3/3 Algebraic K-theory and Trace Methods
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Gerhardt, Teena
Algebraic K-theory is an invariant of rings and ring spectra which illustrates a fascinating interplay between algebra and topology. Defined using topological tools, this invariant has important applications to algebraic geometry, number theory, and geometric topology. One fruitful approach to studying algebraic K-theory is via trace maps, relating algebraic K-theory to (topological) Hochschild homology, and (topological) cyclic homology. In this mini-course I will introduce algebraic K-theory and related Hochschild invariants, and discuss recent advances in this area. Topics will include cyclotomic spectra, computations of the algebraic K-theory of rings, and equivariant analogues of Hochschild invariants.
 Thank you. Since this is my last lecture, I also wanted to take the opportunity to thank the organizers for putting this together and also for making the effort to move this online. I of course wish we could have all been together in France, but it's nice that we were able to do it this way. Anyway, and thank you all for coming to hear the third installment about algebraic k-fury and trace methods. So what I want to do today is I want to talk a little bit more about topological hawk shield homology. We talked a lot about topological cyclic homology on Tuesday, but now I want to dig in a little bit deeper about THH itself. To get started, I want to recall some things that were said already earlier in the week, but that will be very important for us today. So I just want to make sure we're all on the same page with some of these basic constructions. So let's remember all the way back to Monday. And on Monday we talked about how if you have a ring A and you want to study its algebraic k-fury, there's a map relating the algebraic k-fury of the ring and the hawk shield homology of the ring. And that map was called the Dennis trace. And that map was sort of the starting point for the whole trace method and algebraic k-fury approach. So this hawk shield homology is going to be very important for us today. So I know that I've already defined it, but I want to recall the definition just so it's fresh in our memory as we talk a little bit more deeply about it. So what is hawk shield homology? Well, remember that we defined a simplitial abelian group, which we called the cyclic bar construction. So this is the cyclic bar construction. And we said on Monday that what this is, is that in the Q-th level, this is just Q plus one copies of my ring A, tensored together. And it had some face into genus maps. And I even want to recall those face maps because they'll come up again for us today. So what did the face maps do? Well, they take a tensor and most of those face maps just take the i-th and i-th plus first coordinate and multiply them together. So for the most part, these face maps just send this to ai times ai plus one, like so. I used a Q. And that makes sense as long as i is less than Q. But remember that last face map did something different. It brought the last element around to the front and then multiplied. So that last face map sends this to a QA0, tensor A1 through a Q minus one. Okay, those were the face maps that we had on Monday. And then we said that the degeneracies insert the unit after the i-th coordinate. So that was what our degeneracies did. And then we noted that we also had an additional operator, which is called a cyclic operator. That's not part of the simplitial structure, but it is important for us. And that cyclic operator was the operator that just takes my tensor and brings that last factor around to the front. Okay, so we defined this cyclic bar construction on Monday. And then I said, well, what is Hawkshield homology? Well, Hawkshield homology is just the homology of this cyclic bar construction. But what is the boundary map in that chain complex? Well, it's the alternating sum of these face maps. And you can check that that squares to zero. We can take homology and that's called Hawkshield homology. And we also said, well, by the dolde con correspondence, I could alternatively define this as the homotopy groups of the geometric realization of this simplitial object. And that was Hawkshield homology as we had it on Monday. And then we noted something important about it on Monday, which was that it's really not just a simplitial object, it's what we call a cyclic object. So this cyclic bar construction is a cyclic object, which means by the theory of cyclic sets that its geometric realization has an S1 action. Okay, so we talked about that on Monday. And then what did we say about this? Well, what we learned on Monday is that that is an approximation to algebraic K theory, but we can do better by thinking about a topological analog of this theory. So on Monday, we talked about how there is a topological analog, it's called topological Hawkshield homology. And topological Hawkshield homology is related to a K theory via a trace map as well. And that actually factors the Dennis trace. So that first map the map between K theory and THH is often referred to as the topological Dennis trace, or sometimes just the Dennis trace. And this second map is linearization. Okay, and what was the idea, the rough idea of how to define topological Hawkshield homology, well we said on Monday the idea was supposed to be the following. The idea was that, well, in the definition of the cyclic bar constructions I had rings and now I'm going to replace those with rings spectra topological version of rings. My tensor products will become smash products. Instead of working over the integers I'm really working over the sphere spectrum. And if you make those replacements then you'd probably make the following definition of THH, you'd say for our ring spectrum. The topological Hawkshield homology of our is the geometric realization of the cyclic bar construction on our. And then we said when we talk about topological Hawkshield homology of rings for a ring a THH of a is just notation for topological Hawkshield homology of the Eilenberg McLean spectrum of that ring, the ring spectrum associated to that ring. Okay, and then one more thing that we noted on Monday and talked about at length on Tuesday is that this topological Hawkshield homology is an S one. And we saw that that was essential to our applications that was essential to defining topological cyclic homology from it. Okay, so those were all things that we recall on Monday, or I'm recalling things from Monday, excuse me. And the one thing to note about this about the history of this is that it was Box dead who first, excuse me, it was box dead who first constructed topological Hawkshield homology. This is a lot of things that we saw this many years ago and he didn't have some of the nice luxuries that we have today. So, in particular when box dead made this construction he didn't have nice categories of spectra within the box dead product. So what I've written here is this idea. Today we can execute that quite literally, and define topological Hawkshield homology as this cyclic bar construction. But at the time box dead originally constructed THA you couldn't do that so literally because those tools just didn't exist yet. So if you look back at box dead's construction what we now call the box dead model of THA to see that he developed a lot of machinery to work around that he developed what we now refer to as the box dead smash. The interesting thing is that for years up until very recently for K theory applications, we continued to use box dead's model, even though it makes perfect sense now with the current technology to give this kind of cyclic bar construction definition. So why is that. Well, the reason is because it was known that box dead's THH was cyclotomic. And it wasn't known how to put a cyclotomic structure on this definition. So can you put a cyclotomic structure on the cyclic bar construction. In the years of story has changed a bit because of major advances in equivariant homotopy theory and that's sort of the starting point for the new material I want to talk about today is is this cyclic bar construction model for THH and how now we can understand a cyclotomic structure on that. So this comes out of major advances in equivariant stable homotopy theory, coming from work of hill Hopkins and ravenel on the covariant variant one problem. In particular in the context of their work on covariant variant one, they studied extensively what are called norms in equivariant homotopy theory. And so I want to say a little bit about these norms. So as I said, the norms as I'm going to talk about them today are due to coming out of work of hill Hopkins and ravenel building on earlier work so there was earlier work on equivariant norms due to Greenleys and they. Okay, so let me say a little bit about these equivariant these norms and equivariant stable homotopy theory. So what's the idea. Well let's say that we have G a finite group. And H a subgroup of G. The norm functors that hill Hopkins ravenel study have the following form. So we have what are called norm functors. And the norm from H to G is a function, functor excuse me that goes from H spectra to G spectra. So it's a functor that takes as input an H spectrum and it gives you out a G equivariance spectrum. These are symmetric monoidal functors is a very nice thing about them. And in the commutative case they have a nice characterization. So in the commutative case they're characterized as follows. So you can show that the norm from H to G if you input a commutative rings, H spectrum. So a commutative object in H spectra that what you get out is you get a commutative G ring spectrum. And further, this norm functor in the commutative case is left adjoint to the restriction functor. I'll call it I H star. I've used the word restriction a lot of different ways in this lecture series. So what is this functor. Well, the restriction functor here what I mean is just the functor that, you know, we have this G spectrum and we forget down to an H spectrum. So we only remember the H action part of that. Okay, so these norms and equivariance stable homotopy theory it turns out that by studying these deeply hill Hopkins and Ravenel weren't able to get a handle on the career invariant one problem. But that's a very different question than the kinds of questions that we've been looking at. So, how does this connect why does this connect to topological Hockschild homology or this trace methods story. I mean how would you even think. Why would the first question I want to address is why would you even think that there would be a connection there. And the hint that there might be a connection there comes out of a theorem of Hill Hopkins and Ravenel. And hill Hopkins and Ravenel proved the following. They prove that if ours co vibrant. And G is finite. There is an equivalence. And the equivalence is a combination of the two formulas. They constructed map from our to what you get when you take the norm from the trivial group to G of our, and then take the G geometric fixed points of that. See in the very fifth a couple of questions to you. The first questions about do these norms exist for general G or only for finite G. So, he'll hopkins and Robinel constructed these norms for finite G, but we are in a minute going to talk about extending it to a non finite group so in some general sense yes they only exist for finite G and if you want to talk about these norms for a particular group that's not finite. You need to somehow construct what that is which we are going to do in just a moment. And what do I mean when I say commutative yeah I mean, like actual commutative monoids in in the this categories of equivariance vector so this is like a genuine notion of commutative in genuine equivariant homotoby theory. Okay, so what was I saying right so the hill Hopkins and Robinel compute constructed this kind of diagonal map and prove that this gives you an equivalence. And if you look at that well, it looks a bit familiar right it looks like it could be related to that classical definition of cyclotomic spectra that we had on Tuesday. So if you remember what that definition was cyclotomic spectra we're supposed to be things where you take the geometric fixed points and you get back the original spectrum that you start with. Now that's not exactly what's happening here right we have additionally some norm functor in there, but it looks like there could be some kind of relationship, this is giving me some kind of diagonal map, involving geometric fixed points. So this feels reminiscent of the cyclotomic structures. Okay, so I think that question was anonymous but somebody just pointed out that I've said that these norm functors exist for finite groups. And so you could ask, well, can we do this for a group that's not not finite. So in work of the black angle bite. Andrew Bloomberg. Myself, Mike Hill, Tyler Lawson and Mike Mandel. Sorry, that's a lot of people. And let me also mention that there's related work around the same time due to Martin Stoltz. We extended the norm to consider norms to s one. So we, we show that you can extend this. You can extend norms to consider a function, function. Functor, sorry, the norm from the trivial group to s one. Now this makes sense if what you start with is an associative ring spectrum. And this is going to spit out an s one spectrum. So what is this norm functor that we construct. Well, the claim is that the norm from the trivial group to s one should be viewed as the functor that takes a ring spectrum and sends it to the geometric realization of its cyclical construction. So what is the content of saying that that's a norm functor. Well, the claim is that that behaves like a norm. So in particular, if you restrict to the commutative case, you're going to see this thing as the left adjoint of the forgetful functor from s one spectrum down to associate a ring spectrum. I see there's a question I mean strictly associative here. I really do mean associate of rings spectra. So, okay, so we show that you can, you can define a norm in that way, or another way of saying that is well that's like saying we could view topological Hawkshield homology, really we should think of this as an equivalent norm. It's the norm from the trivial group to s one. And then what is the theorem here well the theorem of these same people angle by Bloomberg, Gerhard Hill, Lawson Mandel is that we show that if ours co fibrant. Sorry. For our co fibrant. So, we show that this definition of topological Hawkshield homology as the norm, which is the cyclic bar construction definition actually does have a cyclotomic structure. For many years this was a question, can you put a cyclotomic structure