 Thank you. Thank you for the to the organizers for inviting me. I was really looking forward to being in Paris right now. It would be the first trip I would take after my son was born and it would be the first solo trip. But instead we're here. He's napping in the room next door and I'm giving this talk to you guys. So thank you for the invitation and thank you for putting this together even in this this remote format. So I'm going to tell you about joint progress, joint work with Bergi Yu, Peter May and Mona Merling and this has been a long-term project and it has many pieces and I'm just going to tell you about this this piece about the multiplicative structure of equivariate infant love space machines and you may not know what many of these words mean. So I'm going to start with with an introduction where I'm going to tell you what what the question is and then I'm going to tell you a little bit about what what the answer we're giving us. So what is a what is an infinite loop space machine? Well we should think about it as some black box. This machine where you have some input and I'll say a little bit more what the input looks like and then the output is an infinite space infinite loop space or an omega spectrum and the idea is is that I mean we we all love spectra but one way in which we can think about a spectrum in which can help us produce this infinite loop spaces is thinking that the spaces are spaces that have a multiplicative structure that is associative and communicative up to all higher homo-topies and that that the idea is try to figure out how to encode that in in ways that can actually help us build spectra this way. So the classical examples from the 70s are seagulls gamma spaces and maize infinity spaces which use the idea of operettes which really were first thought about by board members and seagulls and so seagulls gamma spaces and and the infinity spaces they they encode this idea of having a multiplication that's associative and communicative up to all higher homo-topies in very different ways and gamma spaces so I'll tell you a little bit about gamma spaces in a second but with gamma spaces the idea is that you have a functor from the category of finite pointed sets into spaces and the category of finite pointed sets is encoding that that idea of the multiplication versus infinity spaces which are using an operette so the operette is where you're encoding the the multiplication that is associative and communicative up to all higher homo-topies. Both of these examples were used initially to produce a spectra from symmetric monoidal categories and the idea is that if you have a symmetric monoidal category that it has a multiplication that's associative and commutative up to naturalizedomorphisms and that should become associative and commutative up to higher homo-topies once you go to the classified space so let me just write that so both are used to construct spectra from symmetric monoidal categories and this is one of the the beginnings of of k theory and a key property of this construction is that if you take the zero space of the k theory spectrum of asymmetric monoidal category this is the group completion of the classifying space of the symmetric monoidal category and one thing to that's happening here is that the symmetric monoidal category you might not have inverses but a spectrum has has inverses so because the sun is a loop space in particular so so you need to group complete to to make sense of what's going on there's one question that you might ask which is not one of the ones that I'm going to answer in this talk is whether the construction that you give so you have if you give me a symmetric monoidal category you construct a spectrum using seagull's machine or you construct a spectrum using the operatic machine and you would like to know that those spectra are actually equivalent and the answer is yes and that's a celebrated theorem of mayhem thomason but that's not something that I'm going to talk much about in this talk now another aspect of my title is multiplicative so we might want to uh we might want to know what we mean by a multiplicative infinite loop spaceship so let me start by giving you an example so we have the category of finite sets and we really we take a small model of the category of finite sets and one of the things that we know is we have a barrett pretty quillen theorem which says that if you take give me the k theory of the category of finite sets this is the sphere spectrum um and uh we know that the sphere spectrum is not just a spectrum it is actually a ring spectrum so uh we would like to know where we can get that that multiplicative structure and the ring spectrum coming from um um coming from uh the finite sets and well where is it coming from well um finite sets has two monodal structures it has uh these join union which is kind of like addition of of natural numbers and that's the one that we use uh to construct the sphere spectrum as a classifying space of finite sets but it also has cartesian product and it actually turns out that uh cartesian product distributes uh over these join union up to isomorphism so so there's a distributive property that's happening between these two monodal structures and and uh what we know is that um or maybe we could first ask the question is um does the cartesian product uh induce extra structure on the k theory of finite sets and the answer is yes it induces the multiplicative structure the sphere spectrum and so you might wonder if this is a general thing and there's uh a simple version of this of of of an answer to this this question which is um given independently