 So I just wanted to say, first of all, thanks to everyone for being here, and thanks for the invite to speak at the seminar. It's been great. I will sort of preface this by saying that I am more of a homotopy theorist, so if anyone is more of not a homotopy theorist and is confused by anything I'm saying at all, please feel free to jump in and just say, I don't know what those words mean, and I'll try my best to get some indication of what those words mean. All right, so let's go ahead and get started. This is a talk about sort of functor calculus and derivatives, and if right now you don't know what any of those are, hopefully by the end at least you kind of understand at least what they sort of do. The guiding principle that sort of underlies this talk in my work is that if you take the sequence of goodwilly derivatives of the identity functor on a suitably nice model category, so for example the category of topological spaces, category of algebras over an operad, that this sequence should come equipped with extra structure, and that structure is that of an operad, and that it's somehow canonical, that it's sort of ingrained into the derivatives of the identity themselves. So some examples for the category of base spaces, Qing shows in his thesis work that the derivatives of the identity in this category is an operad, and he does this by sort of constructing a very sort of interesting way of describing an operadic structure, and this is sort of a, this is a pretty deep theorem, this is a fairly difficult paper to parse. Another example, and this is sort of my work on the subject, is that oh, if oh is a reduced operad in spectra, then the derivatives of the identity in the category of algebras over this operad is what I'm calling a highly homotopy coherent operad, moreover this highly homotopy coherent operad is equivalent to the one that we started with. So this has been a conjecture for a while, and recently I was able to prove this statement, and more generally there's an approach using sort of infinity categories outlined by Lurie and Michael Qing, so I see something in the chat, okay it's just a live stream URL, perfect. Alright, so maybe these words don't really mean anything at the moment, so let's start off by just sort of saying what is Functor Calculus, what are operads, why is this theorem sort of remarkable, why would you sort of expect these things to look like an operad in any case, so that's going to be sort of the main goal of my talk, and then at the end I'll sort of give some applications of what you can do with my theorem here, pull up point number two. Alright, so Functor Calculus is a theory sort of developed by Tom Goodwillie, and it's sort of, it's called calculus, and really if you think calculus, and you want to say like calculus in the same sense that we teach our undergrad students calculus, there are a lot of similarities here, so if I start with a homotopy meaningful functor from spaces to spaces, so this s underscore dot is just the category of base spaces, then this means like if I have homotopic inputs, I should get homotopic outputs, and let's just assume for simplicity that f is what we're calling reduced, meaning it's going to take the point to a contractible object. So the main construction of Goodwillie's paper, his Calc 3 paper is that of a Taylor tower of n-excisive approximations, which we denote by p underscore f, and natural transformations between these functors, which has the following form. So p1 of f receives a map from p2 of f, receives a map from p3 of f, and so on they assemble into a tower, and moreover this tower, each of the parts receives a map from f itself as well. Okay, so again, I want to stress that we're sort of thinking this as some sort of categorical approach to calculus, and you can really stretch this analogy quite far. The functors p and f themselves can be thought of as polynomials of degree n, and I won't really talk too much about this for the general n, but I will spend a little bit of time in sort of justifying the statement that p1 of f is sort of what we're calling a linear approximation to the functor f that I started with, and you can really make sense of this idea that these are sort of polynomials of degree n, so if anyone's interested in hearing more about that, I can always chat about that in the chat session later. Moreover, why do we use a Taylor series from calculus? Well, one way is that if I have a function that I want to understand better, the idea would be let's break it into smaller parts, each of which we might understand a little better than the function f, and then hope that I can recover the guy that I started with via this process. So the statement here is that if f is nice, the keyword is analytic, this is sort of a strong property to ask, then for sufficiently connected spaces, we can recover the homotopy type of the functor by sort of the limit of the homotopy types of the conceptual pieces in the tower. And everything here, everything in this talk will just sort of be up to homotopy and I'll sort of say more about that as we go along. Right, and you can sort of think of the connectivity of a space, meaning to what degree are its first so many homotopy groups trivial as giving you something like a metric between spaces. So it measures somehow how close two spaces are. In particular, the identity functor on spaces is one connect is one analytic. This is a result of the Blakers-Massie theorems, the more generalized Blakers-Massie theorems from Tom Goodwilly, and that you can recover the homotopy type of a space by this tower for if that space is one connected, and that's, again, just the statement that the identity should be one analytic. Okay, so moving on. So let's give sort of a little bit of an example about what sort of justifies the word linear. We have a pretty good idea of what a linear function is, but if I want to say what a linear functor is, it's not obvious what that definition should be. So this is the sort of, this is what Goodwilly gives, and there's sort of several reasonable approaches to saying why this should be linear, and I'll sort of try to outline that here