on the cyclic bar construction, and it turns out that using this work of hill Hopkins rabanel and these norm functors you can indeed do so. So in other words there's a cyclotomic structure on the cyclic bar construction. After all, in the chat about can we define the s one norm on bear spectra with no ring structure no s one norm in order to be a sensible construction does need to input an associative ring spectrum so that is a bit different than what happens in the classical case, where the norm from h to G for finite groups inputs in each spectrum, not necessarily an H ring spectrum. Yeah, good question. Okay, so and then let me mention that there's a subsequent theorem of Dotto, Milkovich. patch Korea. So gov a and woo. And what do they show well so I've just said that the cyclic bar construction definition of THH has a cyclotomic structure. You'd want to know that it's the same or equivalent to the cyclotomic structure that on the box that model right that we're getting the same theory of topological technology out of these. And that's what Dotto Milkovich patch Korea so gov a and woo show. So they show that the cyclotomic structure that we construct on the cyclic bar construction agrees with or is equivalent to to the one on box that's model, the one that we've used historically for K theory applications. And it's further nice because you know we talked on Tuesday about how Nicholas and Schultz also have this new framework for studying K theory and I didn't say much about their model of THH but they construct THH in an infinity categorical setting as a cyclic bar construction type construction. And Nicholas Schultz to compare their cyclotomic structure also to boxchats. So Dotto Milkovich patch Korea so gov a and woo's results tell you that all three versions of THH have equivalent cyclotomic structures. So that's nice. Okay, so I'm claiming now that we could think of topological Hock shield molybdenum is an equivalent norm and why is that the nice way to think about it. One reason that that's a nice way to think about it is that it lends itself to some nice generalizations. So I want to mention one of those generalizations now, which is coming out of this same work. So we make the following generalization. Let's say we want to study an equivariant ring spectrum so a CN ring spectrum. And then we can define a CN twisted version of topological Hock shield homology. So I'm going to write this like this topological Hock shield homology the CN topological Hock shield homology of R. And then well what should this be. Well recall that I just said that ordinary topological Hock shield homology is supposed to be a norm from the trivial group to ask one. So I'm going to be writing my THH something that already has a cyclic group action. So what is the topological Hock shield homology of that going to be. Well I claim that it's the norm from CN to S one of R. Now, the next natural question is but what is what does that even mean right. I've told you how to define the norm from the trivial group to S one but what is the norm from the cyclic groups the end to S one. So I'm going to construct this well in the commutative case you could set you could define it as a left adjoint to restriction functor. But if you want to take input that's not necessarily commutative you need to give a more concrete construction. So how do we construct this norm. So you can do this using a cyclic bar construction but not a classical cyclic bar construction we're going to have to use a variant of the cyclic bar construction. So I want to use a variant of the cyclic bar construction and I'm going to write this variance in the following way I'm going to write it as be sick CN. This is going to be a CN twisted version of the cyclic bar construction. So at first it's going to seem similar to what we did before. So on the Qth level this thing is going to be Q plus one copies of our ours a spectrum so they're smashed together. And it has the usual degeneracies. So the usual degeneracies by which I mean they just insert a unit in the correct spot after the I coordinate. So face maps are a bit different. So in order to define the face maps for you. I need to first introduce a piece of notation. So I'm going to let G denote the generator. I'm going to put it into the two pi I two pi I over N of CN. And then I'm going to let alpha Q be a map on the Qth level so from my Q plus one copies of our to itself. And this operator alpha Q it does two things. The first is that it cyclically permutes the last factor to the front. And the second is that it acts on the new first factor. By this element little G. So I should have mentioned, maybe up here that when I take this cyclic bar construction that my input are is now a CN ring spectrum. So let me draw. Maybe let me make a little schematic here so I have my Q plus one copies of our and what does this alpha Q do well it takes the last one. It wraps it around to the front. Now it's the new first factor and it acts on it by this generator little G, which makes sense because ours and equivariance spectrum. Okay, that was supposed to be by way of telling you what the face maps are so what are the face maps. Well they're defined as follows. A lot of them are just the same old thing that they were before the I face map, most of them are just multiplication of the I, and I plus first factors. As long as I is less than Q. What is the last one well the last one is something different now. So the last one I'm going to define to be let's do this operator alpha Q. Bring that last factor around the front act on it by little G and now I'm going to multiply the first two factors together, which is also a map D zero. Okay, so that's my new last face map. And the last note is that I claim that this is still a simple shell object meaning that you know we have these simple shell identities that we need to check, and you can check that the simple shell identities are still satisfied. So this is is simple shell. But while you're checking identities you might check for those cyclic identities is that a cyclic object and you'll learn quickly that this is simple shell but not cyclic. And this is like bad news, because if you remember, it was the fact that the cyclic bar construction was cyclic that when we geometrically realized we got an S one action. Now I'm claiming that this thing is supposed to have an S one action because I'm wanting it to be the norm to S one. So how do I understand why it would still have an S one action. So you can check it doesn't satisfy the prop satisfy those identities, but it does have additional structure. So let's see what kinds of things are true. Well this operator alpha q, it generates a CN q plus one action in simplical degree q. And that is that well alpha q is both rotating the q plus one factors and acting by a generator of CN so that's going to generate a CN times q plus one action in simple shell degree to q. And further the face and degeneracy maps satisfy some properties. So we already said by definition if I do alpha q followed by D zero that was my definition of DQ. And if you do alpha q followed by di for some other di where I is between one and q that what you get is di minus one alpha q minus one. And similarly, the alpha satisfies some properties, some relations with respect to the degeneracy switch I won't write down. So if you look down all of those relations what you'll see is that it turns out that this is an example of a familiar object. So this defines what is called a lambda and op object in the sense of box dead. Shang and Madsen. And similarly this structure came up in box dead Shang and Madsen's work on topological cyclic homology in a different way. It came up because they were studying edgewise subdivisions, and, and this kind of structure naturally arises in that context as well. If n equals one, a lambda and op object is just a cyclic object. So this is a generalization of what it means to be cyclic. And the nice thing about the fact that box dead Shang and Madsen have already studied this kind of object in depth is that we can steal some stuff that we know from them. So in particular box dead Shang and Madsen prove that when you geometrically realize this kind of object. So the geometric realization of this kind of object still has an S one action. Okay, which is good news for us because we were hoping to have such an S one action. So what's the definition of this twisted topological Hock shield homology well the definition is as follows that the CN twisted topological Hock shield homology of my CN spectrum are, well I said it's supposed to be the norm from CN to S one of our, and I claim that that the norm can be constructed as this twisted cyclic bar construction on our. There is a question to you. Can you say something about the universal property satisfied by the modified cyclic bar construction. Yeah, that's a good question. Not off the top of my head. Yeah, I'm sorry I should be able to answer that and it's just it's not in my brain right now. If you're interested in learning more about the this kind of like lamb and op object, the place to look for like a lot of understanding of that object is the box of Chang Madsen paper where they originally defined the cyclic atomic trace. I don't, I don't have. Right, I'm sorry, I just can't I can't get there right now with the universal property characterization. So, okay, so I claim that this twisted cyclic bar construction is a construction of this, this norm from CN to S one. And in particular we show that when you restrict to the commutative case that this this twisted cyclic bar construction is the left adjoint to the forgetful functor in the way that you would want. So it has the properties that characterize a norm in the commutative case. I should also note in the interest of honesty that in this definition that I've written down here, I'm omitting some what we call change of universe functors to sort of make the technical equivariance stable homotopy theory correct. And so if you are an expert in that area and are looking for those. They're there I just didn't write them. And if you don't know about change of universe functors I would just for these purposes ignore it. Okay, so this is CN twisted topological Hawkshield homology and then a question that you might immediately come to about this is, well, THH with cyclic atomic and that was important to the story so is this twisted THH still cyclic atomic. So we prove the following that for our co vibrant. And if P is prime to N, then the CN twisted topological Hawkshield homology of R is P cyclic atomic. I don't think I defined P cyclic atomic when we talked about cyclic atomic spectra on Tuesday but P cyclic atomic just means that you only check those cyclic atomic conditions at the prime P. So it's like specific at the prime P. And therefore we can define CN twisted versions of topological cyclic homology as well. So that's nice. The next question I might ask about this theory at first is, well, can you actually compute this twisted cyclic twisted topological Hawkshield homology of anything. So is CN twisted topological Hawkshield homology computable. And what might you want even want to try to compute so maybe it's nice to have an example in mind of like what kind of thing would it be interesting to try to understand. Well, for instance, we could ask, can we understand the C to twisted THH of the spectrum MUR. So what is MUR. Well, MUR is the C to equivalent. Real bored is a spectrum. This was defined by land Weber and Fuji but it's gotten a lot of attention in recent years because it played a really fundamental role in the solution to the curve and very one problem. So this is this is a particular C to aquavarian spectrum that there's a lot of interest in. So you see that there's a question in the Q&A which is, is there a description of the CN relative THH in terms of a factorization homology type construction. Yeah, that's a great question. So, for those who are familiar, ordinary topological Hawkshield homology can be described in terms of the factorization homology of David Ayala and John Francis. The CN twisted THH is has been described by Asapora in terms of his theory of aquavarian factorization homology. So yes, he gives a characterization of these this relative THH in terms of an aquavarian version of factorization homology. Yeah. Okay, so that's the kind of example that we we might want to understand. Now, if you think about this, you'll realize that, you know, we talked a lot on Tuesday about how to compute topological cyclic homology. But I was always sort of assuming in that discussion that we understood THH to begin with, like we described it inductive proper process to build off of THH to understand its fixed points, or in the Nikolaus Schultz model, we understood THH but then we would study its homotopy fixed points or its Tate construction. I didn't talk at all about how to actually compute THH. So I've said very little about that. So before I can talk about this question is this twisted THH computable, we need to take a step back and talk about, well, how do you compute ordinary THH. So let's say something about that. How do we compute ordinary THH. The ordinary topological Hawkshield homology is really, you know, the starting point for sort of modern trace methods. If you can't compute THH of the object you're interested in, you're not going to be able to compute topological cyclic homology or algebraic k-theory. It all starts with THH. So one of the main tools for computing ordinary topological Hawkshield homology is called the Boxstead spectral sequence. And what is the Boxstead spectral sequence? And in the following way, topological Hawkshield homology, remember, was a realization of a cyclic object, which I've been writing as the cyclic bar construction. Now, when you have a cyclic object like that and you study its realization, you get a spectral sequence induced by the skeletal filtration. So that's a standard tool, the skeletal filtration induces a spectral sequence that's going to converge to the homology of the spectrum THH with coefficients in some field. Now, what is the E2 term of that spectral sequence? Well, what Boxstead proved, which is really, I mean, just really nice, is that when you look at the spectral sequence that you get from the skeletal filtration, on E2 you get something familiar. You get ordinary Hawkshield homology. So the E2 term here is the Hawkshield homology of the homology of R. So this is, I think, so beautiful that if you want to study this topological theory of topological Hawkshield homology, we get this spectral sequence whose E2 term is living in the algebraic analog, which is easier to compute. I mean, Hawkshield homology has all the tools of homological algebra at your disposal. So