by elmendorff amandel and may uh which is that the k theory spectrum of uh symmetric by monoidal category is an infinity ring spectrum so what does symmetric by monoidal category means it means precisely a category that has two symmetric monoidal structures one which distributes over the other one up to coherent uh naturalize some more business so sort of like the example of finite sets is a prototypical example but there's other examples for example vector spaces over any field or with this join union as right with uh direct sum and um and tensor product or for example uh um modules over a ring and that's actually what produces k theory as a ring spectrum so we want to do this equivariately so if we forget about multiplicative the first question is well for today we're just going to be working with finite groups so if you give me an equivariate infinitive space machine the first thing that you might want to ask is well what do you want your output to be and here there's there's flavors of of uh equivariate spectra that you might want and for the purposes of this talk we want genuine g-spectra to come out and what that means uh is that we have um uh equivariate infinite spaces and what do i mean by that i.e. we have the loopings with respect to all uh finite dimensional representation spheres and what do i mean by that um that uh for every g or finite dimensional g representation v there exists a space x of v such that your space x is equivalent to the loop space uh with respect to v of x v and what do i mean by that i mean the space of pointed maps from the representation sphere what do i mean by the representation sphere i have my finite dimensional representation i do a one point compactification that's going to be a sphere that has a g action induced by the by the action in the vector space and this space here is the space of all pointed maps from the representation sphere into my g space x v with the g action given by conjugation so this is a a version of loops so it is still just the the um the n dimensional loops where n is the dimension of the of the um this here but it has a g action given by by the representation so so this is what we mean by an equivariate infinite loop space is uh is i basically want the loopings with respect to every possible representation sphere and and of course uh one of the things that you might want to ask as well is what is what is the input and i'm going to tell you a little bit about that um now um there's there's several questions that you might ask at this point one of them is well what are the do we have versions of seagull's gamma spaces and uh infinity operats that work in the setting and the answer is yes for both um the the version of gamma spaces was developed by shimakawa and i'm not going to tell you much more about that because it uh it doesn't really pertain to the story that i'm going to tell you uh today but but it's a very rich theory of how to get the the the gamma spaces to actually work equivariately and uh there is a version of operats which i am going to tell you a little bit more today uh you might want to ask uh is there an equivalence between the two machines the seagullic and the operatic and the answer is yes that's work of jointly with um peter mayan monomerling but again that's not the story that i'm going to tell you today um but the the story that i'm going to tell you today is um what is the categorical input sort of like if you if you want to know how to generalize symmetric monoidal categories into this equivalence setting and another question that that you might ask and that i'm going to tell you about today is about multiplicative structures so that's that's what i'm going to focus on the rest of the talk is answering this these two questions so let's talk about categorical input and in order to tell you about categorical input i first uh need to tell you a little bit about e infinity g over this we're mostly developed by louis may steinberger cos another waiter so um let me just give you the definition and then i'll i'll say a couple of things so an e infinity g operate is an operate in these spaces such that a couple of things the first one uh let's give operate the name let's call it p the jth space of the operate it has a uh an action of the symmetric group and i want that action to be free and the second piece is that for all subgroups lambda of g cross sigma j such that the intersection of lambda with sigma j is trivial we have that the fixed points are contracted now um we have a much better understanding of operas and g spaces now due to the work of of um blumberg and hill and uh and janathan rubin and one thing that we know now is that uh operas in g spaces um might be encoding which transfers we have uh and in particular this this first condition uh sorry the two conditions together tells us that this is an an n infinity operate with all norms basically we have we have for every pair of subgroups of nested subgroups we're going to have maps going the the wrong way uh when we're looking at the structure that we get so that this is this is what this is encoding so uh the idea of of having of having this infinity g structure is that we have sort of like an underlying infinity structure where we have a multiplication that's associated on commutative up to all higher homotopies and on top of that we have this this wrong way maps um so the key theorem here which allows you to construct a spectra out of this is that if y is a p algebra there exists a genuine g spectrum which maybe deserves to be well let's just call it ky such that the zero space of ky is a group completion of y so this is exactly