the composition taken in the other direction the other order and so what what I end up with is actually this isomorphism between the two things which is an extra piece of data and this is some kind of trace sort of quotes isomorphism trace just because it satisfies the basic true property of traces which is that trace of a b is the same of trace trace of b a right and so this is the reason for terminology and one can show that this actually I mean this has to satisfy some pretty compatibility condition and then this this is one the one corresponds since actually I have some kind of function problem the c1 plus these actor trace isomorphisms which satisfy some kind of high compatibility condition but the reason I bothered with the more invariant categorical definition is of course that I want to do also a homotopy version and for that it's not good to say that it's up to some higher things because you have to specify all those characters so it's actually better to use this category lambda c and then definition is that x so a homotopy trace theory is a it's a function from lambda c to spaces now but I only want to consider it after wik equivalence after point wise wik equivalence it sits inside this homotopy category of punctures and it should be locally constant along all the Cartesian maps so all Cartesian maps in lambda c go to wik equivalence constant along Cartesian maps that's the definition let me denote by some denote lambda c the full subcategor spent by homotopy trace theory now if you want you can think about this as some kind of infinity category whatever but actually for me it's not needed it's enough to consider the kind of very naive homotopy category where I just invert point wise wik equivalence and leave it at that without any higher structures so I would not leave it and now so an observation is that so it's not now easy to describe this thing by explicit there's no more reason because there is an infinite number of them but at least so if I have such a thing then for any c and f I get some kind of space but what's more now I can restrict my intention to the situation when this f is not just some random f but actually the identity in them and then this technical problem with zoom okay so if if this f is the identity thing then what I have is actually so there's this projection and I have a whole section of this projection which sends one to c and the identity that actually extends to all the other two three and so on and let me actually denote this by sigma c n goes to so it's just n and then the the function sends everything to so this is a quiver I need to specify objects and I need to specify uh arrows so all the objects would be c and all the arrows would be just the identity right so there's the section and then what I can do I can consider now I can pull back my trace theory x with respect to this section and this will be a locally constant function from lambda to space so this will be this uh locally constant uh cyclic cyclic space and this means that if I have a trace theory then its value at c identity for any c is actually is a comes with the circle action so carry it is a locally constant cyclic object in particular if now I consider not just functions to spaces but functions to spectra then I get this differential automatically so trace theory produces me lots of things with this circle action and so now let me give so this was an abstract general theory but let me give you an example which is of interest to me I mean this is kind of the intended application of the formalism so example I take a two category of algebras and bimorphs so let me denote it by more where this stands for marita so objects associative unit of flat at algebras then morphisms from a to b are sort of a b by modules or formally a opposite modules m flat on winds on one side flat you can compose this so this is a well defined two category so you can consider trace theory then example number two is a much smaller object so one of the basic examples of a two category is a two category of just a single object so when you have a single object you only have the category of its endomorphisms but it comes equipped with a monoidal structure because you can compose them so a two category of a single object has the same thing as a monoidal and so I can consider kind of the part of this marita category which where the only algebra considers k itself so let me denote this by b k mod b here stands for classifying space if you want so there is a single object point can you say why flatness is necessary you can get no it's actually not necessary I mean for the definition it's not necessary but there is a theory which is coming up where this would be important the formula formally you can consider the category I mean this will be a larger two category and you can do that but there is a theorem coming up that's all right so morphisms now of course I mean one two categories a part of another one as I said we just restrict one single algebra which is k itself so there is an obvious kind of reduction function you take a trace theory on the large guy and you restrict it to just the small way by the way when you have the classifying two category of a monoidal category then the trace theory on that guy I called I mean I have paper on this way I called it a trace function and then so the trace function monoidal category is exactly this it's a function plus isomorphism between f of a tensor b and f of b tensor a so this is an extra structure maybe terminology is excessive but it's in the literature so better mention so you can restrict a trace theory to this trace function and then there is a very useful general statement which says that it's actually almost an equivalent so you can recover trace theory from the corresponding trace function theorem there is a left adjoint adjoint not just a left adjoint but a fully faithful adjoint fully faithful I called expansion function while some x trace theories on the bing guy and you can characterize its essential image I don't want to give you the definition but it's some version of homotopy invariant invariance so which basically holds in practice so all trace theories you would want to consider in real life would be in the image so the image so the essential image can be described so this is a bit of a miracle and as you see I slowly sort of I was mostly interested in the originally in commutative algebras but now expanded to my generality non-commutative algebras now I allowed bimodules and there is a reason for that the reason is that you get have this kind of great theorem which roughly speaking tells you that if you generally sufficiently far you allow algebras and bimodules then you can get