it's something that's much more computable than this topological theory. So Boxstead constructed this spectral sequence and did some beautiful calculations with it right off the bat. So Boxstead computed the topological Hawkshield homology of FP and also the topological Hawkshield homology of the integers. And these, I mean, Boxstead did this work quite a few years ago now, but these calculations, particularly the topological Hawkshield homology of FP, are still foundational to so much work we do in K-theory today. So many of those calculational results that I mentioned on Monday take as input Boxstead's work on THH. Okay, so the Boxstead spectral sequence is very powerful and has been foundational to calculations. And so one question is, well, what does this mean in our setting? So I'd want to have an equivariate version of this. Of this Boxstead spectral sequence for twisted THH. That was probably the easiest way or one direct way to get a handle on on calculations here. Okay, so I think about that and I think, well, what does that mean to want that? Well, my Boxstead spectral sequence should compute the topological theory, should compute some homology of twisted THH and that E2 term should be the algebraic analog of twisted THH. And then we realize, we have no idea what that is, right? What is the algebraic analog? So what is the algebraic analog of twisted THH? Well, it's not immediately obvious what that should be, right? In classical theory, we started from the algebra and we made a topological construction analogous to it. Now we've generalized that. And it's not clear anymore what algebra that comes from. So maybe we should revisit what it meant to be the algebraic analog in the classical case and that will hopefully provide us some inspiration. So what did it mean in the classical case? Well, in the classical case, we were looking at rings and we had a relationship between topological Hock-Schild homology and ordinary Hock-Schild homology. That was the linearization map. And we remember that this is notation for THH of the Eilenberg-McLean spectrum. Now, I haven't mentioned so far, but in the classical theory, not only do you have this linearization map relating these two, but it's also the case that in degrees zero, it's an isomorphism. So this linearization map in degrees zero is an isomorphism. So I'd like some analogous story with my twisted THH. I'd like to understand how it relates to some algebraic analog. But if we look at this classical story, you know, I took Hock-Schild homology of a ring and it was related to THH of the Eilenberg-McLean spectrum. And so that brings me to a question which is, well, now I need my input for my twisted theory needs to be equivariant. And so the question is, how do I get a C and ring spectrum as an Eilenberg-McLean spectrum? I'm going to need to do that in order to make sense of this analog. Or a different way maybe of phrasing that question is, what is the equivariant analog of a ring? Okay, so I need to get at those questions if I'm going to be able to understand this kind of equivariant analog. Okay, so we're going to take a little detour to talk about some basic objects and equivariant homotopy theory that we haven't actually heard much about yet this week. And they're called Machyfunctors. So this is going to seem like a detour for a second and it's going to bring us back to this question of what is the equivariant analog of a ring. Now, if you've never seen a Machyfunctor before, the thing that you should have in your head about Machyfunctors is that Machyfunctors are like the abelian groups of equivariant stable homotopy theory. So what do I mean by that? Well, in ordinary homotopy theory, we have a lot of invariants that give us abelian groups. In equivariant homotopy theory, we have a lot of invariants that give us Machyfunctors. So what is a Machyfunctor? Well, I'm going to let my group be finite. So for G finite, a Machyfunctor, M, is actually a pair of functors. So I'm going to call one of them M lower star and one of them M upper star, and they're functors from finite G sets to abelian groups. One of them is covariant and one of them is contra variant. Okay, so I have these two functors. And the, the, they have to satisfy a few properties. So one is that I, the functors have to agree on objects. So M lower star of X has to agree with M upper star of X and that shared value is called M underbar of X. So when you see these underbars, that's indicating that we're working with Machyfunctors. Tina, I need, yes. I'm sorry to have interrupted you. There is a question. What goes wrong if you try to copy the definition of C and twisted THH for G equivariance Z module spacer. I think that the question is that in the ordinary case of topological Hock-Schild homology, you can define topological Hock-Schild homology like a relative version of Hock-Schild, topological Hock-Schild homology for module spectra. And if you do relative T that I'm using the, okay, if you do relative THH for HZ modules in the classical case you get back the algebraic theory of Hock-Schild homology. And the question here is, why can't I do this twisted version for HZ module spectra and would that give me back what I want. You know, I have to admit that I have not thought about the relative, like the A relative version of the twisted theory. So I don't. I don't have a good answer off the top of my head of what kinds of considerations you would need to take there. I just haven't thought through. Thought through how that that we haven't defined that object and I haven't thought through how that definition would work. And another question from Sean Tilson. Oh yeah so Sean says you get some chukla stuff because of the tensor product being derived. So, well I haven't gotten there yet but I'm going to talk about this equilibrium theory of Hock-Schild homology and that can be generalized to this, this version of chukla homology as well. It's not defined in the way that was just mentioned as like thinking of this as a very twisted things over HZ module spectra. And I have not thought about whether that's equivalent. Okay so right I was saying what a Machi functor is so a Machi functor is a pair of functors they have to agree on values. They need to take disjoint union to direct some and they have to satisfy some axioms that I'm not going to write today. So, if you haven't seen Machi functors before one nice sort of diagrammatic thing to keep in mind about Machi functors is the following. In particular, if you have some nested subgroups of your group G, so I have K sitting inside H sitting inside G, we have a projection map right from G mod K to G mod H. So what happens with the Machi functor? Well the Machi functor these G mod K and G mod H those are finite G sets. So I get a value for the Machi functor at G mod H. I get a value for the Machi functor at G mod K. And I have a covariant functor and a contravariant functor relating them so I get maps in both directions. The covariant functor we usually call the transfer from K to H and the contravariant functor is often referred to as the restriction from K to H. And it turns out any finite G set is a direct sum of these orbits, these things of the form G mod K. And so characterizing what happens on these orbits really tells you what happens with the whole Machi functor. Now, we've actually seen a Machi functor already even though we didn't put it in that terminology we've been working with a Machi functor sort of all week, which is the following. If you have X, a G spectrum, you get a G Machi functor, which is the homotopy Machi functor of X. So I want to specify for you what is this homotopy Machi functor do on an orbit G mod H. And this is supposed to give me some abelian group and what does it give me well it gives me the end homotopy group of the H fixed points of my spectrum X. So, on Tuesday all that time we were studying fixed points of THH in particular we were studying a Machi functor. And when I say Machi functors are like the abelian groups of a covariant homotopy theory this is kind of what I have in mind. In ordinary homotopy theory my homotopy groups are going to spit out a billion groups, and an equivalent homotopy theory, the homotopy, the natural way to think about homotopy of an equivalent spectrum is as a Machi functor. So Machi functor constructions are very closely tied to equivalent spectra. So let me note that if you have a G Machi functor. Let's call it M. It has an Eilenberg Maclean spectrum attached to it which is a G spectrum. And we write that as H M. And in what sense is it Eilenberg Maclean well it's a G spectrum so I can take its homotopy Machi functor. And what do I get out in degrees zero, I get my Machi functor back. And in all other degrees I get zero. So that's the sense in which this is Eilenberg Maclean. Okay so that's nice to a Machi functor I can associate an Eilenberg Maclean Equivariant spectrum. So we're going to need a notion of norms for these Machi functors. So, Hill and Mike Hill and Mike Hopkins, give a definition of what it means to take a norm of a Machi functor. And they say the following, if H is a subgroup of my finite group G, and M is an H Machi functor. So I'm going to define what it means to take the norm from H to G of the Machi functor M. And here's their definition. Their definition is, okay so I have my Machi functor, and I just said I could take an Eilenberg Maclean spectrum associated to it now that's an H spectrum. So I have a norm in Equivariant spectra that Hill Hopkins Rabinel norm. So I can take the norm from H to G in spectra. Now I have a G spectrum, but I wanted a G Machi functor. And so to get back to Machi functors I can take Machi functor Pi zero of that. So plus minus on this definition of Machi functor norms is the following. On the upside, it's nice to define it really highlights the close relationship between Machi functors and the Equivariant theory of G spectra. And if you want to prove theorems about the Machi functor norm, this is often the definition that you use. The minus of this definition is if you want to actually compute the norm of a specific Machi functor, this is very difficult to get a handle on a computation this way. So there are much more algebraic constructions of the norm in Machi functors that are due, for instance, to Kristen Mazzur and Rolf Hoyer. And they have more hands on approach of understanding this without going through the Equivariant stable homotopy theory. Okay, but the Hill Hopkins definition is clean and can be useful for us. Another thing we need to note about this is that there's a symmetric monoidal product on this category of G Machi functors which is given by what's called box product. So what is the box product of two G Machi functors. Well, this box product one way of saying what it is is it's closely related to the product in symmetric in Equivariant spectra. So if you take my island berg McLean spectrum of M, my island berg McLean spectrum of n, smash them together to get a G spectrum and then take Machi functor Pi zero to get a G Machi functor. There is a question. Other contra variant maps on PIM some sort of averaging over H mod the mod K. So I've been maybe a little sloppy is the wrong word neglectful up here. So when I talked about this homotopy Machi functor I just told you what it does on each on these orbits. A homotopy Machi functor is more than just of course the information of what happens to the finite G sets it's also these contra variant functors these transfer and restriction maps. In the context of the homotopy Machi functor how do we think about that. So one of them, the restriction map is a nice easy to describe map that's a map given by inclusion of fixed points so it's actually confusingly the map that we called f earlier in the week not the map we called our. There's a clash of notation there but but that map is inclusion of fixed points the other map is in this context sometimes called the well maybe not in this exact context but it's what's called the Aquavariant transfer map. And the way to think about that map maybe is that it comes from. What's called the birth Mueller isomorphism and Aquavariant homotopy theory tells you that you have these kinds of maps as well but it's sort of a unique thing to being in the Aquavariant setting. So you use the birth Mueller isomorphism and also some duality of these orbits to talk about that transfer map. Um, yeah, good question. Okay, so this box product I've given you a definition of the box product and then this is similar to what I was saying about the norms which is that this is a definition that goes through Aquavariant stable homotopy theory. Machi functors are really algebraic objects you're supposed to think of them as living in algebra and lots of people in math use Machi functors that are not interested in stable homotopy theory. Machi functors are useful for studying representation rings and other things. Um, so you can define the box product totally algebraically. But I'm giving you this characterization to show the relationship with the G spectra. Okay, so now I have this symmetric model product on G Machi functors and finally I'm ready to address the question of what is an Aquavariant ring. Well an Aquavariant Abelian group is a Machi functor and Aquavariant ring is called a green functor. So a green functor is a monoid in this category. And then the note is that if I have a green functor for a green functor are its Eilenberg McLean spectrum is a, well, for a green functor are for CN, it's Eilenberg McLean spectrum is a CN ring spectrum. So one of our questions was well how do we get an Aquavariant ring spectrum as an Eilenberg McLean spectrum and the answer is using green functors. So what do I really want for that Aquavariant Analog well it turns out that what I need is a theory of Hawkshield homology for green functors. Okay, and that theory of Hawkshield homology for green functors is defined using that same kind of construction of a twisted cyclic bar construction. So we can do the same construction that we did for spectrum now we can do it in the context of these green functors so where we had a ring spectrum before replace it with my green functor where I had smashed products I replace it with box products, but you do that same twisted construction and it makes sense here. And so what is the definition then of Hawkshield homology for green functors. Well this comes out of work of Andrew Blumberg, myself, Mike Hill and Tyler Lawson. So the definition is the following. If you have h inside g inside s one in our an h green functor. We look at the G twisted Hawkshield homology of our. And we prove that it's the homology of a twisted cyclic bar construction on the norm from h to G of R the Mackie functor norm for our. Okay, so, oh