 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-theory and trace methods. So what I wanna do today is I wanna talk a little bit more about topological Hawkshield homology. We talked a lot about topological cyclic homology on Tuesday, but now I wanna dig in a little bit deeper about THH itself. To get started, I wanna recall some things that were said already earlier in the week, but that will be very important for us today. So I just wanna 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-theory, there's a map relating the algebraic k-theory of the ring and the Hawkshield 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-theory approach. So this Hawkshield homology is gonna be very important for us today. So I know that I've already defined it, but I wanna recall the definition just so it's fresh in our memory as we talk a little bit more deeply about it. So what is Hawkshield homology? Well, remember that we defined a simplicial 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 wanna 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. Well, 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 AQA0, tensor A1 through AQ 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 simplicial 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 doldukon correspondence, I could alternatively define this as the homotopy groups of the geometric realization of this simplicial object. Okay, so 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 simplicial 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 dentist trace. So that first map, the map between K theory and THH is often referred to as the topological dentist trace, or sometimes just the dentist 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 gonna replace those with ring 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 R, a ring spectrum, the topological Hawkshield homology of R is the geometric realization of the cyclic bar construction on R. And then we said, 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 S1 spectrum. Okay, 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 recalled on Monday, or I'm recalling things from Monday, excuse me. But one thing to note about this, about the history of this, is that it was Boxsted who first, oh, excuse me, it was Boxsted who first constructed topological Hawkshield homology. But Boxsted did this many years ago, and he didn't have some of the nice luxuries that we have today. So in particular, when Boxsted made this construction, he didn't have nice categories of spectra with an associative smash 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 Boxsted originally constructed THH, you couldn't do that so literally, because those tools just didn't exist yet. So if you look back at Boxsted's construction, what we now call the Boxsted model of THH, you see that he developed a lot of machinery to work around that. He developed what we now refer to as the Boxsted smash. But the interesting thing is that for years, up until very recently, for K theory applications, we continued to use Boxsted's model, even though it makes perfect sense now with the current technology to give this kind of cyclic bar construction definition. And why is that? Well, the reason is because it was known that Boxsted'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? Now, in recent years, this story has changed a bit because of major advances in equivariant homotropy theory. And that's sort of the starting point for the new material I wanna talk about today is the 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. So 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 wanna say a little bit about these norms. So as I said, the norms as I'm gonna talk about them today are due to a 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 that. Okay, so let me say a little bit about 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 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 commutative object in H spectra, that what you get out is you get a commutative G ring spectrum. Okay. And further, this norm functor in the commutative case is left adjoint to the restriction functor. I'll call it IH 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, 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 Rabinel weren't able to get a handle on the Caverin variant 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 you, the first question I wanna address is why would you even think that there would be a connection there? So the first hint that there might be a connection there comes out of a theorem of Hill, Hopkins and Rabinel. And Hill Hopkins and Rabinel prove the following. They prove that if R is co-fibrant and G is finite, there is an equivalence as follows. They construct a map from R to what you get when you take the norm from the trivial group to G of R and then take the G geometric fixed points of that. Tina, we received a couple of questions to you. The first questions about do these norms exist for general G or only for finite G? That's a great question. I feel like I almost planted that question. So Hill Hopkins and Rabinel constructed these norms for finite G, but we are in a minute gonna talk about extending it to a non-finite group. So in some general sense, yes, they only exist for finite G and if you wanna 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 the, this category is an equivariant spectra. So this is like a genuine notion of commutative in genuine equivariant homotopy theory. Okay, so what was I saying? So the Hill Hopkins and Rabinel constructed this kind of diagonal map and proved 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 were supposed to be things where you take the geometric fixed points and you get back the original spectrum that you started with. Now that's not exactly what's happening here, right? We have additionally some normed 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 normed functors exist for finite groups. And so you could ask, well, can we do this for a group that's not finite? So in work of Beagleck-Angeltweight, Andrew Blumberg, 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 S1. So we show that you can extend this. You can extend norms to consider a function, functor, sorry, the norm from the trivial group to S1. Now this makes sense if what you start with is an associative ring spectrum and this is gonna spit out an S1 spectrum. So what is this normed functor that we construct while the claim is that the norm from the trivial group to S1 should be viewed as the functor that takes a ring spectrum and