3/3 Algebraic K-theory and Trace Methods
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Twin primeMusical ensembleAlgebraSquare numberCategory of beingObject (grammar)Network topologyHomologieRing (mathematics)Coordinate systemSpectrum (functional analysis)Algebraische K-TheorieCartesian coordinate systemTensorLengthElement (mathematics)ChainSummierbarkeitSphereMultiplication signDivisorPositional notationMany-sorted logicPoint (geometry)Smash-ProduktOperator (mathematics)TensorproduktAnalogyHomotopiegruppeMaß <Mathematik>TheoryAlgebraic structureModel theorySet theoryApproximationKozyklusMereologyIntegerCorrespondence (mathematics)Energy levelAbelsche GruppeCuboidComputer animation
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Gamma functionIcosahedronGroup actionGeometryTheoremHill differential equationEquivalence relationNormal (geometry)TheoryCurveConnected spaceÄquivariante AbbildungTable (information)Spectrum (functional analysis)MereologySeries (mathematics)FunktorArithmetic meanDifferent (Kate Ryan album)Point (geometry)CommutatorModel theoryObject (grammar)Algebraic structureStability theoryMany-sorted logicHomologieRing (mathematics)Network topologyFunctional (mathematics)KreiskörperMonoidEndliche GruppeBuildingDiagonalSubgroupObservational studyModulformAngleMultiplication signClassical physicsRight angleCategory of beingFinitismusMoment (mathematics)HomotopieContent (media)Computer animationMeeting/Interview
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DivisorOperator (mathematics)Alpha (investment)Spectrum (functional analysis)MultiplicationElement (mathematics)Network topology2 (number)HomologieRing (mathematics)PermutationMaß <Mathematik>Order (biology)Positional notationEnergy levelCartesian coordinate systemPearson product-moment correlation coefficientFunktorNormal (geometry)Classical physicsKreiskörperVarianceZyklische GruppeGroup actionEndliche GruppeStatistical hypothesis testingAlgebraic structureGastropod shellFiber bundleRange (statistics)ResultantEquivalence relationModel theoryInfinityMorley's categoricity theoremTheoryCuboidRight angleGoodness of fitTheoremIdentical particlesObject (grammar)Arithmetic meanAnalogyAlgebraische K-TheorieÄquivariante AbbildungNatural numberDifferent (Kate Ryan album)Coordinate systemComputer animation
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Hydraulic jumpExploratory data analysisCategory of beingSpectrum (functional analysis)Group actionRight angleObject (grammar)AutocovarianceAlpha (investment)Universe (mathematics)Normal (geometry)MathematicsReal numberLatent heatHomologieNetwork topologyHomotopieÄquivariante AbbildungDivisorDegree (graph theory)TheoryTerm (mathematics)Condition numberMultiplication signFunktorAlgebraic structureKreiskörperOperator (mathematics)Theory of relativityAreaDescriptive statisticsStability theoryDirection (geometry)Perturbation theoryComputabilityMusical ensembleRing (mathematics)CuboidAlgebraische K-TheorieSmash-ProduktTopologischer RaumMany-sorted logicAnalogyArithmetic meanTensorSequencePrime idealApproximationGroup representationMoment (mathematics)Complete metric spacePolynomialGenerating set of a groupCalculationLimit (category theory)Point (geometry)Receiver operating characteristicAlgebraGreen's functionGoodness of fitOrder (biology)Intercept theoremArithmetic progressionProduct (business)Classical physicsIdentical particlesAdditionStudent's t-testComputer animation
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Twin primeMaß <Mathematik>Abelsche GruppeHomotopieFunktorDegree (graph theory)Linear mapTheoryObject (grammar)Analogy2 (number)SequenceTerm (mathematics)Spectrum (functional analysis)Äquivariante AbbildungMany-sorted logicCoefficientRing (mathematics)Invariant (mathematics)Homologische AlgebraPoint (geometry)Order (biology)AlgebraComputabilityHomologieNetwork topologyLinearizationArithmetic meanPositional notationCalculationClassical physicsAlgebraische K-TheorieProcess (computing)ResultantDivisorIntegerFinitismusGroup actionStability theoryLimit (category theory)Theory of relativityConnected spaceCuboidMusical ensembleArithmetic progressionNatural numberEquivalence relationCategory of beingDifferent (Kate Ryan album)Acoustic shadowMorley's categoricity theoremFree groupRight angleEndliche GruppeIsomorphieklasseVarianceKörper <Algebra>Model theoryComputer animation