the kind of thing that you want uh meaning that these are the right definition uh for uh operas if what we want is to produce genuine g spectra so this this theorem tells you that that's precisely right and again you need a group completion because y might not have inverses but we know that an infinite loop space should have inverses so that's that's what's happening here now this is not the answer about what's the categorical input because here uh you're starting with an algebra um you're starting with an operating g spaces which means that y is a g space with extra structure but if you actually want to have some categorical input well what you can do is you can go back uh one step and define what it means to have a category called g uh infinity g opera so a categorical infinity g opera uh p is a it's an opera in g categories such that when you take the classifying space uh this is an infinity g opera in spaces in g spaces so um there's a a very uh specific example that i want to look at which is is the following so i'm going to let oh be the non-equivariant with that important disease um categorical barrett eckles opera so what is this uh so the jth piece of this opera is supposed to be a category and i'm going to you know that category as sigma j tilde and i'll tell you what this is so this is a category with objects given by sigma j itself is a metric group and um yeah yeah it's a category just a category with a g action or on each object or on each morphism uh that's a good question so uh a g category is uh a category where uh there's an action by g via functors so you can think about this that g is acting on the set of objects and g is acting on the set of morphisms in a way that is compatible with uh source and target yeah another question is to cut the category of uh categories with a g action yeah so i guess this is like this is a similar question right so so yeah so uh maybe i should i should say so uh when i say g categories is this is this is the category of all small categories with a g action um and as i said this is via functors and uh the the morphisms are all g-equivariant uh functors as well so that that's what that is of course there's also you you could also think oh well maybe i'm only acting on the morphisms or maybe i'm only acting on the objects and this would you could sort of like think about this as as being in here too when your action is trivial uh but yeah i probably should have said something about that um all right so here i was in the middle again so uh the sigma j is the category with objects given by sigma j and with exactly one morphism between any two objects so if you forget about the fact that i want a sigma j i write sigma j action here uh this is going to be contractible because basically the fact that i'm adding um oh this question so i saw the question uh g is a finite group uh there are some some uh things that i can say about the groups uh compactly groups but i'm not going to say them today so so for the purposes of this talk you just just assume that that g is a finite group um so as i was saying here if you forget about the fact that i want a sigma j a write sigma j action uh the fact that i have exactly one morphism between any two objects makes this contractible uh and but i also have a sigma j action just given by multiplication by by sigma j and that's going to be a free action as well so when i take the classifying space of this uh of this operand i get an infinity operand in in um in spaces which is precisely the the what it's called the barrett echoes operand well one cool thing about this operand is that the algebras are um precisely permutative categories and i don't think i've said this before this just means strict symmetric and this is this is one way in which you can think about how to start with us uh a permutative category and produce a spectrum is you start with a category uh you think about a permutative category you think about it as an o algebra and then when you take the classifying space you're going to get an algebra or b o which is then uh an infinity space and then you can produce a spectrum out of it so how am i going to turn this into an equilibrium operand i'm going to define o sub g to be the following and here i am going to explain what everything i'm writing is so i'm going to write uh the um cat g tilde um so in particular the g f p's of this operand is cat g tilde o j so what am i doing here i'm taking the functor category uh of all functors from g tilde g tilde is the same construction as i just explained before you start with a category whose objects are given by g and uh then you add exactly one more face in between any two objects so you have this g tilde here uh which you should think about as a categorical version of e g um of the total space of the of the canonical g bundle and um i look at all the functors from g tilde into o j and all natural transformations so that's that's a category but then this is going to have a g action so this is going to be a g category so g action by precomposition so um g tilde here is a g cat is an example of a g category with action given by left multiplication and then uh when i precompose i'm going to get a g action on this whole thing um you can sort of think about that that's the g action that you get by uh conjugation when this category here has the trivial g action so that that's what's up you should think about this as a as a way of inducing up the fact that i had uh this uh trivial g category here but then i'm going to i'm going to induce an action given by this this g tilde here and here's the main theorem about this this gadget this is a theorem in