yeah so is the box products symmetric yeah this is the symmetric monoidal structure in this category. So the symmetric monoidal structures. In order to talk about that you really want to be working more with 10 borrow functions and not just with Mackie functors and I'm kind of, I don't want to go there right now. But yes it is a symmetric monoidal structure and if you're working with some borrow structures, the tomorrow functors you get even more than that. So, I claim that this is the equivariant, or sorry the algebraic analog of my twisted THH and the theorem is that well we have a linearization map. Relating these theories so if I look at the H twisted THH of my Eilenberg McLean spectrum. So if I take its homotopy Mackie functor that maps to this twisted version of topological Haxial Tomology for green functors. And this is further in isomorphism if K is equal to zero, which is what we wanted to see from the, which is what we wanted to see from the perspective of what happens in the classical case. So one of our goal was to define an equivariant version of the boxed at spectral sequence so another like point of proof that this is the right algebraic analog would be if you had a boxed at spectral sequence, computing twisted THH with E2 term in this Haxial Tomology for green functors. And indeed there is such a spectral sequence. So in work of Catherine Adamic. Myself, Catherine has in bar client and Hannah Gekong. So that we construct such an equivariant spectral boxed spectral sequence. So we construct an equivariant boxed spectral sequence for twisted THH. And it has E2 term in the Haxial Tomology for green functors. And it turns out that this Haxial Tomology for green functors really is, you know, the right algebraic analog that you're looking for. And it turns out that this spectral sequence can be used computationally. So part part of this work of these authors I just mentioned, is that we use this equivariant boxed at spectral sequence to compute the equivariant homology of the C2 twisted THH of MUR. And we use this equation that we learned earlier with coefficients in what's called the constant Mach-e-Funkter F2. So this is an equivariant version of homology. If you're not familiar with that, I'm not going to dive into what exactly that means, but that's the natural notion of homology to consider in this equivariant setting. Okay, I'm almost out of time, but I want to, I want to close by addressing one question. I want to sort of bring this full circle. In the beginning of the week, we were talking about k theory of rings, and now I've been talking about these equivariant analogs of THH. And so a question you might have is, well, can these equivariant theories tell us anything about the classical story? So can we learn about the classical story this way? And so I want to connect it back. So what does this tell us about the classical story? Here's one thing to say about it. Well, why would there be any connection? So here's one reason we might expect to have a connection to the classical story. The classical story was about rings. And now I've moved into this world of green functors and Mach-e functors, but a ring is actually a green functor. A classical ring is a green functor for the trivial group. Okay, so what does that mean? Well, it means that we get some new trace maps out of this story. So I had a trace map from algebraic k theory to topological Hock-Shield homology, we lifted through the fixed points. And if you use this new linearization map relating the equivariant THH to this twisted version of Hock-Shield homology, what you find is that you get a trace map from the algebraic k theory of a classical ring to the Cp to the n twisted Hock-Shield homology of that ring, evaluated at the orbit Cp to the n mod Cp to the n. So I'm not going to unpack how you get that trace map exactly, but it follows directly from that linearization map that we had a moment ago. So what is this thing? Well, the way to think about this is that this is the algebraic analog of fixed points of THH. So this is a purely algebraic object that is going to serve as an analog of fixed points of THH. Now, in order for those fixed points of THH to be useful in order to study TC, I needed to not only know about the fixed points themselves, but I needed to know about those two operators on them, that f and that r. The f map is already part of the Mach-E-Funkter. It's like built into this story automatically because it is the restriction map in those Mach-E-Funkters. So I don't have to worry about that, but the r map is something outside the Mach-E-Funkter structure. The r map, which was confusingly called the restriction in this context, that was the map that depended on the cyclotomic structure. And once you have that r map in the classical theory, there is an object called topological restriction homology, which is what you get when you take the limit across the r maps of THH. You can frame it this way on Monday, but this TR, this topological restriction homology is like between the fixed points of THH and topological cyclic homology. It's one of the things you compute on the way. So what I'd like to know is, well, can I do this in the algebraic setting? Do I have an analog of that restriction map, the map that came from the cyclotomic structure? And so what we show in Blumberg, myself, Hill, and Lawson is that you do get such an analog of this restriction map. So we define geometric fixed points for Mach-E-Funkters. And we show that you get a type of cyclotomic structure on the Hock-Shield homology of Green-Funkters. And in particular, Can you get this tracing out via the universal property of algebraic k-theory as in Blumberg, Gevner, Tabawata? I'd have to think about whether there's a way to characterize it in terms of the, probably, I'd have to think about whether there's a way to characterize it in terms of the universal properties. That's not how we define it. We define it directly through the trace from k-theory, through the topological denistrace, basically, and the fact that the topological denistrace lifts through fixed points. But I have not thought about whether there is a universal characterization of it. So I'm not sure. Right. So the last thing I was saying is that we define what it means to take geometric fixed points for Mach-E-Funkters. We prove that there's a type of cyclotomic structure on Hock-Shield homology for Green-Funkters. And what it gives you is it gives you an algebraic version of TR, which we call little TR. And what is this? Well, it's some limit over these algebraic versions of the restriction map of these C-P to the N-twisted Hock-Shield homology for the ring A, evaluated at this orbit. And as an example, you could compute, for instance, the algebraic TR of Fp. So we do this calculation and what goes into that? Well, you need to understand the Mach-E-Funkter norms on Fp, the twisted cyclic bar construction on Fp, and then the cyclotomic structure for that. And it turns out what you see is that you get the p-addicts in degrees zero and zero everywhere else. And so in this case, this algebraic approximation is really a good approximation because that agrees exactly with the topological restriction homology, the topological theory, and the p-completion of algebraic k-theory of Fp. Okay, so in this case of Fp, it captures, you know, all the information basically, that of course in general will not be true. Some algebraic analog of this topological theory. Okay, I'm a little bit over time, so I'm going to stop there and it has been a pleasure to be with you this week and I hope that I have given, especially the early queer people, some idea about what trace methods and k-theory is about and some sort of interesting developments that have been happening in this area. So I will stop there. Many thanks indeed. And let's thank this Tina for a wonderful mini course. There is a question. Can you say roughly what the bread and homology of THH, M-U-R is? And does it split into pieces where some of these summands are familiar? Yeah, I should know how to split it into pieces. So I can tell you what the, you know, if I look it up, I can tell you what the answer is. I still remember when I was a PhD student many years ago now, the first time that I asked my advisor something that was in a paper that he wrote, and he said, oh, I don't know, I'll look it up. And that was like for me a very liberating moment that even my advisor like didn't remember everything he'd ever written down. So I don't feel bad about looking up this answer in my own paper. Right, so what is the answer for the C2-equivariate homology of the topological hawkshield homology of M-U-R? So this is the C2-twisted hawkshield homology. So in this case, we get a really nice answer. So I haven't, okay, there were a few things I didn't say. So one of the things I didn't say is that that equivariate box-ded spectral sequence, when you start working in these equivariate worlds, you end up having multiple gradings. You get a Z graded spectral sequence. So a spectral, you get Z graded theories where they're graded by the integers like we're used to. And you also get theories graded by representation rings. And so it turns out that the representation ring object is the more natural thing to consider in many cases. So what I'm writing down is the ROC2 graded equivariate homology. And for those of you who are not, who are new to equivariate stable homotopy theory, let me introduce you to a really special convention, which is that a five-pointed star is usually an equivariate grading. That means a representation, and an asterisk is an integer grading. So something useful to know. So that grading is now graded by representations. And what this is, is it's the Eilenberg-McLean, the equivariate homotopy groups of the Eilenberg-McLean spectrum of F2 with polynomial generators on that. And then it's box over HF2 star of the exterior thing over HF2 star of Z1, Z2, etc. So I don't know if that was helpful or not. The degree of BI is I times the representation, the regular representation, and the degree of Zi is one plus that. So that's what the answer is for the equivariate homology of that. And I don't know if you find that helpful or not. But yeah. Okay, so one way of seeing, so the question is, one way of seeing topological Hock-Schild homology is as a relative smash product over a tensor A op, and is there an analog of this for this CN twisted THH? Yes, you can think of the CN twisted THH in terms of these relative smash products. And maybe the thing to say is that, so if I'm interested in, let's say the H twisted THH of R, and I'm interested in that as like a G spectrum. So if I'm, I think I did that in the opposite direction I meant to. If I want to look at the G spectrum, restrict that to a G spectrum, then you can write this as a G spectrum from H to G of R smash over that thing op. So the, no, sorry, that the enveloping thing, the that smash that op. But then what you have over here is you have a twisted version in of the norm. So yes, there is a way to characterize it in terms of these relative smash products but you pick up a little bit of a twist so it's a bit different than the classical one. The next question I see is, is there a trace map from the equivariant k theory to this twisted THH? Yeah, that's a great question. So that's in a very natural question to ask is like, well, I've talked about, you know, I, at the end I was saying, well, you can get trace maps out of a ring but can we naturally get trace maps from some kind of equivariant version of k theory to this twisted THH. The answer to that is yes. I have work in progress with some of the people I mentioned, Catherine Adamic, Catherine Haas, and I'm going to ask, in Mark Lang and Hannah Gekong, where we're looking at like, what is the right kind of k theory in order to get that sort of trace map and maybe I won't say too much about that since this work in progress. I don't want to make any bold claims yet. I don't have like known notions of equivariant k theory. So a lot of people have considered different notions of equivariant algebraic k theory, and I don't yet have like a connection between the CN relative THH and those different known theories of equivariant k theory. Is there a reason we don't have to use a drive limit for TR? I mean, so this limit that I'm talking about here, these things that I am taking the limit of now are all just a billion groups and so here it is just like an ordinary group of billion groups that ends up being the right thing to take their, I don't know, maybe that's not a very satisfying answer, but that's what's happening in this case. Let me read your question. If G is a finite group acting on a commutative ring R, can we cook up a G timbara functor? Perhaps with the fixed point. G is a finite group acting on a commutative ring R. I'm not sure exactly what you're asking. So you want like a group acting on a classical ring, can we cook up a G timbara functor? I don't know off the top of my head. Yeah, I'm not sure what the answer to your question is. I apologize. What kind of marine invariance property is this twisted THH have? That's another great question. So the, right, so these Hock-Shield theories in general, Hock-Shield homology, topological Hock-Shield homology, any sort of Hock-Shield theory, one thing that you might want to ask for is that it's marine invariance, right? That's something that's really common amongst Hock-Shield theories and something that's sort of important to those theories. So in some cases you can prove that directly, but there's a recent work of coming out of work of Cape Ponto and Ponto Shulman, and now there's a larger group of collaborators, Ponto, Campbell and Ponto, etc. So we talked about Hock-Shield homology and topological Hock-Shield homology as bi-categorical shadows. And that shadow approach, whatever that means, I don't want to go into what that means, but it's, we heard a little bit about it in the first lecture this morning, if you were there, that kind of idea of a shadow. If you know that THH is a shadow, then marine invariance comes for free, because it turns out marine invariance is like a natural notion of equivalence on bi-categories. So the question is about twisted THH, and this is a great question. In that same work in progress that I mentioned of Adamic and myself and Hauss and Clang and Kong, we have shown that you can view this twisted topological Hock-Shield homology as sort of an equivariant shadow. So in particular, you get marine invariance also in this case for free. And so, yeah, it aligns nicely with what you'd expect from one of these topological Hock-Shield, or Hock-Shield theories and you do get marine invariance. So that's a nice property to know that you have of twisted THH. Any other questions or comments to Tina? If not, then, merci beaucoup for everything for a wonderful mini-course. Thanks. Thank you Tina again.