sends it to the geometric realization of its cyclic bar construction. Okay, so what is the content of saying that that's a normed functor? Well, the claim is that that behaves like a norm. So in particular, if you restrict to the commutative case, you're gonna see this thing as the left adjoint of the forgetful functor from S1 spectra down to associative ring spectra. I see there's a question. I mean, strictly associative here. I really do mean associative ring spectra. Okay, so we show that 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 equivariate norm. It's the norm from the trivial group to S1. And then what is the theorem here? Well, the theorem of these same people, Anglebyte, Blumberg, Gerhard, Hill, Lawson, Mandel, is that we show that if R is co-fibrant, sorry, for R co-fibrant, that this definition of topological Hawkshield homology as the norm, which is the cyclic bar construction definition, actually does have a cyclotomic structure. So 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-Rabinall 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. I see a question in the chat about, can we define the S1 norm on bare spectra with no ring structure? No, the S1 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 an H 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, Malkovich, Pachkoria, Sagave, and Wu. 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 wanna know that it's the same or equivalent to the cyclotomic structure on the Boxed model, right? That we're getting the same theory of topological cyclic homology out of these. And that's what Dotto, Malkovich, Pachkoria, Sagave, and Wu show. So they show that the cyclotomic structure that we construct on the cyclic bar construction agrees with or is equivalent to the one on Boxed's model. The one that we've used historically for K theory applications. Now that's further nice because, we talked on Tuesday about how Nikolaus 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 Nikolaus Schultz compared their cyclotomic structure also to Boxed's. So Dotto, Malkovich, Pachkoria, Sagave, and Wu'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-Schild homology as an equivariant norm. And why is that the nice way to think about it? Well, one reason that that's a nice way to think about it is that it lends itself to some nice generalizations. So I wanna 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. Now we can define a CN twisted version of topological Hock-Schild homology. So I'm gonna write this like this. Topological Hock-Schild homology, the CN topological Hock-Schild homology of R. And then what should this be? Well, recall that I just said that ordinary topological Hock-Schild homology is supposed to be a norm from the trivial group to S1. Now I'm feeding my THH something that already has a cyclic group action. So what is the topological Hock-Schild homology of that gonna be? Well, I claim that it's the norm from CN to S1 of R. Now, the next natural question is, but what does that even mean, right? I've told you how to define the norm from the trivial group to S1, but what is the norm from the cyclic groups CN to S1? So how can we construct this? Well, in the commutative case, you could set, you could define it as a left adjoint to a restriction functor. But if you wanna take input that's not necessarily commutative, you need to give a more concrete construction. So how do we construct this norm? Well, I claim that 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 wanna use a variant of the cyclic bar construction, and I'm gonna write this variant in the following way. I'm gonna write it as B6 CN. This is gonna be a CN twisted version of the cyclic bar construction. So here's how we define this twisted cyclic bar construction. So at first it's gonna seem similar to what we did before. So on the Q-th level, this thing is gonna be Q plus one copies of R, R's 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-th coordinate. But the 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 gonna let G denote the generator E to the two pi I, two pi I over N of CN. And then I'm gonna let alpha Q be a map on the Q-th level. So from my Q plus one copies of R 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 R is now a CN ring spectrum. Okay, so let me draw maybe, let me make a little schematic here. So I have my Q plus one copies of R 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 R is an equilibrium 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. But what is the last one? Well, the last one is something different now. So the last one I'm gonna 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 gonna multiply the first two factors together which is also the map D zero. Okay, so that's my new last face map. So my first note is that I claim that this is still a simplicial object, meaning that we have these simplicial identities that we need to check. And you can check that the simplicial identities are still satisfied. So this is simplicial. But while you're checking identities, you might check for those cyclic identities. Is it a cyclic object? And you'll learn quickly that this is simplicial but not cyclic. And that initially seems 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 S1 action. Now I'm claiming that this thing is supposed to have an S1 action because I'm wanting it to be the norm to S1. So how do I understand why it would still have an S1 action? Well, it's not cyclic. You can check, it doesn't 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 CNQ plus one action in simplicial degree Q. Why is that? Well, alpha Q is both rotating the Q plus one factors and acting by a generator of CN. So that's gonna generate a CN times Q plus one action in simplicial degree 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. But you can check that 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 degeneracies, which I won't write down. So if you write 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 n op object in the sense of Boxstead, Shang and Madsen. So interestingly, this structure came up in Boxstead, Shang and Madsen's work on topological cyclic homology in a different way. It came up because they were studying edgewise subdivisions and this kind of structure naturally arises in that context as well. So if n equals one, a lambda n 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 Boxstead, 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, Boxstead, Shang and Madsen prove that when you geometrically realize this kind of object, sorry, that geometric realization