a paper by gioume and merlin which is that o g is a categorical infinity g so this tells us that algebras over these upper ad are are a good categorical input so we might as well give them a name so this definition in a paper by gioume a genuine commutative g category is an algebra over o g so this is this is uh the categorical input that we're going to consider uh and i want to give you a couple of remarks uh about about this these gadgets uh the first one is that since o is a sub-operate of o g via the constant functors so here if you think about what my o g is i can look at all the functors that are constant and that's going to give me a version of o that sits inside o g so that tells me that any o g algebra has an underlying naive permutative structure so this is useful to to to realize that that what we're giving here is something on top of a permutative uh structure now another thing that uh that's important to notice so one reason we like uh permutative categories or symmetric monocategories is because we can define them with finitely many data and finitely many axioms so what do i mean by this if you want to tell me that a category is a permutative category you just have to give me the monoidal product and the the unit and then you have to give me the symmetry which is an extra piece of data but then uh you give me a few axioms that they have to satisfy there's finitely many and then you can actually check them and you get um you you get to know that what you have is a permutative category um this is this is uh basically the same thing as saying that the operate o is finitely generated in a in a very precise way um one would like to do the same with o g and it turns out that that cannot be uh done so o g is not finitely generated for groups of order uh greater than one so o g algebras can be described using finitely many data and axioms uh this is actually work from of my undergraduate students two summers ago um it turns out though that one can can find some sub operas of o g that are actually finitely generated and that are also in infinity g operas but that that's again uh not the story that i'm telling you today um now you might be wondering how am i going to come up with examples of jno ng uh categories and it turns out that i actually have a big family of examples so if c is a naive permutative g category and in particular an example is if it's just a permutative category with no g action um then i can take the functor category from g tilde into c and this is an o g algebra so i can produce many many examples this way and one question that we have that we haven't answered yet is whether all of all the examples are of this form um so putting together this theorem and how the definition comes about we have um a theorem uh that says that if um c is a genuine permutative g category then uh uh there exists a genuine g spectrum kc such that the zero space is a group completion right just went down many pages sorry about that so um so now so now we know how to go from from some categorical input to um g spectra and and we get to the point of asking well how about multiplication what what can we say about that and the first thing that you you want to ask when when when you you're confronted with this question is what do you mean by multiplicative structure how do we really encode that um when we were doing things non-equivariately we were talking about having uh two one other structures that one distributed over the other one but actually i was hiding a lot under the rug at this time and this is the this is the moment of the talk in which i'm going to tell you a little bit more about that so uh one of the things that that when you think about multiplicative structures for example in vector spaces is that you want to talk about the tensor product right like if if something preserves a tensor product but the tensor product is really the universal object that encodes bilinear maps and and that's what's going to happen here so there's there's a lot of things that i could say about what happens when you're working with symmetric monoidal categories but uh one of the things that you might want to ask is well what does it mean to have a bilinear map of symmetric monoidal categories and of course because you're working with categories you're not going to ask for bilinearity to be on the nose but only up to isomorphism or maybe even just up to a map going in one direction or the other um and then you may ask is there a universal object that um that represents bilinear maps and the answer is well yes if you think about things in terms of a two category of things but not really if you're only thinking about things one category and you get into uh into issues about well maybe you should really be thinking about things yeah in terms of infinity categories to to really make sense of a lot of this but there's another direction in which to do this which is let's forget about having a universal uh object and instead just consider the collection of all bilinear maps and if i'm going to consider all bilinear maps i might as well consider all multi linear maps and just think about a multi category instead and that's the that's the approach that i'm going to take so to talk about multiplicative structure i'm going to tell you about the multi category of symmetric monoidal categories and i'm not going to tell you much but just enough so i can i can move and and tell you about how to do this equivariately symmetric monoidal so um so not equivariately we have that if c d and e are symmetric monoidal categories uh bilinear functor so let's give it a name so it's an a functor f that goes from