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Gamma functionQuadrilateralCategory of beingFunktorDegree (graph theory)OrbitSpectrum (functional analysis)FinitismusSet theorySummierbarkeitDirection (geometry)Heat transferÄquivariante AbbildungTheoryGroup actionProjective planeDivisorDiagramSubgroupMany-sorted logicAxiomMoving averageEndliche GruppeHomologieModule (mathematics)ComputabilityNormal (geometry)Object (grammar)Latent heatTheoremGastropod shellNetwork topologyClosed setTensorproduktAbelsche GruppeAlgebraHomotopieTheory of relativityPerturbation theoryRight angleHill differential equationMultiplication signPoint (geometry)ModulformHomotopiegruppeComputer animation
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Heat transferFunktorSpectrum (functional analysis)HomotopieSequenceTheoryOrder (biology)MonoidAlgebraic structureInclusion mapProduct (business)CuboidTerm (mathematics)Many-sorted logicGreen's functionStability theoryCategory of beingTheoremAnalogyMathematicsSymmetric matrixObject (grammar)Point (geometry)Proof theoryHomologieÄquivariante AbbildungGoodness of fitOrbitIsomorphieklassePerspective (visual)MereologyRing (mathematics)Linear mapSet theoryNetwork topologyGroup representationDuality (mathematics)Abelsche GruppeNormal (geometry)Positional notationComputabilityHill differential equationDivisorAlgebraWave packetComputer animation
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Twin primeSigma-algebraGame theoryPoint (geometry)SequenceAnalogyFunktorCoefficientOrder (biology)Algebraic structureSpectrum (functional analysis)Term (mathematics)CircleApproximationSet theoryCategory of beingUniverse (mathematics)Complete metric spaceTheoryLogical constantNatural numberRing (mathematics)Äquivariante AbbildungAlgebraische K-TheorieNetwork topologyAlgebraOperator (mathematics)Green's functionMereologyConnected spaceHomologieMultiplication signClassical physicsMany-sorted logicDegree (graph theory)Linear mapLimit (category theory)KreiskörperObject (grammar)OrbitGroup actionNormal (geometry)CalculationMoment (mathematics)Fiber bundleAlgebraic varietyComputer animation
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Invariant (mathematics)Spectrum (functional analysis)Acoustic shadowNormal (geometry)Direction (geometry)Term (mathematics)Group representationGroup actionMany-sorted logicCategory of beingFree groupDegree (graph theory)Theory of relativityTensorConnected spaceOrder (biology)HomologieNetwork topologyCuboidArithmetic progressionGenerating set of a groupEquivalence relationNatural numberPolynomialRight angleÄquivariante AbbildungLimit (category theory)AlgebraRing (mathematics)FunktorGradientCommutatorMoment (mathematics)Endliche GruppeAlgebraische K-TheorieAnalogyAbelsche GruppeMultiplication signStudent's t-testTheoryDifferent (Kate Ryan album)Smash-ProduktHomotopiegruppeObject (grammar)IntegerSequenceComputer animation
Transcript: English(auto-generated)
00:17
Thank you. Since this is my last lecture, I also wanted to take the opportunity to thank
00:23
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 want to do today is I want
00:45
to talk a little bit more about topological Hochschild 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
01:00
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-theory, there's a map relating the algebraic K-theory of the ring and the Hochschild homology of the ring, and that map was called
01:25
the Dennis Trace. And that map was sort of the starting point for the whole trace method and algebraic K-theory approach. So this Hochschild 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
01:40
just so it's fresh in our memory as we talk a little bit more deeply about it. So what is Hochschild 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,
02:02
this is just q plus one copies of my ring A tensored together, and it had some face into generacy 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.