of this kind of object still has an S1 action. Okay, which is good news for us because we were hoping to have such an S1 action. Okay, so then what is the definition of this twisted topological Hawkshield homology? Well, the definition is as follows. That the CN twisted topological Hawkshield homology of my CN spectrum are, well, I said it's supposed to be the norm from CN to S1 of R. And I claim that that norm can be constructed as this twisted cyclic bar construction on R. Tina, 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 this kind of like lamb and object, the place to look for like a lot of understanding of that object is the Buckshitt-Shang-Madsen paper where they originally defined the cyclotomic trace. But yeah, I don't have, right, I'm sorry, I just 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 norm from CN to S1. And in particular, we show that when you restrict to the commutative case, that 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, equivariate 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 was cyclatomic and that was important to the story. So is this twisted THH still cyclatomic? So we proved the following that for our cofibrant and if P is prime to N, then the CN twisted topological hawkshield homology of R is P cyclatomic. Now, I don't think I defined P cyclatomic when we talked about cyclatomic spectra on Tuesday, but P cyclatomic just means that you only check those cyclatomic 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 even wanna 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 C2 twisted THH of the spectrum MUR? So what is MUR? Well, MUR is the C2 aqua variant real boredism spectrum. This was defined by Landweber 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 covariant rate one problem. So this is a particular C2 aqua variant spectrum that there's a lot of interest in. I see that there's a question in the Q and 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. This CN twisted THH has been described by Asaporov in terms of his theory of aqua variant factorization homology. So yes, he gives a characterization of this relative THH in terms of an aqua variant version of factorization homology. Yeah. Okay, so that's the kind of example that we might wanna understand. Now, if you think about this, you'll realize that 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 an 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? Ordinary topological Hock-Schultz homology is really 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 gonna 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 Hock-Schultz homology is called the Boxstead spectral sequence. And what is the Boxstead spectral sequence? Well, it works in the following way. Topological Hock-Schultz 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 Hock-Schultz homology. So the E2 term here is the Hock-Schultz homology of the homology of R. So this is, I think, so beautiful that if you want to study this topological theory of topological Hock-Schultz homology, we get this spectral sequence whose E2 term is living in the algebraic analog, which is easier to compute. I mean, Hock-Schultz 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 Hock-Schultz homology of FP and also the topological Hock-Schultz homology of the integers. And I mean, Boxstead did this work quite a few years ago now, but these calculations, particularly the topological Hock-Schultz 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 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. It should compute some homology of twisted THH and the 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 a 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 Hawkshield homology and ordinary Hawkshield 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 Hawkshield 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 CN ring spectrum as an Eilenberg-McLean spectrum? I'm gonna 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 this kind of equivariant analog. Okay, so we're gonna 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 Macky Functors. So this is gonna seem like a detour for a second and it's gonna bring us back to this question of what is the equivariant analog of a ring? Now, if you've never seen a Macky Functor before, the thing that you should have in your head about Macky Functors is that Macky Functors 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 Macky Functors. So what is a Macky Functor? Well, I'm gonna let my group be finite. So for G finite, a Macky Functor, M is actually a pair of functors. So I'm gonna 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 contravariant. Okay, so I have these two functors and they have to satisfy a few properties. So one is that 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 under bar of X. So when you see these under bars, that's indicating that we're working with Macky Functors. Yes. 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-modules, Peter? Right, so 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 topological Hock-Schild homology for module spectra. And if you do relative THH for HZ-modules in the classical case, you get back the algebraic theory of Hock-Schild homology. And so I think 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, yeah, 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, 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 shookless stuff because of the tensor product being derived. Yeah, so, well I haven't gotten there yet, but I'm gonna talk about this equilibrium theory of Hock-Schild homology and that can be generalized to this version of shookless homology as well. It's not defined in the way that was just mentioned as like thinking of this as equilibrium twisted things over HZ-module spectra. And I have not thought about whether that's equivalent. Okay, so right, I was saying what a Mackie functor is. So a Mackie functor is a pair of functors that have to agree on values. They need to take disjoint union to direct sum and they have to satisfy some axioms that I'm not going to write today. So if you haven't seen Mackie functors before, one nice sort of diagrammatic thing to keep in mind about Mackie functors is the following. So 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 from G mod K to G mod H. So what happens with the Mackie functor? Well, the Mackie functor, these G mod K and G mod H, those are finite G sets. So I get a value for the Mackie functor at G mod H. I get a value for the Mackie 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 Mackie