c cross d into e um and so is a functor such that well why would i mean by bilinearity if i take f of c one plus c two comma d i want that to be well not necessarily equal but isomorphic to f of c one d plus f of c two d and similarly on the other coordinate and of course uh if you if you remember the definition of what it means to have a monoidal functor we want this isomorphisms to be natural in our variables c one c two and d um and so on but i also we want them to be coherent and what do we mean by coherent well they should behave well with respect to the associativity isomorphism the symmetry but also with each other so there's one axiom that is going to tell you how they what happens if they cross um and so let me just write that down because it's important so the isomorphisms must be natural and coherent and of course you you uh this is bilinear but one can similarly define k linear maps and in the case of k linear you're going to have k different isomorphisms that you want them to be natural and you want to have some uh sort of um coherence between them so um just as an example going back to an example that we talked about before if you think about finite sets with cartesian product sorry uh with this join union cartesian product is the example is an example of a bilinear map and this is precisely uh the same thing as saying that uh the cartesian product distributes over these join union and so here's here's a statement which is that uh we can form a multi-category of symmetric monoidal categories where the k area maps are k now if you've never seen multi-categories the idea is that instead of just having morphisms from one source to one target you have morphisms that have sources that have the sources that is a k to pull of objects and here and here this is precisely what i'm doing here and you're supposed to be able to compose appropriately so that's that's exactly what's happening um so we want to be able to do this um for o g algebras and and here you we do since we don't have an explicit description of what an o g algebra structure means in terms of generators and relations uh this is precisely this this thing that i was telling you before that o g is not finally generated well we really need to think about the whole operate and do it all and so we might as well just do it for an arbitrary operate so uh what we're doing here is is actually we're taking some something that the category theories have been doing for a while and they have actually worked with uh they've done similar things with with monets and then we just translated them to operas and and this is what we get so let p be an operate in the category of g categories and let uh c d and e be p algebras a bilinear map um let's call it f from the pair c d into e consists of a g functor same as before from the product of c and d into e and g natural transformations so these are supposed to be uh replacing the distributivity isomorphisms that we saw before but now we have an operate so we have to encode a lot more things so g natural transformations for n greater or equal to zero so i'm just going to draw one of them and then there's a second one that i won't uh draw but we we have as well so here we have p of n cross c to the n cross d and uh i'm going to take the action on c that sends this to c cross d here i'm going to do nothing to p of n and then i'm not going to write what this map is doing but i'll i'll tell you in words so here i have an n to pole of objects in c and an object in d and i'm just going to give you uh so i'm going to take the diagonal in d so for example i have c one through c well actually let me just write down it's going to be easier so if i have c one through c n and d here this is going to go to the pair c one d c and d so i basically just repeat the object uh from d enough times to get pairs and actually you can see that that's something that i that i need here too uh in this equation d appears once on this side and appears twice on this side so so that's uh exactly what what's happening here too uh now what can i do here well i can do identity cross f to the power of n that lands in p n cross d to the n and here i have the action of p on e that lands in e and then here i have f so if i want a distributivity on the nose i would want this diagram to commute but since i'm in categories i don't want that and instead i want my isomorphism here that i'm going to call delta one there's also a delta two that is mediating between uh c cross p n cross d to the n so that that would be the other one i'm not going to write it down but you can imagine what it's doing so it's sort of like doing it in the second variable um so we have these two and then and then you have uh some coherence conditions and the coherence conditions are supposed to tell you how this uh got to interact as you vary n for example so if you're using the upright structure to to to look at what's going on uh but also what happened what happens when you have the two variables and and you use kind of swap what's going on so that those are the kinds of axioms that you that you have um and coming up with the with the sort of like figuring out exactly what the axioms are and how they they come about um is um is one of the tricky things in this business so um it turns out though that i sort of lied a little bit to you so i want to have a remark here so uh the the coherence relation that relates delta one and delta two requires an extra condition on p called pseudocommutativity it's right here pseudocommutativity and basically the idea uh that the upright should have a way for me to swap uh uh variables and if people have questions