rid of algebras so if you know the trace theory for just key but with coefficients in an arbitrary vector space then you recover your trace theory for all algebras and all bimodules and then what you're interested in is of course some algebra plus the identity and all plus the diagonal bimodules but it pays to generalize the story because then it reduces to just key okay so this is a general theorem and of course even if you construct a trace theory for some other methods it's very useful to compare between two different points so if you have some map and you want to show that it's an isomorphism it's enough to do it on the trace function level just for key because once it's a fully faithful embedding so once an isomorphism there it's an isomorphism everywhere I don't have to really prove it the machine gives it to you okay and now the punchline the punchline is of course is that my width k theory can be promoted to Hamilton-Petrer's theory in fact it's almost obvious so let me show how this is done so width k theory so it used to be defined just for a single algebra now I need an algebra a and then a bimodule the model over a opposite what I do and it's flat on one one side so what I do I consider the tensor algebra of m over a just the usual tensor algebra I can't truncate it at any m so I take the two sided ideal generated by m to the power n plus one and higher and I take the portion where I define completed k theory of tensor algebra as before simply just as a well a multibillion right respect to m the truncating guys and then I observe as before that there is an augmentation map to a which is split and so I define the definition right so it's a statement that is completed with another algebra splits into k theory of a plus something else which I want to call width k theory of a with coefficients in m and then it also carries those endomorphism defined exactly as before because you know this is k theory so now I'm thinking about modules not so the formal power series is of course the tensor algebra of the diagonal bimodule but for any bimodule I can consider models of this algebra and then by the same twisting which I had last time and which I had an opportunity to recall in the beginning this lecture fortunately because of the question by Willie it works again in exactly the same way and defines those endomorphisms again square to e n squares to n e n and if now my k is p local then the whole thing you can show that this is p local you can take the kernel so you get p typical and lemma is that all these guys are naturally homogenous theories so let me do the big one but the small one is homotopy race theory this two category of algebras and I call it lemma not a proposition I mean it's kind of important for the business but it's also very very simple if you almost know this so how do we do this so we know what we want when I have an algebra and the bimodule so we know how the wheel is about race theory on this fiber lambda c1 so how to define it on lambda cn right so here we have so what's an object here it's again a cure so you have some algebras here 0 a1 and so on a n and you have some bimodules 0 0 1 if you want m 1 2 and so on what you do you take just but now I have some flexibility right I mean I didn't say that my a is k or anything it can be anything so I just take a which is just the direct sum of those a i's it's an algebra and m which is the direct sum of those m you know i plus one it's a bimodule respect to this bimodule structure which is obvious here if you if you write it in block form it will have zeros everywhere except for the permutation cycle so it particularly will have nothing on the diagonals but you'll have something when you off diagonal by one and then so what you can do you can consider you can consider with k theory of a and m and then it's very easy to prove that this actually canonical isomorphic to with k theory of say a zero with coefficients in the product so you have a cycle you can take the product that m 0 1 times a1 and 1 2 times so on until then and the reason is basically quillings devisage so k theory is invariant of a category in this case the category is the category of so let me go back the category here is the category of modules over this algebra but this the same is basically the presentations of the quillers so what's the model here or every vertex you have some p which is a module over a so pi which is a model over a i and then m's act and then you can consider the subcategory of modules such that its value at zero is zero and this category would have a final filtration with such a certain grid coefficients are just modules over a i which means that its k theory would be just the k theory the sum of k theories of a i where i is different from z all the m's would disappear because you know when you have a matrix algebra when you have an algebra which is written in a matrix form and everything is upper triangular it's only their terms on the diagonal that matter it's a basic property of k theory which says that it has this kind of additivity property when you have an extension of two categories then k theory only depends on the categories and not on the extension date so you can forget the extension date and then of course so there is a subcategory where the term at zero is just zero and the portion by that is exactly modules of free algebra over i mean a zero and then this module which is the composition so the point is that modules over tensor algebra is a category which is very simple it has a homological dimension one and so in this case you can analyze it completely you can split it into two parts and you see that the difference between k groups are just k groups of the terms which you remove and when i go to wit k theory which i remind you is something where again i already took the constant term out then the result is that the two things are completely obvious so i could write this down probably but to take too much space with the simplified technology so let me just say that this follows by David Sachs and this this only works because the tensor algebras are so simple it's homological dimension and for me this is the main reason which kind of to say that the whole theory wants to be non-commutative because if you stay in the commutative world then you can also consider free commutative variables but those would be more difficult i mean algebra of polynomials and n variables has homological dimension n but if you look at non-commuting polynomials in any variables could be infinite number of them it's still homological dimension so it looks like this work so the upshot is that my wit k theory more or less directly without any effort Mitya before erasing