02:24
So for the most part, these face maps just send this to ai times ai plus one, like so. And that makes sense, as long as i is less than q. But remember that last face map brought the last element around to the front and then multiplied.
02:44
So that last face map sends this to aq a0 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. Okay, so that was what our degeneracies did.
03:04
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.
03:22
Okay, so we defined this cyclic bar construction on Monday, and then I said, well, what is Hochschild homology? Well, Hochschild 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
03:41
of these face maps. You can check that that squares to zero. We can take homology, and that's called Hochschild homology. And we also said, well, by the Doldkhan correspondence, I could alternatively define this as the homotopy groups of the geometric realization of this simplicial object.
04:02
Okay, so that was Hochschild 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,
04:21
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,
04:42
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 Hochschild homology. And topological Hochschild homology is related to K-theory via a trace map as well.
05:04
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.
05:26
Okay, and what was the idea, the rough idea, of how to define topological Hochschild 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 ring spectra.
05:41
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,
06:03
the topological Hochschild homology of R is the geometric realization of the cyclic bar construction on R. And then we said when we talk about topological Hochschild homology of rings, for a ring A, THH of A is just notation for topological Hochschild homology
06:24
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 Hochschild homology is an S1 spectrum.
06:41
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 were called 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 Boxstedt who first, excuse me,
07:04
it was Boxstedt who first constructed topological Hochschild homology. But Boxstedt did this many years ago, and he didn't have some of the nice luxuries that we have today. So in particular, when Boxstedt made this construction, he didn't have nice categories of spectra with an associative smash product.
07:23
So what I've written here is this idea. Today we can execute that quite literally and define topological Hochschild homology as this cyclic bar construction. But at the time Boxstedt originally constructed THH, you couldn't do that so literally, because those tools just didn't exist yet.
07:42
So if you look back at Boxstedt's construction, what we now call the Boxstedt 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 Boxstedt smash. But the interesting thing is that for years, up until very recently, for K-theory applications,
08:03
we continued to use Boxstedt'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 Boxstedt's THH was cyclotomic,
08:21
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 homotopy theory, and that's sort of the starting point for the new material I want to talk about today,
08:41
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 curverin variant 1 problem. So in particular, in the context of their work on curverin variant 1,
09:03
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,
09:23
building on earlier work. So there was earlier work on equivariant norms due to Greenleece and May. Okay, so let me say a little bit about these norms in equivariant stable homotopy theory. So what's the idea? Well, let's say that we have G, a finite group,
09:43
and H, a subgroup of G. The norm functors that Hill-Hopkins and 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,
10:01
that goes from H spectra to G spectra. So it's a functor that takes its input in H spectrum and it gives you out a G equivariance spectrum. These are symmetric monoidal functors. There's a very nice thing about them. And in the commutative case, they have a nice characterization.
10:21
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 ring H spectrum, so a commutative object in H spectra, that what you get out is you get a commutative G ring spectrum.
10:47
And further, this norm functor in the commutative case is left adjoint to the restriction functor. I'll call it IH star.
11:03
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. OK, so these norms in equivariance stable homotopy theory,
11:22
it turns out that by studying these deeply, Hill, Hopkins, and Ravenel were able to get a handle on the Carverian 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 Hochschild homology or this trace method story?
11:40
I mean, the first question I want to 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 Ravenel. And Hill, Hopkins, and Ravenel proved the following. They proved that if R is co-fibrant and G is finite,
12:06
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
12:22
and then take the G geometric fixed points of that. See, and we received a couple of questions to you. The first question's about do these norms exist for general G or only for finite G? Oh, that's a great question. I feel like I almost planted that question. So Hill, Hopkins, and Ravenel
12:41
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?
13:01
Yeah, I mean like actual commutative monoids in these categories of equivariant spectra. So this is like a genuine notion of commutative in genuine equivariant homotopy theory. Okay, so what was I saying, right?
13:21
So the Hill, Hopkins, and Ravenel 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,
13:40
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 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.
14:01
So this feels reminiscent of 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 finite? So in work of Wieglich-Angeltweit,
14:23
Andrew Blomberg, 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.