in a second. So a function, a functor is linear, also known as one excise, if f takes homotopy push out squares to homotopy pullback squares. So again, everything is up to homotopy. If this is at all sort of uncomfortable, just sort of think that we're implicitly, you can implicitly work in the category of CW complexes in pointed topological spaces. If you do so, then push out squares are homotopy push out squares, everything is automatically derived. And, you know, if you if you feel uncomfortable with this word homotopy showing up so frequently, feel free to just sort of ignore it and you'll still get kind of the main idea of what goes into some of these constructions. I haven't chosen to sort of rely on things being homotopical too strongly. So what does this sort of mean? Well, if I start with a homotopy push out square, then I apply f, the resultant square is a homotopy pullback, meaning I can recover f of a in terms of the data of these three guys over here. All right, some, some remarks. So what are maybe examples of such functions that seems like kind of an odd request for homotopy pullback or push outs to turn into pullbacks? Well, sort of the canonical example that I'll focus on in this talk is that the stabilization functor from spaces, the guy that if I take homotopy groups, this gives me the comparison from unstable homotopy groups to stable homotopy groups is one excise it. And here I'll sort of just sketch a reason why. Well, remember that there is an adjunction between based spaces. And this category of spectra, you think of spectra as sort of the stabilization of spaces. It's sort of the analogous process of taking a module and resolving it by a chain complex. So the two functors that go into this adjunction, one is the suspension spectrum. And one is the underlying sort of loop space. So it's, you know, if I take a spectrum, I can recover a space out of this. And this is in fact a derived adjunction. So if I start with a push out square, push out in spaces and hit this with the left adjoint. So this is in in based spaces. Now I'm in spectra. Left adjoints will preserve the fact that something is a push out. But now I've landed in spectra. And if I am in spectra is also stable. So stability means that push out squares are the same as pullback squares. And then when I hit this with loops infinity, well loops infinity is a right adjoint. So it'll retain the fact that in top, this is a pullback square. So this is sort of a sort of maybe a very abstract argument to sort of why this guy would satisfy this one excisiveness property. But perhaps this feels a little too too abstract. So I'm going to unfortunately clear all those. And spotlight. So another example of a one excisive functor is just pick any homology theory. So say integral homology is one excisive. And that's sort of because another way of unraveling this statement is that you get an associated Myer-Vatoris sequence. So this is exactly the statement of excision that all homology theories are required to satisfy. And then this notion of higher degree polynomials is some sort of higher degree excision that we require our functors to satisfy. So sort of a non example is that the identity functor on spaces is not linear, right? If I think identity function, it's very compelling to think that the identity should always just be linear. That's not the case. Because if I just give myself an arbitrary pushout square in spaces, there's very little reason to expect why it should be a pullback square. And in general, it's often not going to be a pullback square. So this is just a remark that the category of spaces is not stable. And that if I hit this guy with P1, meaning find the best one excisive approximation to the identity, in fact, that best one excisive approximation is just the stabilization functor. So that this guy is sort of crucial in understanding the Taylor Tower for the identity functor in spaces. Okay. Last remark is just if I have a reduced functor, then the associated first degree or linear polynomial to this guy will always have the form as follows. I take some spectrum. I smash it with a suspension spectrum and take the underlying loop space. And you can note that since spectra is stable, that the functor that just smashes with some spectrum is going to be linear. So this spectrum E is determined by the functor F up to choice of homotopy. And this is what we're going to call the first derivative. So it's something that sort of plays the role of classifying linear functors. So I've noticed that it's often a good idea to stop more frequently and ask for questions. I'll try to do this close to the end of each slide. Please again, feel free to interrupt me if there's anything. Cool. All right. So I promise some more insight about these things called derivatives. So where do these derivatives show up? Well, we have our Taylor tower. And we think of these as being the sort of nth degree polynomial approximations to just an arbitrary functor. So if I want to get to the derivative, the nth derivative, well, first, let's strip off the parts that are n minus 1 degree approximations from the nth degree approximation. There I'm due to good willy is that this guy dn, the homotopy fiber of pnf to pn minus 1f. So just think fiber. It's essentially killing all the things that show up that are elements of n minus 1 in the nth part. This is sort of a deep theorem that if I look at this layer, this homogeneous layer, then it's characterized up to homotopy by a spectrum which we're denoting by partial n of f. The spectrum comes with a natural sigma n action. This is just the nth symmetric group. I know every little subfield tends to have their own notation for the nth symmetric group. So in homotopy theory, we tend to write sigma n. And it will always take the following form. So the value of dn of f on a space x is comatopic to the underlying loop space of the nth derivative smashed with the suspension spectrum of x smashed n times. And both of those have naturally occurring sigma n actions. The spectrum here comes with one. And then if I take something and smash it n times, I can just permute those guys. So