 Thank you.  Since this is my last lecture,  I also wanted to take the opportunity to thank the organizers for putting this  together and also for making the effort to move this online.  I of course wish we could have all been together in France,  but it's nice that we were able to do it this way.  Anyway, and thank you all for coming to hear the third installment about  algebraic k-fury and trace methods.  So what I want to do today is I want to talk a little bit more about topological  hawk shield homology. We talked a lot about topological cyclic homology on Tuesday,  but now I want to dig in a little bit deeper about THH itself.  To get started, I want to recall some things that were said already earlier in the week,  but that will be very important for us today.  So I just want to make sure we're all on the same page with some of these basic  constructions. So let's remember all the way back to Monday.  And on Monday we talked about how if you have a ring A and you want to study its  algebraic k-fury, there's a map relating the algebraic k-fury of the ring and the  hawk shield homology of the ring. And that map was called the  Dennis trace. And that map was sort of the starting point for the whole trace  method and algebraic k-fury approach.  So this hawk shield homology is going to be very important for us today.  So I know that I've already defined it, but I want to recall the definition just so  it's fresh in our memory as we talk a little bit more deeply about it.  So what is hawk shield homology? Well, remember that we defined a  simplitial abelian group, which we called the cyclic bar construction.  So this is the cyclic bar construction.  And we said on Monday that what this is, is that in the Q-th level,  this is just Q plus one copies of my ring A, tensored together.  And it had some face into genus maps. And I even want to recall those face maps  because they'll come up again for us today.  So what did the face maps do? Well, they take a tensor and most of those  face maps just take the i-th and i-th plus first coordinate and multiply them together.  So for the most part, these face maps just send this to ai times ai plus one,  like so. I used a Q. And that makes sense as long as i is less than Q.  But remember that last face map did something different.  It brought the last element around to the front and then multiplied.  So that last face map sends this to a QA0, tensor A1 through a Q minus one.  Okay, those were the face maps that we had on Monday.  And then we said that the degeneracies insert the unit after the i-th coordinate.  So that was what our degeneracies did.  And then we noted that we also had an additional operator, which is called a cyclic operator.  That's not part of the simplitial structure, but it is important for us.  And that cyclic operator was the operator that just takes my tensor and brings that last factor around to the front.  Okay, so we defined this cyclic bar construction on Monday.  And then I said, well, what is Hawkshield homology?  Well, Hawkshield homology is just the homology of this cyclic bar construction.  But what is the boundary map in that chain complex? Well, it's the alternating sum of these face maps.  And you can check that that squares to zero.  We can take homology and that's called Hawkshield homology.  And we also said, well, by the dolde con correspondence, I could alternatively define this as the homotopy groups of the geometric realization of this simplitial object.  And that was Hawkshield homology as we had it on Monday. And then we noted something important about it on Monday, which was that it's really not just a simplitial object, it's what we call a cyclic object.  So this cyclic bar construction is a cyclic object, which means by the theory of cyclic sets that its geometric realization has an S1 action.  Okay, so we talked about that on Monday.  And then what did we say about this? Well, what we learned on Monday is that that is an approximation to algebraic K theory, but we can do better by thinking about a topological analog of this theory.  So on Monday, we talked about how there is a topological analog, it's called topological Hawkshield homology.  And topological Hawkshield homology is related to a K theory via a trace map as well.  And that actually factors the Dennis trace.  So that first map the map between K theory and THH is often referred to as the topological Dennis trace, or sometimes just the Dennis trace.  And this second map is linearization.  Okay, and what was the idea, the rough idea of how to define topological Hawkshield homology, well we said on Monday the idea was supposed to be the following.  The idea was that, well, in the definition of the cyclic bar constructions I had rings and now I'm going to replace those with rings spectra topological version of rings.  My tensor products will become smash products. Instead of working over the integers I'm really working over the sphere spectrum.  And if you make those replacements then you'd probably make the following definition of THH, you'd say for our ring spectrum.  The topological Hawkshield homology of our is the geometric realization of the cyclic bar construction on our.  And then we said when we talk about topological Hawkshield homology of rings for a ring a THH of a is just notation for topological Hawkshield homology of the Eilenberg McLean spectrum of that ring, the ring spectrum associated to that ring.  Okay, and then one more thing that we noted on Monday and talked about at length on Tuesday is that this topological Hawkshield homology is an S one.  And we saw that that was essential to our applications that was essential to defining topological cyclic homology from it.  Okay, so those were all things that we recall on Monday, or I'm recalling things from Monday, excuse me.  And the one thing to note about this about the history of this is that it was Box dead who first, excuse me, it was box dead who first constructed topological Hawkshield homology.  This is a lot of things that we saw this many years ago and he didn't have some of the nice luxuries that we have today. So, in particular when box dead made this construction he didn't have nice categories of spectra within  the box dead product. So what I've written here is this idea. Today we can execute that quite literally, and define topological Hawkshield homology as this cyclic bar construction. But at the time box dead originally constructed THA you couldn't do that so literally because those tools just didn't exist yet.  So if you look back at box dead's construction what we now call the box dead model of THA to see that he developed a lot of machinery to work around that he developed what we now refer to as the box dead smash.  The interesting thing is that for years up until very recently for K theory applications, we continued to use box dead's model, even though it makes perfect sense now with the current technology to give this kind of cyclic bar construction definition.  So why is that. Well, the reason is because it was known that box dead's THH was cyclotomic. And it wasn't known how to put a cyclotomic structure on this definition. So can you put a cyclotomic structure on the cyclic bar construction.  In the years of story has changed a bit because of major advances in equivariant homotopy theory and that's sort of the starting point for the new material I want to talk about today is is this cyclic bar construction model for THH and how now we can understand a cyclotomic structure on that.  So this comes out of major advances in equivariant stable homotopy theory, coming from work of hill Hopkins and ravenel on the covariant variant one problem.  In particular in the context of their work on covariant variant one, they studied extensively what are called norms in equivariant homotopy theory. And so I want to say a little bit about these norms.  So as I said, the norms as I'm going to talk about them today are due to coming out of work of hill Hopkins and ravenel building on earlier work so there was earlier work on equivariant norms due to Greenleys and they.  Okay, so let me say a little bit about these equivariant these norms and equivariant stable homotopy theory. So what's the idea. Well let's say that we have G a finite group.  And H a subgroup of G.  The norm functors that hill Hopkins ravenel study have the following form. So we have what are called norm functors.  And the norm from H to G is a function, functor excuse me that goes from H spectra to G spectra.  So it's a functor that takes as input an H spectrum and it gives you out a G equivariance spectrum. These are symmetric monoidal functors is a very nice thing about them.  And in the commutative case they have a nice characterization.  So in the commutative case they're characterized as follows.  So you can show that the norm from H to G if you input a commutative rings, H spectrum. So a commutative object in H spectra that what you get out is you get a commutative G ring spectrum.  And further, this norm functor in the commutative case is left adjoint to the restriction functor.  I'll call it I H star.  I've used the word restriction a lot of different ways in this lecture series. So what is this functor. Well, the restriction functor here what I mean is just the functor that, you know, we have this G spectrum and we forget down to an H spectrum.  So we only remember the H action part of that.  Okay, so these norms and equivariance stable homotopy theory it turns out that by studying these deeply hill Hopkins and Ravenel weren't able to get a handle on the career invariant one problem. But that's a very different question than the kinds of questions that we've been looking at.  So, how does this connect why does this connect to topological Hockschild homology or this trace methods story.  I mean how would you even think. Why would the first question I want to address is why would you even think that there would be a connection there.  And the hint that there might be a connection there comes out of a theorem of Hill Hopkins and Ravenel.  And hill Hopkins and Ravenel proved the following. They prove that if ours co vibrant.  And G is finite.  There is an equivalence.  And the equivalence is a combination of the two formulas. They constructed map from our to what you get when you take the norm from the trivial group to G of our, and then take the G geometric fixed points of that.  See in the very fifth a couple of questions to you.  The first questions about do these norms exist for general G or only for finite G.  So, he'll hopkins and Robinel constructed these norms for finite G, but we are in a minute going to talk about extending it to a non finite group so in some general sense yes they only exist for finite G and if you want to talk about these norms for a particular group that's not finite.  You need to somehow construct what that is which we are going to do in just a moment.  And what do I mean when I say commutative yeah I mean, like actual commutative monoids in in the this categories of equivariance vector so this is like a genuine notion of commutative in genuine equivariant homotoby theory.  Okay, so what was I saying right so the hill Hopkins and Robinel compute constructed this kind of diagonal map and prove that this gives you an equivalence.  And if you look at that well, it looks a bit familiar right it looks like it could be related to that classical definition of cyclotomic spectra that we had on Tuesday.  So if you remember what that definition was cyclotomic spectra we're supposed to be things where you take the geometric fixed points and you get back the original spectrum that you start with.  Now that's not exactly what's happening here right we have additionally some norm functor in there, but it looks like there could be some kind of relationship, this is giving me some kind of diagonal map, involving geometric fixed points.  So this feels reminiscent of the cyclotomic structures.  Okay, so I think that question was anonymous but somebody just pointed out that I've said that these norm functors exist for finite groups. And so you could ask, well, can we do this for a group that's not not finite.  So in work of the black angle bite.  Andrew Bloomberg.  Myself, Mike Hill, Tyler Lawson and Mike Mandel. Sorry, that's a lot of people.  And let me also mention that there's related work around the same time due to Martin Stoltz.  We extended the norm to consider norms to s one. So we, we show that you can extend this.  You can extend norms to consider a function, function.  Functor, sorry, the norm from the trivial group to s one.  Now this makes sense if what you start with is an associative ring spectrum.  And this is going to spit out an s one spectrum.  So what is this norm functor that we construct. Well, the claim is that the norm from the trivial group to s one should be viewed as the functor that takes a ring spectrum and sends it to the geometric realization of its cyclical construction.  So what is the content of saying that that's a norm functor. Well, the claim is that that behaves like a norm. So in particular, if you restrict to the commutative case, you're going to see this thing as the left adjoint of the forgetful functor from s one spectrum down to associate a ring  spectrum. I see there's a question I mean strictly associative here.  I really do mean associate of rings spectra.  So, okay, so we show that you can, you can define a norm in that way, or another way of saying that is well that's like saying we could view topological Hawkshield homology, really we should think of this as an equivalent norm.  It's the norm from the trivial group to s one.  And then what is the theorem here well the theorem of these same people angle by Bloomberg, Gerhard Hill, Lawson Mandel is that we show that if ours co fibrant.  Sorry.  For our co fibrant.  So, we show that this definition of topological Hawkshield homology as the norm, which is the cyclic bar construction definition actually does have a cyclotomic structure.  For many years this was a question, can you put a cyclotomic structure on the cyclic bar construction, and it turns out that using this work of hill Hopkins rabanel and these norm functors you can indeed do so.  So in other words there's a cyclotomic structure on the cyclic bar construction. After all,  in the chat about can we define the s one norm on bear spectra with no ring structure no s one norm in order to be a sensible construction does need to input an associative ring spectrum so that is a bit different than what happens in the classical case, where the norm from  h to G for finite groups inputs in each spectrum, not necessarily an H ring spectrum. Yeah, good question.  