14:43
So we show that you can extend this. You can extend norms to consider a functor, the norm from the trivial group to S1. Now, this makes sense if what you start with is an associative ring spectrum.
15:05
And this is gonna spit out an S1 spectrum. So what is this norm functor that we construct? Well, the claim is that the norm from the trivial group to S1 should be viewed as the functor that takes a ring spectrum
15:21
and sends it to the geometric realization of its cyclic bar construction. Okay, 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 gonna see this thing as the left adjoint of the forgetful functor from S1 spectra
15:42
down to associative ring spectra. I see there's a question. I mean strictly associative here. I really do mean associative ring spectra. So 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
16:00
topological Hochschild homology. Really, we should think of this as an equivariant 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, Engelpike, Blumberg, Gerhard, Hill, Lawson, Mandel, is that we show that if R is co-fibrant,
16:26
for R co-fibrant, that this definition of topological Hochschild as the norm, which is the cyclic bar construction definition, actually does have a cyclotomic structure.
16:47
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-Rabinel and these norm functors, you can indeed do so.
17:00
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,
17:21
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,
17:41
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 want to know that it's the same or equivalent to the cyclotomic structure
18:03
on the Boxted 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
18:20
agrees with, or is equivalent to, to the one on Boxted'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 Niklaus and Schultze
18:41
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 Niklaus and Schultze compare their cyclotomic structure also to Boxted's. So Dotto, Malkovich, Pachkoria, Sagave, and Wu's results
19:02
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 Hochschild 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
19:21
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 generalizations. Let's say we want to study an equivariant ring spectrum, so a CN ring spectrum.
19:43
Now we can define a CN twisted version of topological Hochschild homology. So I'm going to write this like this. Topological Hochschild homology, the CN topological Hochschild homology of R. And then, well, what should this be?
20:02
Well, recall that I just said that ordinary topological Hochschild 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 Hochschild homology of that going to be? Well, I claim that it's the norm from CN to S1 of R.
20:24
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 group CN to S1? So how can we construct this? Well, in the commutative case, you could define it as a left adjoint
20:42
to a 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? Well, I claim that you can do this using a cyclic bar construction,
21:00
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 variant in the following way. I'm going to write it as BSicCN. This is going to be a CN twisted version
21:21
of the cyclic bar construction. So here's how we define this twisted 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 R, R is a spectrum, so they're smashed together. And it has the usual degeneracies.
21:41
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 going to let g denote the generator
22:02
e to the 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 R to itself. And this operator alpha q,
22:20
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,
22:41
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
23:01
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 equivariant spectrum.
23:21
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-th face map, most of them, are just multiplication of the i-th and i plus first factors.
23:42
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 to the front, act on it by little g. And now I'm gonna multiply the first two factors together,
24:02
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, you know, we have these simplicial identities that we need to check. And you can check that the simplicial identities are still satisfied.
24:21
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
24:41
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? Well, it's not cyclic. You can check. It doesn't satisfy those identities,
25:02
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 simplicial degree q.
25:20
Why 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 simplicial degree q. And further, the face and degeneracy maps satisfy some properties.
25:42
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,
26:00
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
26:24
a Lambda-n-op object in the sense of Bocksted, Sheng, and Madsen. So interestingly, this structure came up in Bocksted, Sheng, and Madsen's work
26:41
on topological cyclic homology in a different way. It came up because they were studying edge-wise 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.
27:03
And the nice thing about the fact that Bocksted, Sheng, and Madsen have already studied this kind of object in depth is that we can steal some stuff that we know from that. So in particular, Bocksted, Sheng, and Madsen prove that when you geometrically realize this kind of object,
27:20
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
27:42
topological Hochschild homology? Well, the definition is as follows. That the CN twisted topological Hochschild homology of my CN spectrum R, 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
28:02
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.