about that i can tell you a little bit more what this uh pseudocommutativity is uh now one can similarly define k linear maps so instead of just having um one distributivity we're going to have uh sorry instead of having two distributivities we're going to have k of them and then they're going to have to interact properly with each other and maybe i should say that the one linear maps are precisely the same thing that we've always known as as um monoidal uh functor right so where you have you don't expect to have a strict monoidal functor but just only something apt to isomorphism and um here's one theorem which is that um there is a multi-category of p-algebra in k linear maps meaning we can compose this appropriately and they satisfy all the conditions that you want for that and uh the main theorem that i wanted to tell you about today is that um there is a multi functor really good question uh that's what the uh what delta two involve a smaller simple coherence diagram than var delta one uh no it's it's the diagram is is basically of the same shape uh you still need to you still need to uh use that diagonal to to have several copies of things in c so it is it is not simpler it it just uh yeah i am just too lazy to write it down um so yeah going back to what i was saying here there is a multi functor um from the uh the category of algebras the category of g spectra and so uh i should tell you well let me finish uh the statement and then i'll um i'll say a little bit more about what i mean um um equivalent to um the classical so uh maybe i'll start by explaining the last sentence sentence that i wrote equivalent to the classical construction what i mean is that the spectrum that you get is uh naturally equivalent to the one that you that you you constructed using lewis main steinberger uh what do i mean by this multi category here the multi category of g spectra well g spectra has a monoidal structure given by smash product and that means that we have a multi category structure uh because of that so what this really is saying uh so let's give this a name let's call it k sub g this is equivalent um and here i should have a g here too is that if you give me um so remark if um f from c e to e is a bilinear map then we're going to get k f which is going to be a map of g spectra from k c smash k d into k e so that's what we mean by having a multi linear functor that if you give me a bilinear map then you're going to get a map from the smash product into uh the the k theory of the target and similarly for k linear things so this is actually uh what's really useful is uh for example if you have you can encode in your multi category things like being a module uh so one one category being a module over the other one or having um a ring structure and things like that and then when you have a multi function it means that you're going to be able to translate this from one side to the other so anytime that you have some sort of algebraic structure here you're going to end up uh with um an algebraic structure here as well so a ring object here will give you a ring object here and so on uh one problem with uh or one issue with this construction is that it's not symmetric meaning that uh i i cannot tell if i i cannot swap the order of things uh in the other this functor here so even though this is a symmetric multi category and this is a symmetric multi category the functor is not reserving that that symmetric structure um yeah so uh just to finish uh i'm going to give you one last corollary which is that um there is a multiplicative equivariant variation of barrette pretty quillen uh meaning that if you give me a just a g set um then i can produce um the suspension spectrum for a it's a digital digital base point by uh taking the k theory of finite g sets over a and i can do this in a multiplicative way uh so so uh so that's that's one of the main consequences of having this this multi functor here um and with that i think i'm gonna i'm gonna finish and and see if anyone has uh questions okay many thanks indeed any questions so shan has a question um so so one way is is this um so shan's question is what are some good ways of producing uh such og algebra as that we can apply this k okay functor two well one of them is uh uh this thing that i talked about if you give me an o algebra you can you can induce it up to have a g action uh so that's one way uh there's also for example for this barret pretty quillen theorem there's a way of starting with a g set and producing a free g algebra on it uh so that's that's another way but uh this is a good a very good question one of the things that we want more is more examples of of og algebras and maybe maybe one thing to think about is that maybe some things that we want will not come with a structure of an og algebra or maybe just something equivalent um uh shan is also asking is it obvious that we get a equivariant k theory this way and the answer is uh this is this is the work in mona merlin's thesis and and she should probably be able to answer a lot more about this um brian sheen is asking about uh the theorem not being symmetric and uh he's asking if there's a quick example more than having a quick example like you you can also you can actually just write down and see that the symmetry fails uh and and one thing that's interesting is that it's not that so it's not symmetric on the nose but we know it's symmetric up to homotopy so there should be a way of encoding this but we we just haven't been able to to work it out in a way that is that is satisfactory any other questions or comments if not let's thank andrelika for an interesting talk