Okay, so and then let me mention that there's a subsequent theorem of Dotto, Milkovich.  patch Korea.  So gov a and woo.  And what do they show well so I've just said that the cyclic bar construction definition of THH has a cyclotomic structure. You'd want to know that it's the same or equivalent to the cyclotomic structure that on the box that model right that we're getting the same theory of topological  technology out of these. And that's what Dotto Milkovich patch Korea so gov a and woo show. So they show that the cyclotomic structure that we construct on the cyclic bar construction agrees with or is equivalent to  to the one on box that's model, the one that we've used historically for K theory applications.  And it's further nice because you know we talked on Tuesday about how Nicholas and Schultz also have this new framework for studying K theory and I didn't say much about their model of THH but they construct THH in an infinity categorical setting as a cyclic bar  construction type construction.  And Nicholas Schultz to compare their cyclotomic structure also to boxchats. So Dotto Milkovich patch Korea so gov a and woo's results tell you that all three versions of THH have equivalent cyclotomic structures.  So that's nice.  Okay, so I'm claiming now that we could think of topological Hock shield molybdenum is an equivalent norm and why is that the nice way to think about it.  One reason that that's a nice way to think about it is that it lends itself to some nice generalizations. So I want to mention one of those generalizations now, which is coming out of this same work.  So we make the following generalization. Let's say we want to study an equivariant ring spectrum so a CN ring spectrum.  And then we can define a CN twisted version of topological Hock shield homology.  So I'm going to write this like this topological Hock shield homology the CN topological Hock shield homology of R.  And then well what should this be. Well recall that I just said that ordinary topological Hock shield homology is supposed to be a norm from the trivial group to ask one.  So I'm going to be writing my THH something that already has a cyclic group action. So what is the topological Hock shield homology of that going to be. Well I claim that it's the norm from CN to S one of R.  Now, the next natural question is but what is what does that even mean right. I've told you how to define the norm from the trivial group to S one but what is the norm from the cyclic groups the end to S one.  So I'm going to construct this well in the commutative case you could set you could define it as a left adjoint to restriction functor. But if you want to take input that's not necessarily commutative you need to give a more concrete construction.  So how do we construct this norm.  So you can do this using a cyclic bar construction but not a classical cyclic bar construction we're going to have to use a variant of the cyclic bar construction.  So I want to use a variant of the cyclic bar construction and I'm going to write this variance in the following way I'm going to write it as be sick CN.  This is going to be a CN twisted version of the cyclic bar construction.  So at first it's going to seem similar to what we did before. So on the Qth level this thing is going to be Q plus one copies of our ours a spectrum so they're smashed together.  And it has the usual degeneracies. So the usual degeneracies by which I mean they just insert a unit in the correct spot after the I coordinate.  So face maps are a bit different. So in order to define the face maps for you. I need to first introduce a piece of notation.  So I'm going to let G denote the generator.  I'm going to put it into the two pi I two pi I over N of CN. And then I'm going to let alpha Q be a map on the Qth level so from my Q plus one copies of our to itself.  And this operator alpha Q it does two things.  The first is that it cyclically permutes the last factor to the front.  And the second is that it acts on the new first factor.  By this element little G.  So I should have mentioned, maybe up here that when I take this cyclic bar construction that my input are is now a CN ring spectrum.  So let me draw. Maybe let me make a little schematic here so I have my Q plus one copies of our and what does this alpha Q do well it takes the last one.  It wraps it around to the front.  Now it's the new first factor and it acts on it by this generator little G, which makes sense because ours and equivariance spectrum.  Okay, that was supposed to be by way of telling you what the face maps are so what are the face maps. Well they're defined as follows.  A lot of them are just the same old thing that they were before the I face map, most of them are just multiplication of the I, and I plus first factors.  As long as I is less than Q.  What is the last one well the last one is something different now. So the last one I'm going to define to be let's do this operator alpha Q.  Bring that last factor around the front act on it by little G and now I'm going to multiply the first two factors together, which is also a map D zero.  Okay, so that's my new last face map.  And the last note is that I claim that this is still a simple shell object meaning that you know we have these simple shell identities that we need to check, and you can check that the simple shell identities are still satisfied.  So this is is simple shell.  But while you're checking identities you might check for those cyclic identities is that a cyclic object and you'll learn quickly that this is simple shell but not cyclic.  And this is like bad news, because if you remember, it was the fact that the cyclic bar construction was cyclic that when we geometrically realized we got an S one action.  Now I'm claiming that this thing is supposed to have an S one action because I'm wanting it to be the norm to S one.  So how do I understand why it would still have an S one action.  So you can check it doesn't satisfy the prop satisfy those identities, but it does have additional structure. So let's see what kinds of things are true.  Well this operator alpha q, it generates a CN q plus one action in simplical degree q.  And that is that well alpha q is both rotating the q plus one factors and acting by a generator of CN so that's going to generate a CN times q plus one action in simple shell degree to q.  And further the face and degeneracy maps satisfy some properties.  So we already said by definition if I do alpha q followed by D zero that was my definition of DQ.  And if you do alpha q followed by di for some other di where I is between one and q that what you get is di minus one alpha q minus one.  And similarly, the alpha satisfies some properties, some relations with respect to the degeneracy switch I won't write down.  So if you look down all of those relations what you'll see is that it turns out that this is an example of a familiar object. So this defines what is called a lambda and op object in the sense of box dead.  Shang and Madsen.  And similarly this structure came up in box dead Shang and Madsen's work on topological cyclic homology in a different way. It came up because they were studying edgewise subdivisions, and, and this kind of structure naturally arises in that context as well.  If n equals one, a lambda and op object is just a cyclic object. So this is a generalization of what it means to be cyclic. And the nice thing about the fact that box dead Shang and Madsen have already studied this kind of object in depth is that we can steal some stuff that we know from them.  So in particular box dead Shang and Madsen prove that when you geometrically realize this kind of object.  So the geometric realization of this kind of object still has an S one action.  Okay, which is good news for us because we were hoping to have such an S one action.  So what's the definition of this twisted topological Hock shield homology well the definition is as follows that the CN twisted topological Hock shield homology of my CN spectrum are, well I said it's supposed to be the norm from CN to S one of our, and I claim that that  the norm can be constructed as this twisted cyclic bar construction on our.  There is a question to you. Can you say something about the universal property satisfied by the modified cyclic bar construction. Yeah, that's a good question.  Not off the top of my head. Yeah, I'm sorry I should be able to answer that and it's just it's not in my brain right now.  If you're interested in learning more about the this kind of like lamb and op object, the place to look for like a lot of understanding of that object is the box of Chang Madsen paper where they originally defined the cyclic atomic trace.  I don't, I don't have.  Right, I'm sorry, I just can't I can't get there right now with the universal property characterization.  So, okay, so I claim that this twisted cyclic bar construction is a construction of this, this norm from CN to S one. And in particular we show that when you restrict to the commutative case that this this twisted cyclic bar construction is the left adjoint to the forgetful  functor in the way that you would want. So it has the properties that characterize a norm in the commutative case.  I should also note in the interest of honesty that in this definition that I've written down here, I'm omitting some what we call change of universe functors to sort of make the technical equivariance stable homotopy theory correct.  And so if you are an expert in that area and are looking for those. They're there I just didn't write them. And if you don't know about change of universe functors I would just for these purposes ignore it.  Okay, so this is CN twisted topological Hawkshield homology and then a question that you might immediately come to about this is, well, THH with cyclic atomic and that was important to the story so is this twisted THH still cyclic atomic.  So we prove the following that for our co vibrant.  And if P is prime to N, then the CN twisted topological Hawkshield homology of R is P cyclic atomic.  I don't think I defined P cyclic atomic when we talked about cyclic atomic spectra on Tuesday but P cyclic atomic just means that you only check those cyclic atomic conditions at the prime P. So it's like specific at the prime P.  And therefore we can define CN twisted versions of topological cyclic homology as well.  So that's nice. The next question I might ask about this theory at first is, well, can you actually compute this twisted cyclic twisted topological Hawkshield homology of anything.  So is CN twisted topological Hawkshield homology computable.  And what might you want even want to try to compute so maybe it's nice to have an example in mind of like what kind of thing would it be interesting to try to understand. Well, for instance, we could ask, can we understand the C to twisted THH of the spectrum MUR.  So what is MUR.  Well, MUR is the C to equivalent.  Real bored is a spectrum.  This was defined by land Weber and Fuji but it's gotten a lot of attention in recent years because it played a really fundamental role in the solution to the curve and very one problem.  So this is this is a particular C to aquavarian spectrum that there's a lot of interest in.  So you see that there's a question in the Q&A which is, is there a description of the CN relative THH in terms of a factorization homology type construction. Yeah, that's a great question. So, for those who are familiar, ordinary topological Hawkshield homology can be described in terms of the factorization  homology of David Ayala and John Francis. The CN twisted THH is has been described by Asapora in terms of his theory of aquavarian factorization homology. So yes, he gives a characterization of these this relative THH in terms of an  aquavarian version of factorization homology. Yeah. Okay, so that's the kind of example that we we might want to understand.  Now, if you think about this, you'll realize that, you know, we talked a lot on Tuesday about how to compute topological cyclic homology.  But I was always sort of assuming in that discussion that we understood THH to begin with, like we described it inductive proper process to build off of THH to understand its fixed points, or in the Nikolaus Schultz model, we understood THH but then we would study its homotopy fixed  points or its Tate construction.  I didn't talk at all about how to actually compute THH. So I've said very little about that. So before I can talk about this question is this twisted THH computable, we need to take a step back and talk about, well, how do you compute ordinary THH.  So let's say something about that.  How do we compute ordinary THH.  The ordinary topological Hawkshield homology is really, you know, the starting point for sort of modern trace methods. If you can't compute THH of the object you're interested in, you're not going to be able to compute topological cyclic homology or algebraic k-theory.  It all starts with THH.  So one of the main tools for computing ordinary topological Hawkshield homology is called the Boxstead spectral sequence.  And what is the Boxstead spectral sequence?  And in the following way, topological Hawkshield homology, remember, was a realization of a cyclic object, which I've been writing as the cyclic bar construction.  Now, when you have a cyclic object like that and you study its realization, you get a spectral sequence induced by the skeletal filtration.  So that's a standard tool, the skeletal filtration induces a spectral sequence that's going to converge to the homology of the spectrum THH with coefficients in some field.  Now, what is the E2 term of that spectral sequence?  Well, what Boxstead proved, which is really, I mean, just really nice, is that when you look at the spectral sequence that you get from the skeletal filtration, on E2 you get something familiar.  You get ordinary Hawkshield homology. So the E2 term here is the Hawkshield homology of the homology of R.  So this is, I think, so beautiful that if you want to study this topological theory of topological Hawkshield homology, we get this spectral sequence whose E2 term is living in the algebraic analog, which is easier to compute.  I mean, Hawkshield homology has all the tools of homological algebra at your disposal. So it's something that's much more computable than this topological theory.  So Boxstead constructed this spectral sequence and did some beautiful calculations with it right off the bat. So Boxstead computed the topological Hawkshield homology of FP and also the topological Hawkshield homology of the integers.  And these, I mean, Boxstead did this work quite a few years ago now, but these calculations, particularly the topological Hawkshield homology of FP, are still foundational to so much work we do in K-theory today.  