28:22
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 lambda n object, the place to look for like a lot of understanding of that object is the Bakshat-Sheng-Matson paper
28:41
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
29:00
is a construction of this, 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
29:20
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 equivariant stable homotopy theory correct. And so if you are an expert in that area and are looking for those,
29:40
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 Hochschild homology. And then a question that you might immediately come to about this is, well, THH was cyclotomic and that was important to the story. So is this twisted THH
30:01
still cyclotomic? So we proved the following that for R-cofibrin and if p is prime to n, then the Cn twisted
30:20
topological Hochschild homology of R is p-cyclotomic. Now, I don't think I defined p-cyclotomic when we talked about cyclotomic spectra on Tuesday, but p-cyclotomic just means that you only check those cyclotomic conditions at the prime p. So it's like specific at the prime p.
30:40
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,
31:02
twisted topological Hochschild homology of anything? So is Cn twisted topological Hochschild 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,
31:21
we could ask, can we understand the C2 twisted THH of the spectrum MUR? So what is MUR? Well, MUR is the C2 equivariant real boardism spectrum.
31:41
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 variant 1 problem. So this is, this is a particular C2 equivariant spectrum that there's a lot of interest in.
32:01
I 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 Hochschild homology can be described in terms of the factorization homology of David Aiella and John Francis.
32:21
The, this Cn twisted THH is, has been described by Asaf Horev in terms of his theory of equivariant factorization homology. So yes, he gives a characterization of these, this relative THH in terms of an equivariant version of factorization homology.
32:41
Yeah. Okay, so that's the kind of example that 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
33:00
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 Nagelau-Schulze 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.
33:21
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?
33:41
Ordinary topological Hochschild 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
34:01
Hochschild homology is called the Bocksted spectral sequence. And what is the Bocksted spectral sequence? Well, it works in the following way. Topological Hochschild 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
34:21
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
34:40
with coefficients in some field. Now, what is the E2 term of that spectral sequence? Well, what Bocksted 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
35:00
Hochschild homology. So the E2 term here is the Hochschild homology of the homology of R. So this is, I think, so beautiful that if you want to study this topological theory of topological Hochschild homology, we get this spectral sequence whose E2 term is living in the algebraic analog,
35:24
which is easier to compute. I mean, Hochschild 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 Bocksted constructed this spectral sequence and did some beautiful calculations with it right off the bat.
35:41
So Bocksted computed the topological Hochschild homology of Fp and also the topological Hochschild homology of the integers. And these, I mean, Bocksted did this work quite a few years ago now, but these calculations, particularly the topological Hochschild homology of Fp,
36:01
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 Bocksted's work on THH. Okay, so the Bocksted spectral sequence is very powerful and has been foundational to calculations.
36:20
And so one question is, well, what does this mean in our setting? So I'd want to have an equivariant version of this, of this Bocksted spectral sequence for twisted THH. That was probably the easiest way or one direct way to get a handle on calculations here.
36:43
Okay, so I think about that and I think, well, what does that mean to want that? Well, my Bocksted 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
37:02
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?
37:21
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.
37:40
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 Hochschild homology and ordinary Hochschild homology. That was the linearization map. And we remember that this is notation for THH of the Eilenberg-McLean spectrum.
38:02
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 degree zero, it's an isomorphism. So this linearization map in degree zero is an isomorphism. So I'd like some analogous story
38:21
with my twisted THH. I'd like to understand how it relates to some algebraic analog. But if we look at this classical story, I took Hochschild homology of a ring and it was related to THH of the Eilenberg-McLean spectrum. And so that brings me to a question,
38:40
which is, well, now 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 going to need to do that in order to make sense of this analog. Or a different way,
39:01
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
39:22
kind of equivariant analog. Okay, so we're going to take a little detour to talk about some basic objects in equivariant homotopy theory that we haven't actually heard much about yet this week. And they're called Mackey functors. So this is going to seem like a detour for a second.
39:40
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 Mackey functor before, the thing that you should have in your head about Mackey functors is that Mackey 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
40:00
that give us abelian groups. In equivariant homotopy theory, we have a lot of invariants that give us Mackey functors. So what is a Mackey functor? Well, I'm going to let my group be finite. So for G finite, a Mackey functor, M, is actually a pair of functors.
40:24
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 contravariant.