functor. Now, we've actually seen a Mackie functor already, even though we didn't put it in that terminology, we've been working with a Mackie functor sort of all week, which is the following. If you have X, a G spectrum, you get a G Mackie functor, which is the homotopy Mackie functor of X. So I wanna specify for you what is this homotopy Mackie 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 nth 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 Mackie functor. And when I say Mackie functors are like the abelian groups of aquavariant homotopy theory, this is kind of what I have in mind. In ordinary homotopy theory, my homotopy groups are gonna spit out abelian groups. And an aquavariant homotopy theory, the natural way to think about homotopy of an aquavariant spectrum is as a Mackie functor. So Mackie functor constructions are very closely tied to aquavariant spectra. So let me note that if you have a G Mackie functor, let's call it M, it has an Eilenberg-McLean spectrum attached to it, which is a G spectrum. And we write that as HM. And in what sense is it Eilenberg-McLean? Well, it's a G spectrum, so I can take its homotopy Mackie functor. And what do I get out? In degrees zero, I get my Mackie functor back. And in all other degrees, I get zero. So that's the sense in which this is Eilenberg-McLean. Okay, so that's nice to a Mackie functor I can associate an Eilenberg-McLean aquavariant spectrum. And we're also gonna need a notion of norms for these Mackie functors. So Mike Hill and Mike Hopkins give a definition of what it means to take a norm of a Mackie functor. And they say the following, if H is a subgroup of my finite group G and M is an H Mackie functor, they wanted to define what it means to take the norm from H to G of the Mackie functor M. And here's their definition. Their definition is, okay, so I have my Mackie functor and I just said I could take an Eilenberg-McLean spectrum associated to it. Now that's an H spectrum. I have a norm, an aquavariant spectra, the 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 Mackie functor. And so to get back to Mackie functors, I can take Mackie functor pi zero of that. So the plus minus on this definition of Mackie functor norms is the following. On the upside, it's nice to define. It really highlights the close relationship between Mackie functors and the aquavariant theory of G spectra. And if you wanna prove theorems about the Mackie functor norm, this is often the definition that you use. The minus of this definition is if you wanna actually compute the norm of a specific Mackie 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 Mackie functors that are due, for instance, to Kristen Mazur and Rolf Hoyer. And they have more hands-on approach of understanding this without going through the aquavariant stable homotopy theory. Okay, but the Hill-Hopkins definition is clean and can be useful for us. So one other thing we need to note about this is that there's a symmetric monoidal product on this category of G Mackie functors, which is given by what's called box product. So what is the box product of two G Mackie functors? Well, this box product, one way of saying what it is, is it's closely related to the product in symmetric, in aquavariant spectra. I can take my Eilenberg-McLean spectrum of M, my Eilenberg-McLean spectrum of N, smash them together to get a G spectrum and then take Mackie functor pi zero to get a G Mackie functor. Tina, there is a question. Are the contravariant maps on PIM some sort of averaging over H mod K? Yeah, okay. So I've been maybe a little sloppy is the wrong word, neglectful up here. So when I talked about this homotopy Mackie functor, I just told you what it does on each, on these orbits. A homotopy Mackie functor is more than just, of course, the information of what happens to the finite G sets. It's also these contravariant functors, these transfer and restriction maps. In the context of the homotopy Mackie functor, how do we think about that? Well, 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 R. There's a clash of notation there, but that map is inclusion of fixed points. The other map is in this context sometimes called, well, maybe not in this exact context, but it's what's called the equivariant transfer map. And the way to think about that map maybe is that it comes from what's called the birthmular isomorphism and equivariant 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 equivariant setting. So you use the birthmular isomorphism and also some duality of these orbits to talk about that transfer map. 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 equivariant stable homotopy theory. Macky functors are really algebraic objects. You're supposed to think of them as living in algebra and lots of people in math use Macky functors that are not interested in stable homotopy theory. Macky functors are useful for studying representation rings and other things. 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 monoidal product on G Macky functors. And finally, I'm ready to address the question of what is an equivariant ring? Well, an equivariant abelian group is a Macky functor and equivariant 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 R, its Eilenberg-McLean spectrum is a, for a green functor R for CN, its Eilenberg-McLean spectrum is a CN ring spectrum. So one of our questions was, well, how do we get an equivariant ring spectrum as an Eilenberg-McLean spectrum? And the answer is using green functors. So what do I really want for that equivariant 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 could do the same construction that we did for spectra. Now we can do it in the context of these green functors. So where we had a ring spectrum before, I replaced it with my green functor, where I had smashed products, I replaced 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. And the definition is the following. If you have H inside G inside S1 and R and H green functor, we look at the G twisted Hawkshield homology of R. 