So many of those calculational results that I mentioned on Monday take as input Boxstead's work on THH.  Okay, so the Boxstead spectral sequence is very powerful and has been foundational to calculations. And so one question is, well, what does this mean in our setting?  So I'd want to have an equivariate version of this.  Of this Boxstead spectral sequence for twisted THH.  That was probably the easiest way or one direct way to get a handle on on calculations here.  Okay, so I think about that and I think, well, what does that mean to want that? Well, my Boxstead spectral sequence should compute the topological theory, should compute some homology of twisted THH and that E2 term should be the algebraic analog of twisted THH.  And then we realize, we have no idea what that is, right? What is the algebraic analog? So what is the algebraic analog  of twisted THH?  Well, it's not immediately obvious what that should be, right? In classical theory, we started from the algebra and we made a topological construction analogous to it. Now we've generalized that.  And it's not clear anymore what algebra that comes from.  So maybe we should revisit what it meant to be the algebraic analog in the classical case and that will hopefully provide us some inspiration.  So what did it mean in the classical case? Well, in the classical case, we were looking at rings and we had a relationship between topological Hock-Schild homology and ordinary Hock-Schild homology.  That was the linearization map.  And we remember that this is notation for THH of the Eilenberg-McLean spectrum.  Now, I haven't mentioned so far, but in the classical theory, not only do you have this linearization map relating these two, but it's also the case that in degrees zero, it's an isomorphism.  So this linearization map in degrees zero is an isomorphism.  So I'd like some analogous story with my twisted THH. I'd like to understand how it relates to some algebraic analog.  But if we look at this classical story, you know, I took Hock-Schild homology of a ring and it was related to THH of the Eilenberg-McLean spectrum.  And so that brings me to a question which is, well, now I need my input for my twisted theory needs to be equivariant.  And so the question is, how do I get a C and ring spectrum as an Eilenberg-McLean spectrum?  I'm going to need to do that in order to make sense of this analog.  Or a different way maybe of phrasing that question is, what is the equivariant analog of a ring?  Okay, so I need to get at those questions if I'm going to be able to understand this kind of equivariant analog.  Okay, so we're going to take a little detour to talk about some basic objects and equivariant homotopy theory that we haven't actually heard much about yet this week.  And they're called Machyfunctors.  So this is going to seem like a detour for a second and it's going to bring us back to this question of what is the equivariant analog of a ring.  Now, if you've never seen a Machyfunctor before, the thing that you should have in your head about Machyfunctors is that Machyfunctors are like the abelian groups of equivariant stable homotopy theory.  So what do I mean by that?  Well, in ordinary homotopy theory, we have a lot of invariants that give us abelian groups. In equivariant homotopy theory, we have a lot of invariants that give us Machyfunctors.  So what is a Machyfunctor?  Well, I'm going to let my group be finite. So for G finite, a Machyfunctor, M, is actually a pair of functors.  So I'm going to call one of them M lower star and one of them M upper star, and they're functors from finite G sets to abelian groups.  One of them is covariant and one of them is contra variant.  Okay, so I have these two functors.  And the, the,  they have to satisfy a few properties. So one is that I, the functors have to agree on objects.  So M lower star of X has to agree with M upper star of X and that shared value is called M underbar of X. So when you see these underbars, that's indicating that we're working with Machyfunctors.  Tina, I need, yes.  I'm sorry to have interrupted you. There is a question. What goes wrong if you try to copy the definition of C and twisted THH for G equivariance Z module spacer.  I think that the question is that in the ordinary case of topological Hock-Schild homology, you can define topological Hock-Schild homology like a relative version of Hock-Schild, topological Hock-Schild homology for module spectra.  And if you do relative T that I'm using the, okay, if you do relative THH for HZ modules in the classical case you get back the algebraic theory of Hock-Schild homology.  And the question here is, why can't I do this twisted version for HZ module spectra and would that give me back what I want.  You know, I have to admit that I have not thought about the relative, like the A relative version of the twisted theory.  So I don't.  I don't have a good answer off the top of my head of what kinds of considerations you would need to take there. I just haven't thought through.  Thought through how that that we haven't defined that object and I haven't thought through how that definition would work.  And another question from Sean Tilson.  Oh yeah so Sean says you get some chukla stuff because of the tensor product being derived.  So, well I haven't gotten there yet but I'm going to talk about this equilibrium theory of Hock-Schild homology and that can be generalized to this, this version of chukla homology as well.  It's not defined in the way that was just mentioned as like thinking of this as a very twisted things over HZ module spectra.  And I have not thought about whether that's equivalent.  Okay so right I was saying what a Machi functor is so a Machi functor is a pair of functors they have to agree on values. They need to take disjoint union  to direct some  and they have to satisfy some axioms that I'm not going to write today.  So, if you haven't seen Machi functors before one nice sort of diagrammatic thing to keep in mind about Machi functors is the following.  In particular, if you have some nested subgroups of your group G, so I have K sitting inside H sitting inside G, we have a projection map right from G mod K to G mod H.  So what happens with the Machi functor? Well the Machi functor these G mod K and G mod H those are finite G sets.  So I get a value for the Machi functor at G mod H.  I get a value for the Machi functor at G mod K.  And I have a covariant functor and a contravariant functor relating them so I get maps in both directions.  The covariant functor we usually call the transfer from K to H and the contravariant functor is often referred to as the restriction from K to H.  And it turns out any finite G set is a direct sum of these orbits, these things of the form G mod K. And so characterizing what happens on these orbits really tells you what happens with the whole Machi functor.  Now, we've actually seen a Machi functor already even though we didn't put it in that terminology we've been working with a Machi functor sort of all week, which is the following.  If you have X, a G spectrum, you get a G Machi functor, which is the homotopy Machi functor of X.  So I want to specify for you what is this homotopy Machi functor do on an orbit G mod H. And this is supposed to give me some abelian group and what does it give me well it gives me the end homotopy group of the H fixed points of my spectrum X.  So, on Tuesday all that time we were studying fixed points of THH in particular we were studying a Machi functor. And when I say Machi functors are like the abelian groups of a covariant homotopy theory this is kind of what I have in mind.  In ordinary homotopy theory my homotopy groups are going to spit out a billion groups, and an equivalent homotopy theory, the homotopy, the natural way to think about homotopy of an equivalent spectrum is as a Machi functor.  So Machi functor constructions are very closely tied to equivalent spectra. So let me note that if you have a G Machi functor.  Let's call it M. It has an Eilenberg Maclean spectrum attached to it which is a G spectrum.  And we write that as H M. And in what sense is it Eilenberg Maclean well it's a G spectrum so I can take its homotopy Machi functor.  And what do I get out in degrees zero, I get my Machi functor back.  And in all other degrees I get zero. So that's the sense in which this is Eilenberg Maclean.  Okay so that's nice to a Machi functor I can associate an Eilenberg Maclean Equivariant spectrum.  So we're going to need a notion of norms for these Machi functors. So, Hill and Mike Hill and Mike Hopkins, give a definition of what it means to take a norm of a Machi functor.  And they say the following, if H is a subgroup of my finite group G, and M is an H Machi functor.  So I'm going to define what it means to take the norm from H to G of the Machi functor M.  And here's their definition. Their definition is, okay so I have my Machi functor, and I just said I could take an Eilenberg Maclean spectrum associated to it now that's an H spectrum.  So I have a norm in Equivariant spectra that Hill Hopkins Rabinel norm. So I can take the norm from H to G in spectra. Now I have a G spectrum, but I wanted a G Machi functor. And so to get back to Machi functors I can take Machi functor Pi zero of that.  So plus minus on this definition of Machi functor norms is the following. On the upside, it's nice to define it really highlights the close relationship between Machi functors and the Equivariant theory of G spectra.  And if you want to prove theorems about the Machi functor norm, this is often the definition that you use.  The minus of this definition is if you want to actually compute the norm of a specific Machi functor, this is very difficult to get a handle on a computation this way.  So there are much more algebraic constructions of the norm in Machi functors that are due, for instance, to Kristen Mazzur and Rolf Hoyer.  And they have more hands on approach of understanding this without going through the Equivariant stable homotopy theory.  Okay, but the Hill Hopkins definition is clean and can be useful for us.  Another thing we need to note about this is that there's a symmetric monoidal product on this category of G Machi functors which is given by what's called box product.  So what is the box product of two G Machi functors.  Well, this box product one way of saying what it is is it's closely related to the product in symmetric in Equivariant spectra.  So if you take my island berg McLean spectrum of M, my island berg McLean spectrum of n, smash them together to get a G spectrum and then take Machi functor Pi zero to get a G Machi functor.  There is a question.  Other contra variant maps on PIM some sort of averaging over H mod the mod K.  So I've been maybe a little sloppy is the wrong word neglectful up here.  So when I talked about this homotopy Machi functor I just told you what it does on each on these orbits.  A homotopy Machi functor is more than just of course the information of what happens to the finite G sets it's also these contra variant functors these transfer and restriction maps.  In the context of the homotopy Machi functor how do we think about that.  So one of them, the restriction map is a nice easy to describe map that's a map given by inclusion of fixed points so it's actually confusingly the map that we called f earlier in the week not the map we called our.  There's a clash of notation there but but that map is inclusion of fixed points the other map is in this context sometimes called the well maybe not in this exact context but it's what's called the Aquavariant transfer map.  And the way to think about that map maybe is that it comes from.  What's called the birth Mueller isomorphism and Aquavariant homotopy theory tells you that you have these kinds of maps as well but it's sort of a unique thing to being in the Aquavariant setting.  So you use the birth Mueller isomorphism and also some duality of these orbits to talk about that transfer map.  Um, yeah, good question.  Okay, so this box product I've given you a definition of the box product and then this is similar to what I was saying about the norms which is that this is a definition that goes through Aquavariant stable homotopy theory.  Machi functors are really algebraic objects you're supposed to think of them as living in algebra and lots of people in math use Machi functors that are not interested in stable homotopy theory.  Machi functors are useful for studying representation rings and other things.  Um, so you can define the box product totally algebraically.  But I'm giving you this characterization to show the relationship with the G spectra.  Okay, so now I have this symmetric model product on G Machi functors and finally I'm ready to address the question of what is an Aquavariant ring.  Well an Aquavariant Abelian group is a Machi functor and Aquavariant ring is called a green functor.  So a green functor is a monoid in this category.  And then the note is that if I have a green functor for a green functor are its Eilenberg McLean spectrum is a, well, for a green functor are for CN, it's Eilenberg McLean spectrum is a CN ring spectrum.  So one of our questions was well how do we get an Aquavariant ring spectrum as an Eilenberg McLean spectrum and the answer is using green functors.  So what do I really want for that Aquavariant Analog well it turns out that what I need is a theory of Hawkshield homology for green functors.  Okay, and that theory of Hawkshield homology for green functors is defined using that same kind of construction of a twisted cyclic bar construction.  So we can do the same construction that we did for spectrum now we can do it in the context of these green functors so where we had a ring spectrum before replace it with my green functor where I had smashed products I replace it with box products, but you do that same twisted construction and it makes sense here.  And so what is the definition then of Hawkshield homology for green functors. Well this comes out of work of Andrew Blumberg, myself, Mike Hill and Tyler Lawson.  So the definition is the following. If you have h inside g inside s one in our an h green functor.  We look at the G twisted Hawkshield homology of our.  And we prove that it's the homology of a twisted cyclic bar construction on the norm from h to G of R the Mackie functor norm for our.  Okay, so, oh yeah so is the box products symmetric yeah this is the symmetric monoidal structure in this category.  So the symmetric monoidal structures.  In order to talk about that you really want to be working more with 10 borrow functions and not just with Mackie functors and I'm kind of, I don't want to go there right now.  