the white boards there is a question was the lambda c n plus square bracket n plus one instead of square bracket m yes because i start from zero right yes sorry it wasn't plus one it's all this this this numbering thing yeah sorry about it okay so the upshot w and also the same is true for the typical right a w k a m so it's a it's a homogenous theory and so its study reduces so in order to study it for all a and m you only need to consider w of k m for arbitrary m w k right and now there is a miracle coming up specific for the case for now up to now k could be anything but now let me specify well i mean it should be p local if you want to take your position but now let me reduce the case where k is a perfect field so if k perfect characteristic p and there is a theorem and this i should attribute this is definitely due to large and this is actually fantastic so wit k theory i mean his terminology is different but it's a statement of k m is zero unless so you only get anything in a single degree and everything else is just here i mean the way he proves it's really high tech so he basically uses this uh dandas macarthia theorem which is a version of good william theorem which shows how k theory changes when you're doing intensible deformations in terms of tc and then he he does the computation with a tensor algebra and i mean it's a non-trivial highly retrieval computation and then everything just cancels out if we were to know a direct proof of this the whole theory was simplified mostly but i don't know directly i mean it looks deceptively simple essentially you're computing a completed k theory of tensor algebra but tensor algebra has a more homological dimension one so you expect there to be something very simple but i don't know any argument which would go like that without the honest computation and then in degree zero it's of course highly non-trivial and so to finish today let me just roughly speak and tell you what you get what's the shape of things you get on degree zero i mean not zero but one i mean there's a shift by one that's k one degree one so essentially what i get is a function from vector spaces to a billion groups so in fact we have a billion groups so we get let me denote it just simply by w some kind of you know polynomial function of two vectors that's a function from k vector to a billion groups it actually comes with the with the filtration so it's actually complete it's an inverse limit of a certain natural tower and the terms of the tower you cannot really describe them directly i think the definition is the easiest way to describe them of course it's just k one so it's not that difficult it's basically just a realization of of the groups of matrices i think actually there is a recent reference by nick allows and maybe crowds i'm not sure where this is worked out in writing so it's an archive preprint from this year i also have a paper on this which is not finished but the point is that it's an iterated extension so there's w n of m w and plus one of m and the kernel is the cyclic power so it's m for the power n co-invited with respect to sigma where the sigma is just the order n cycle fermentation there's a quotient by z mod n z that's a factor from vector spaces to a billion groups which is built out by some kind of twisted you know cyclic power of course if if if it were split this would just be zero horses homology of the tensor algebra and this thing is a twisted version of that so it has a filtration such that it should associate great quotient with that and if you want the p-typical guys then they correspond to powers of powers of p instead of all n's so there is right correct correct so thank you Yuri there's nick allows crowds but another paper by nick allows crowds and other people so crowds on nick allows right so for p-typical guys it's the same but you would only get powers of n would be a power of p so there's also some natural filtration so the rate of cyclic power and so this is something which is very very concrete and down so when i said in my first lecture that i'm not going to prove comparison between wittke theory and coaxial witt and there are witt but i will show you that wittke theory is computable this is basically what i meant so by general machine of trace series it reduces to the computation for just k and in that case it's a very very concrete functor there is nothing even hypothetical about it anymore it's a functor from vector spaces to a billion lips which you can analyze by hand construct direct and so on so forth okay i think that's enough for today and on my last lecture on monday i think i will discuss a little bit the cyclotomic structure which this thing has which we saw in tina's lectures on tuesday but which in my lectures hasn't yet come up so it will come up on monday okay thank you very much okay thank you very much indeed and let's thank the speaker for an interesting talk any questions at all or comments we received a question from uh uh remy at the end of the proof of the lemma how has was the trace ultimately defined okay so i need to define the point is that it's uh inconvenient to define the trace i directly define trace theory which is a value of so it's a functor on this lambda c and the trace comes out when you when you do this comparison so basically if you want the trace you take a and b you take a plus b and then m plus n which is the off diagonal by module you compute with k theory of that and you see that it has comparison with two guys with k theory of a of coefficients in the product so let me maybe write this let me do the computation i mean not do the computation but write write the statement you have a b m and m you do this and there is a direct comparison map which is just you know it's a functor between categories from this to this of a of coefficients in m tensor b m just because the category of modules over this tensor algebra which is in the first uh line uh contains is a full subcategory the category of models of the tensor algebra of m times n over a and then this map is on and then we do the same at b and that traces composition of this guy and then verse to the other so that that's roughly how it goes okay thank you another question uh in the definition of the category lambda capital what happens to the story if you allow maps of any degree and not just of degree one what happens will appear on one cyclotonic structure happens it's a very good question but it's exactly what i am going to discuss in my third lecture okay it's a very good question we deserve the one hour answer any other questions or comments if not let's thank Dmitry again and next lecture will be by Tina Gerhardt at half past three Paris time thank you