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 R. Okay, so, oh yeah. So is the box products symmetric? Yeah, this is the symmetric monoidal structure in this category. G symmetric monoidal structures, in order to talk about that, you really want to be working more with Timbara functors 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 Timbara functors, you get even more than that. Okay, so I claim that this is the equivariant, they're 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 and I take its homotopy Mackie functor, that maps to this twisted version of topological hoaxial 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 our goal, part of our goal was to define an equivariant version of the boxed-ed spectral sequence. So another like point of proof that this is the right algebraic analog would be if you had a boxed-ed spectral sequence computing twisted THH with E2 term in this hoaxial tomology for green functors. And indeed there is such a spectral sequence. So in work of Catherine Adamic, myself, Catherine Haas in Barclang and Hannah Giacong, we show that we construct such an equivariant boxed-ed spectral sequence. So we construct an equivariant boxed-ed spectral sequence for twisted THH and it has E2 term in the hoaxial tomology for green functors. So that's saying that this hoaxial tomology for green functors really is the right algebraic analog that you're looking for. And it turns out that this spectral sequence can be used computationally. So part of this work of these authors I just mentioned is that we use this equivariant boxed-ed spectral sequence to compute the equivariant homology of the C2-twisted THH of MUR. That example I mentioned 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 gonna 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 wanna close by addressing one question. I wanna sort of bring this full circle. So at 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 wanna connect it back. So what does this tell us about the classical story? Well, 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-Schild 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-Schild 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-Schild homology of that ring evaluated at the orbit CP to the N, mod CP to the N. So I'm not gonna 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 gonna 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's an object called topological restriction homology, which is what you get when you take the limit across the R maps of THH. I didn't 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 and the Hock-Schild homology of Green-Funkters. And in particular, Can you get this trace map 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 Dennis trace, basically, and the fact that the topological Dennis trace 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 Mackey-Funkters. We prove that there's a type of cyclotomic structure on Hock-Schild 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 CP to the N twisted Hock-Schild 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 Mackey-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 all the information, basically. That, of course, in general, will not be true. It's some algebraic analog of this topological theory. Okay, I'm a little bit over time, so I'm gonna stop there. And it has been a pleasure to be with you this week, and I hope that I've given, especially the early career people, some idea about what trace methods and k-theory is about and some sort of interesting recent developments that have been happening in this area. So I will stop there. Many thanks indeed. 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 Hock-Shield homology of M-U-R? So this is the C2-twisted Hock-Shield 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, et cetera. 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. Right. Okay. So one way of seeing, so the question is one way of seeing topological homology is 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. 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 it's the norm from H to G of R smash over that thing op. So the, you know, I'm sorry, no, sorry, that the enveloping thing, that smash that op. But then what you have over here is you have a twisted version 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 story. The next question I see is, 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 equivalent 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 Hass, and Mark Lang and Hannah Geocong, where we're looking at like what is the right kind of K theory to get that sort of trace map. And maybe I won't say too much about that since it's work in progress. I don't want to make any bold claims yet, but we have constructed, we're working on a trace map relating that. I don't know how it relates to like known notions of equivalent K theory. So a lot of people have considered different notions of equivalent algebraic K theory. And I, I don't yet have like a connection between the CN relative THH and those different known theories of equivalent K theory. Is there a reason we don't have to use a drive limit for T. R. 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 limit of a billion groups that ends up being the right thing to take there. 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. Sure, exactly what you're asking. 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, I'm not sure what the answer to your question is. I apologize. What kind of read invariance properties this twisted THH have. That's another great question. So the right. So these Hawkshield theories in general, Hawkshield homology, topological Hawkshield homology, any sort of Hawkshield theory. One thing that you might want to ask for is that it's Merida invariance. Right. That's something that's really common amongst Hawkshield theories and something that's sort of important to those theories. In many 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 up Ponto, Campbell and Ponto, et cetera, where they've talked about Hawkshield homology and topological Hawkshield homology as bi categorical shadows. And that shadow approach, whatever that means, I don't want to go into what that means, but, 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 Merida invariance comes for free because it turns out Merida invariance is like a natural notion of equivalence on bi categories. So the question is about twisted THH and is a great question. In that same work in progress that I mentioned of Adamic and myself and House and Clang and Kong, we have shown that you can view this twisted topological Hawkshield homology as sort of an equivariant shadow. And so in particular, you get Merida invariance also in this case for free. And so yeah, it aligns nicely with what you'd expect from one of these topological Hawkshield or Hawkshield theories. And you do get Merida invariance. So that's a nice property to know that you have a twisted THH. Any other questions or comments to Tina? If not, then Merci beaucoup for everything for a wonderful mini course. Thanks. Let's thank Tina again.