But yes it is a symmetric monoidal structure and if you're working with some borrow structures, the tomorrow functors you get even more than that.  So, I claim that this is the equivariant, or sorry the algebraic analog of my twisted THH and the theorem is that well we have a linearization map.  Relating these theories so if I look at the H twisted THH of my Eilenberg McLean spectrum.  So if I take its homotopy Mackie functor that maps to this twisted version of topological Haxial Tomology for green functors.  And this is further in isomorphism if K is equal to zero, which is what we wanted to see from the, which is what we wanted to see from the perspective of what happens in the classical case.  So one of our goal was to define an equivariant version of the boxed at spectral sequence so another like point of proof that this is the right algebraic analog would be if you had a boxed at spectral sequence, computing twisted THH with E2 term in this Haxial Tomology for green functors.  And indeed there is such a spectral sequence. So in work of Catherine Adamic.  Myself, Catherine has in bar client and Hannah Gekong.  So that we construct such an equivariant spectral boxed spectral sequence.  So we construct an equivariant boxed spectral sequence for twisted THH.  And it has E2 term in the Haxial Tomology for green functors.  And it turns out that this Haxial Tomology for green functors really is, you know, the right algebraic analog that you're looking for. And it turns out that this spectral sequence can be used computationally.  So part part of this work of these authors I just mentioned, is that we use this equivariant boxed at spectral sequence to compute the equivariant homology of the C2 twisted THH of MUR.  And we use this equation that we learned earlier with coefficients in what's called the constant Mach-e-Funkter F2.  So this is an equivariant version of homology. If you're not familiar with that, I'm not going to dive into what exactly that means, but that's the natural notion of homology to consider in this equivariant setting.  Okay, I'm almost out of time, but I want to, I want to close by addressing one question. I want to sort of bring this full circle.  In the beginning of the week, we were talking about k theory of rings, and now I've been talking about these equivariant analogs of THH.  And so a question you might have is, well, can these equivariant theories tell us anything about the classical story? So can we learn about the classical story this way?  And so I want to connect it back. So what does this tell us about the classical story?  Here's one thing to say about it. Well, why would there be any connection? So here's one reason we might expect to have a connection to the classical story.  The classical story was about rings. And now I've moved into this world of green functors and Mach-e functors, but a ring is actually a green functor.  A classical ring is a green functor for the trivial group.  Okay, so what does that mean? Well, it means that we get some new trace maps out of this story.  So I had a trace map from algebraic k theory to topological Hock-Shield homology, we lifted through the fixed points.  And if you use this new linearization map relating the equivariant THH to this twisted version of Hock-Shield homology, what you find is that you get a trace map from the algebraic k theory of a classical ring to the Cp to the n twisted Hock-Shield homology of that ring,  evaluated at the orbit Cp to the n mod Cp to the n.  So I'm not going to unpack how you get that trace map exactly, but it follows directly from that linearization map that we had a moment ago.  So what is this thing? Well, the way to think about this is that this is the algebraic analog of fixed points of THH.  So this is a purely algebraic object that is going to serve as an analog of fixed points of THH.  Now, in order for those fixed points of THH to be useful in order to study TC, I needed to not only know about the fixed points themselves, but I needed to know about those two operators on them, that f and that r.  The f map is already part of the Mach-E-Funkter. It's like built into this story automatically because it is the restriction map in those Mach-E-Funkters.  So I don't have to worry about that, but the r map is something outside the Mach-E-Funkter structure. The r map, which was confusingly called the restriction in this context, that was the map that depended on the cyclotomic structure.  And once you have that r map in the classical theory, there is an object called topological restriction homology, which is what you get when you take the limit across the r maps of THH.  You can frame it this way on Monday, but this TR, this topological restriction homology is like between the fixed points of THH and topological cyclic homology. It's one of the things you compute on the way.  So what I'd like to know is, well, can I do this in the algebraic setting? Do I have an analog of that restriction map, the map that came from the cyclotomic structure?  And so what we show in Blumberg, myself, Hill, and Lawson is that you do get such an analog of this restriction map. So we define geometric fixed points for Mach-E-Funkters.  And we show that you get a type of cyclotomic structure on the Hock-Shield homology of Green-Funkters.  And in particular,  Can you get this tracing out via the universal property of algebraic k-theory as in Blumberg, Gevner, Tabawata?  I'd have to think about whether there's a way to characterize it in terms of the, probably, I'd have to think about whether there's a way to characterize it in terms of the universal properties.  That's not how we define it. We define it directly through the trace from k-theory, through the topological denistrace, basically, and the fact that the topological denistrace lifts through fixed points.  But I have not thought about whether there is a universal characterization of it. So I'm not sure.  Right. So the last thing I was saying is that we define what it means to take geometric fixed points for Mach-E-Funkters. We prove that there's a type of cyclotomic structure on Hock-Shield homology for Green-Funkters.  And what it gives you is it gives you an algebraic version of TR, which we call little TR.  And what is this? Well, it's some limit over these algebraic versions of the restriction map of these C-P to the N-twisted Hock-Shield homology for the ring A, evaluated at this orbit.  And as an example, you could compute, for instance, the algebraic TR of Fp.  So we do this calculation and what goes into that? Well, you need to understand the Mach-E-Funkter norms on Fp, the twisted cyclic bar construction on Fp, and then the cyclotomic structure for that.  And it turns out what you see is that you get the p-addicts in degrees zero and zero everywhere else.  And so in this case, this algebraic approximation is really a good approximation because that agrees exactly with the topological restriction homology, the topological theory, and the p-completion of algebraic k-theory of Fp.  Okay, so in this case of Fp, it captures, you know, all the information basically, that of course in general will not be true. Some algebraic analog of this topological theory.  Okay, I'm a little bit over time, so I'm going to stop there and it has been a pleasure to be with you this week and I hope that I have given, especially the early queer people, some idea about what trace methods and k-theory is about and some sort of interesting  developments that have been happening in this area. So I will stop there.  Many thanks indeed. And let's thank this Tina for a wonderful mini course.  There is a question.  Can you say roughly what the bread and homology of THH, M-U-R is? And does it split into pieces where some of these summands are familiar?  Yeah, I should know how to split it into pieces. So I can tell you what the, you know, if I look it up, I can tell you what the answer is.  I still remember when I was a PhD student many years ago now, the first time that I asked my advisor something that was in a paper that he wrote, and he said,  oh, I don't know, I'll look it up. And that was like for me a very liberating moment that even my advisor like didn't remember everything he'd ever written down.  So I don't feel bad about looking up this answer in my own paper.  Right, so what is the answer for the C2-equivariate homology of the topological hawkshield homology of M-U-R? So this is the C2-twisted hawkshield homology.  So in this case, we get a really nice answer. So I haven't, okay, there were a few things I didn't say. So one of the things I didn't say is that that  equivariate box-ded spectral sequence, when you start working in these equivariate worlds, you end up having multiple gradings. You get a Z graded spectral sequence. So a spectral, you get Z graded theories where they're graded by the integers like we're used to. And you also get theories graded by representation rings.  And so it turns out that the representation ring object is the more natural thing to consider in many cases. So what I'm writing down is the ROC2 graded equivariate homology.  And for those of you who are not, who are new to equivariate stable homotopy theory, let me introduce you to a really special convention, which is that a five-pointed star is usually an equivariate grading.  That means a representation, and an asterisk is an integer grading. So something useful to know. So that grading is now graded by representations. And what this is, is it's the Eilenberg-McLean, the equivariate  homotopy groups of the Eilenberg-McLean spectrum of F2 with polynomial generators on that. And then it's box over HF2 star of the exterior thing over HF2 star of Z1, Z2, etc.  So I don't know if that was helpful or not. The degree of BI is I times the representation, the regular representation, and the degree of Zi is one plus that.  So that's what the answer is for the equivariate homology of that. And I don't know if you find that helpful or not.  But yeah.  Okay, so one way of seeing, so the question is, one way of seeing topological Hock-Schild homology is as a relative smash product over a  tensor A op, and is there an analog of this for this CN twisted THH? Yes, you can think of the CN twisted THH in terms of these relative smash products.  And maybe the thing to say is that, so if I'm interested in, let's say the H twisted THH of R, and I'm interested in that as like a G spectrum.  So if I'm, I think I did that in the opposite direction I meant to. If I want to look at the G spectrum, restrict that to a G spectrum, then you can write this as  a G spectrum from H to G of R smash over that thing op. So the, no, sorry, that the enveloping thing, the that smash that op.  But then what you have over here is you have a twisted version in of the norm. So yes, there is a way to characterize it in terms of these relative smash products but you pick up a little bit of a twist so it's a bit different than the classical  one. The next question I see is, is there a trace map from the equivariant k theory to this twisted THH? Yeah, that's a great question.  So that's in a very natural question to ask is like, well, I've talked about, you know, I, at the end I was saying, well, you can get trace maps out of a ring but can we naturally get trace maps from some kind of  equivariant version of k theory to this twisted THH. The answer to that is yes. I have work in progress with some of the people I mentioned, Catherine Adamic, Catherine Haas,  and I'm going to ask, in Mark Lang and Hannah Gekong, where we're looking at like, what is the right kind of k theory in order to get that sort of trace map and maybe I won't say too much about that since this work in progress. I don't want to make any bold claims yet.  I don't have like known notions of equivariant k theory. So a lot of people have considered different notions of equivariant algebraic k theory, and I don't yet have like a connection between the CN relative THH and those different known theories of  equivariant k theory. Is there a reason we don't have to use a drive limit for TR? I mean, so this limit that I'm talking about here, these things that I am taking the limit of now are all just a billion groups and so here it is just like an ordinary  group of billion groups that ends up being the right thing to take their, I don't know, maybe that's not a very satisfying answer, but that's what's happening in this case.  Let me read your question. If G is a finite group acting on a commutative ring R, can we cook up a G timbara functor? Perhaps with the fixed point.  G is a finite group acting on a commutative ring R.  I'm not sure exactly what you're asking. So you want like a group acting on a classical ring, can we cook up a G timbara functor?  I don't know off the top of my head.  Yeah, I'm not sure what the answer to your question is. I apologize.  What kind of marine invariance property is this twisted THH have? That's another great question.  So the, right, so these Hock-Shield theories in general, Hock-Shield homology, topological Hock-Shield homology, any sort of Hock-Shield theory, one thing that you might want to ask for is that it's marine invariance, right?  That's something that's really common amongst Hock-Shield theories and something that's sort of important to those theories.  So in some cases you can prove that directly, but there's a recent work of coming out of work of Cape Ponto and Ponto Shulman, and now there's a larger group of collaborators, Ponto, Campbell and Ponto, etc.  So we talked about Hock-Shield homology and topological Hock-Shield homology as bi-categorical shadows. And that shadow approach, whatever that means, I don't want to go into what that means, but it's, we heard a little bit about it in the first lecture this morning, if you were there, that kind of idea of a shadow.  If you know that THH is a shadow, then marine invariance comes for free, because it turns out marine invariance is like a natural notion of equivalence on bi-categories.  So the question is about twisted THH, and this is a great question.  In that same work in progress that I mentioned of Adamic and myself and Hauss and Clang and Kong, we have shown that you can view this twisted topological Hock-Shield homology as sort of an equivariant shadow.  So in particular, you get marine invariance also in this case for free. And so, yeah, it aligns nicely with what you'd expect from one of these topological Hock-Shield, or Hock-Shield theories and you do get marine invariance.  So that's a nice property to know that you have of twisted THH.  Any other questions or comments to Tina?  If not, then, merci beaucoup for everything for a wonderful mini-course.  Thanks.  Thank you Tina again.
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