40:41
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
41:00
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 Mackey functors. Tina, sorry to have interrupted you. There is a question. What goes wrong if you try to copy the definition of C and twist,
41:21
THH for G equivariant Z modules, Petra? Right, so I think that the question is that in the ordinary case of topological Hochschild homology, you can define topological Hochschild homology
41:40
like a relative version of topological Hochschild homology for module spectra. And if you do relative THH for HZ modules in the classical case, you get back the algebraic theory of Hochschild homology. So I think the question here is why can't I do this twisted version
42:00
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
42:20
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 Shukla stuff
42:41
because of the tensor product being derived. Yeah, so well, I haven't gotten there yet, but I'm going to talk about this equivariant theory of Hochschild homology and that can be generalized to this version of Shukla homology as well. It's not defined in the way that was just mentioned as like thinking of this as equivariant twisted things
43:00
over HZ module spectra. And I have not thought about whether that's equivalent. Okay, so right, I was saying what a Mackey functor is. So a Mackey functor is a pair of functors that have to agree on values. They need to take disjoint union to direct sum
43:21
and they have to satisfy some axioms that I'm not going to write today. So if you haven't seen Mackey functors before, one nice sort of diagrammatic thing to keep in mind about Mackey functors is the following. So in particular, if you have some nested subgroups of your group G,
43:41
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 Mackey functor? Well, the Mackey functor, these G mod K and G mod H, those are finite G sets. So I get a value for the Mackey functor at G mod H. I get a value
44:01
for the Mackey 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
44:21
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 Mackey functor. Now, we've actually seen
44:40
a Mackey functor already, even though we didn't put it in that terminology, we've been working with a Mackey functor sort of all week, which is the following. If you have X, a G spectrum, you get a G Mackey functor, which is the homotopy Mackey functor of X.
45:03
So I want to specify for you, what is this homotopy Mackey 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.
45:22
So on Tuesday, all that time, we were studying fixed points of THH. In particular, we were studying a Mackey functor. And when I say Mackey functors are like the abelian groups of equivariant homotopy theory, this is kind of what I have in mind. In ordinary homotopy theory, my homotopy groups are going to spit out abelian groups.
45:40
And in equivariant homotopy theory, the homotopy, the natural way to think about homotopy of an equivariant spectrum is as a Mackey functor. So Mackey functor constructions are very closely tied to equivariant spectra. So let me note that if you have a G Mackey functor,
46:02
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 Mackey functor.
46:22
And what do I get out? In degree zero, I get my Mackey 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 Mackey functor, I can associate an Eilenberg-McLean equivariant spectrum.
46:41
And we're also going to need a notion of norms for these Mackey functors. So Mike Hill and Mike Hopkins give a definition of what it means to take a norm of a Mackey functor. And they say the following. If H is a subgroup of my finite group G, and M is an H Mackey functor,
47:05
they wanted to define what it means to take the norm from H to G of the Mackey functor M. And here's their definition. Their definition is, okay, so I have my Mackey functor and I just said I could
47:20
take an Eilenberg-McLean spectrum associated to it. Now that's an H spectrum. I have a norm in equivariant 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 Mackey functor. And so to get back to Mackey functors, I can take Mackey functor pi zero of that.
47:43
So the plus minus on this definition of Mackey functor norms is the following. On the upside, it's nice to define. It really highlights the close relationship between Mackey functors and the equivariant theory of G spectra. And if you want to prove theorems about the Mackey functor norm, this is often the definition
48:00
that you use. The minus of this definition is if you want to actually compute the norm of a specific Mackey 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 Mackey functors that are due, for instance, to Kristin Mazur and Rolf Heuer.
48:21
And they have more hands-on approach of understanding this without going through the equivariant. Stable homotopy theory. OK, 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
48:41
monoidal product on this category of G Mackey functors, which is given by what's called box product. So what is the box product of two G Mackey functors? Well, this box product, one way of saying what it is, is it's closely related to the product in equivariant spectra.
49:00
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 Mackey functor pi zero to get a G Mackey functor. Tina, there is a question. Are the contravariant maps on P i n
49:20
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 Mackey functor, I just told you what it does on these orbits. A homotopy Mackey functor
49:40
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 Mackey functor, how do we think about that? Well, one of them, the restriction map is a nice, easy to describe map.
50:01
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 the, well, maybe not in this exact context,
50:20
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 birth, Mueller 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 birth,
50:40
Mueller 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.
51:02
Maki functors are really algebraic objects. You're supposed to think of them as living in algebra, and lots of people in math use Maki functors that are not interested in stable homotopy theory. Maki functors are useful for studying representation rings and other things. So you can define the box product totally algebraically, uh, but I'm giving you
51:22
this characterization to show the relationship with the G-spectra. Okay, so now I have this symmetric monoidal product on G Maki functors, and finally I'm ready to address the question of what is an equivariant ring. Well, an equivariant abelian group is a Maki functor, and equivariant ring is called a Green functor.
51:42
So a Green functor is a monoid in this category. And then the note is that if I have a Green functor, or a Green functor R, its Eilenberg-McLean spectrum is a, well, I'll say it,
52:01
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
52:21
for that equivariant analog? Well, it turns out that what I need is a theory of Hochschild homology for Green functors. Okay, and that theory of Hochschild homology for Green functors is defined using that same kind of construction
52:41
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 replace it with my Green functor, where I had smash products, I replace it with box products. But if you do that same twisted construction,
53:01
and it makes sense here. And so what is the definition then of Hochschild 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,
53:23
and H Green functor, we look at the G twisted Hochschild 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.
53:42
The Mackie functor norm for R. Okay, so is the box product symmetric? Yeah, this is the symmetric monoidal structure in this category. G symmetric monoidal structures.
54:01
In order to talk about that, you really want to be working more with Tambora functors and not just with Mackie functors. And I think 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 Tambora structures, Tambora functors, you get even more than that. Okay, so I claim that
54:22
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, and I take its homotopy Mackie functor
54:42
that maps to this twisted version of topological Hochschild homology for Green functors. And this is further in isomorphism if K is equal to zero, which is what we wanted to see from the perspective
55:03
of what happens in the classical case. So our goal, part of our goal was to define an equivariant version of the box-dead spectral sequence. So another like point of proof that this is the right algebraic analog would be if you had a box-dead spectral sequence computing twisted THH with E2 term in this
55:21
Hochschild homology for Green functors. And indeed there is such a spectral sequence. So in work of Catherine Adamic, myself, Catherine Hess, Inbar Klang, and Hannah Jia Kong, we show that we construct such an equivariant box-dead spectral sequence. So we construct
55:41
an equivariant box-dead spectral sequence for twisted THH. And it has E2 term in the Hochschild homology for Green functors. So that's saying that this Hochschild homology for Green functors really is, you know, the right algebraic analog
56:01
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 box-dead spectral sequence to compute the equivariant homology of the C2 twisted THH
56:21
of MUR, that example I mentioned earlier, with coefficients in what's called the constant Maki functor 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,
56:41
but I want to close by addressing one question. I want to 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?
57:00
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? 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.
57:21
And now I've moved into this world of green functors and maki 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
57:42
out of this story. So I had a trace map from algebraic K-theory to topological Hochschild homology, we lifted through the fixed points. And if you use this new linearization map relating the equivariant THH
58:01
to this twisted version of Hochschild 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 Hochschild homology of that ring evaluated at the orbit CP to the N mod CP to the N.
58:20
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.
58:41
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,
59:00
that F and that R. The F map is already part of the Maki functor. It's like built into this story automatically because it is the restriction map in those Maki functors. So I don't have to worry about that. But the R map is something outside the Maki functor structure. The R map, which was confusingly
59:20
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,
59:41
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?
01:00:00
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 Mackey functors.
01:00:27
And we show that you get a type of cyclotomic structure in the Hochschild homology of Green functors. And in particular, can you get this trace map
01:00:46
via the universal property of algebraic K-theory as in Blumberg, Geppner, Tebawada? I'd have to think about whether there's a way to characterize it in terms of the, probably.
01:01:01
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
01:01:21
is a universal characterization of it. So I'm not sure. So the last thing I was saying is that we define what it means to take geometric fixed points for Mackey functors. We prove that there's a type of cyclotomic structure on Hochschild homology for Green functors. And what it gives you is it gives you
01:01:41
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 Hochschild homology for the ring A evaluated at this orbit.
01:02:06
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 functor norms on FP, the twisted cyclic bar
01:02:21
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 degree 0 and 0 everywhere else. And so in this case, this algebraic approximation is really a good approximation because that agrees exactly
01:02:40
with the topological restriction homology, the topological theory, and the P completion of algebraic K theory of FP. So in this case of FP, it captures all the information, basically. That, of course, in general will not be true. So that's some algebraic analog of this topological theory.
01:03:02
OK, 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've given, especially the early queer 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.
01:03:20
So I will stop there. Many thanks indeed. Let's thank Tina for a wonderful mini-course. There is a question. Can you say roughly what the Breton homology of THH-MUR is?
01:03:43
And does it split into pieces where some of these surmands are familiar? Tina? Yeah, I should know how to split it into pieces.
01:04:04
So I can tell you what the, 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.
01:04:21
And he said, oh, I don't know. I'll look it up. And that was, for me, a very liberating moment that even my advisor 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 equivariant homology of the topological Hochschild homology of MUR?
01:04:44
So this is the C2 twisted Hochschild homology. So in this case, we get a really nice answer. So I haven't, OK, there were a few things I didn't say. So one of the things I didn't say is that that equivariant boxed spectral sequence, when you start working in these equivariant worlds,
01:05:02
you end up having multiple gradings. You get a z-graded spectral sequence. So 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.
01:05:21
So what I'm writing down is the ROC2-graded equivariant homology. And for those of you who are new to equivariant stable homotopy theory, let me introduce you to a really special convention, which is that a five-pointed star is usually
01:05:40
an equivariant 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 equivariant homotopy groups of the Eilenberg-McLean spectrum of F2 with polynomial generators on that.
01:06:04
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 regular representation.
01:06:24
And the degree of zi is 1 plus that. So that's what the answer is for the equivariant homology of that. And I don't know if you find that helpful or not. But yeah. Right, OK. So the question is, one way of seeing topological Hochschild
01:06:43
homology as a relative smash product over a tensor AOP. 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.
01:07:02
Maybe the thing to say is that 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
01:07:21
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, no, sorry, that enveloping thing,
01:07:40
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, is there
01:08:01
a trace map from the equivariant K theory to this twisted THH? Yeah, that's a great question. So that's a very natural question to ask is like, well, I've talked about, 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
01:08:20
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 Hess, Inbar Klang, and Hana Jia Kong, 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 it's a work in progress, I don't
01:08:41
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 equivariant K theory. So a lot of people have considered different notions of equivariant algebra K theory. And I don't yet have like a connection between the CN relative THH and those different known
01:09:03
theories of equivariant K theory. Is there a reason we don't have to use a derived 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 abelian groups.
01:09:23
And so here it is just like an ordinary limit of abelian 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 Yuri's question.
01:09:40
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.
01:10:04
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?
01:10:24
I don't know off the top of my head. Yeah, I'm not sure what the answer to your question is. I apologize.
01:10:41
What kind of marine invariance properties does twisted THH have? That's another great question. So these Hochschild theories in general, Hochschild homology, topological Hochschild homology, any sort of Hochschild theory, one thing that you might want to ask for is that it's meridian invariance, right? That's something that's really common
01:11:01
amongst Hochschild 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 K. Ponto and Ponto-Schulman, and now there's a larger group of collaborators, Campbell and Ponto, et cetera, where they've talked about Hochschild homology
01:11:20
and topological Hochschild homology as bicategorical shadows. And that shadow approach, whatever that means, I don't want to go into what that means, but 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 meridian invariance comes for free because it turns out meridian invariance
01:11:41
is like a natural notion of equivalence on bicatteories. 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 House and Klang and Kong, we've shown that you can view
01:12:02
this twisted topological Hochschild homology as sort of an equivariant shadow. And so in particular, you get meridian invariance also in this case for free. And so, yeah, it aligns nicely with what you'd expect from one of these topological Hochschild theories, and you do get meridian invariance. So that's a nice property to know
01:12:22
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, and thank Tina again.