I can take the covariance with respect to that action. And this classifies this homogeneous layer. And again, we call this guy partial n of f, the nth derivative of f, in that it sort of classifies the stuff that I've added to get from n minus 1 to n. So a remark, again, it's not, it's quite remarkable that you can sort of get such a strong analogy between functors in some sort of homotopical sense and functions in some sort of like classic calculus sense. I want to just say that it's quite remarkable that this formula really for the nth homogeneous layer is so similar to the nth monomial that shows up in a Taylor series. In particular, I have something that looks like a derivative times some input with a n-fold product of itself. And then maybe the biggest stretch here is, all right, I've divided out by some sort of action. Sigma n has n factorial elements. And I've divided out by n factorial here. So you can really sort of think of this as like I'm doing some sort of like analysis-like approach to analyzing functors. And sort of the really cool part is that this gives you a lot of homotopical information. In particular, we can compute this derivative from this homogeneous layers by something called cross effects. I won't say too much about this. In the classical sense, this is just some process that you can say, plug in these values for x and then subtract a bunch of times, and then you just end up with the number that is the derivative. If you sort of abstract this argument, you can perform some similar task to this homogeneous layer to get just the homotopy type of this derivative out. So let's look at some examples. We already saw that the first order linear approximation to the identity is the stabilization guy sort of mirroring it with this statement here. Well, the guy that its first derivative has to be is the unit for the smash product. And I guess I should say something about working with sort of spectra, if this is unfamiliar, you can really think of spectra as just sort of a homotopical version of working with like modules over z, so like Abelian groups, where s is playing the role of the integers z, and a spectrum is sort of a module over this sphere spectrum s. In particular, smash is like tensor of z modules, and s is just the unit for this tensor. Less obvious is sort of how you would try to compute the higher order derivatives. If you try to do it by definitions, it quickly becomes very difficult. So a result of Johnson, and then later Aaron and Mahowad, is that if I give myself a larger than one integer, that the derivatives of the identity are related to this thing called the partition post set complex. And I won't say too much about this. It's essentially just you take the set, the post set of all partitions of the set one through n, and you arrange them by sort of refinement, and you can turn this into a simplicial set. Once you realize this simplicial set, it's sort of the dual of this that is the derivatives here. In particular, if you go through this argument, you'll see that the second derivative of the identity is loops sort of the desuspension of the sphere spectrum with a trivial action by sigma two. So overall, if I start with a functor, I'm going to sort of acquire a symmetric sequence of spectra. That just means I have an object reach positive integer, and it comes equipped with an action by the symmetric group. And the sort of goal is to sort of investigate what extra structure this sequence possesses. So we already sort of saw that at least in the case of the identity functor, we're sort of offering an interpretation between stable homotopy at the first layer. And for a one connected space, if I go up the tower, I'll recover the type of x itself. So this is some sort of crucial object that tells us something about the interpolation between unstable homotopy and stable homotopy. So operands. If operands are totally new, they can be also extremely overwhelming. So I'll give sort of a crash course in just what we mean by an operad and sort of how to think about such objects. So an operad is sort of a useful tool for describing spectra with extra algebraic structure. And when you work in the world of spectra, well, you might be interested in knowing what are sort of ring spectra. So by that, we mean what are if I have a spectrum are, you know, if it admits sort of a self map like that, r smash r down to r, which is suitably associative and unital, we think of such as sort of a ring, right? If I have a z module that has a map from r tensor r down to itself, this is exactly what a ring is. So we can encode this algebraic structure via these things called operands. But what's remarkable about operands is that often when you work with spectra, asking for something to be a strictly commutative or associative ring spectrum is just not it's just too much, it's too much to request. So we can also encode structures such as a infinity ring spectra. And these are things that are associative only up to some coherent choice of homotopes. And I'll get back to this sort of later on some examples of a infinity ring spectra and sort of a infinity objects as they come up relating to my work here. And another category of interest would be those of en ring spectra. So for n varying between one and infinity, it doesn't matter too much what that specifically means when n varies. Just think of this as varying levels of homotopy commutative ring spectra, where my commutivity relations, you know, two ways of encoding like a times b and b times a are equivalent up to some sort of coherent choice of homotopes between those sort of outputs. Cool. So a definition for what an operad fully is comes from May and also Borman-Fucht. So an operad O in a symmetric minoidal category consists of, well, it should consist of a symmetric sequence. So I have objects for each positive integer that come equipped with the symmetric group action. I have a unit map that sort of tells me what it means to just take an object and multiply it by one. And this is sort of the main crux of what goes into an operad. I need sort of action maps from O to itself that satisfy associativity and whatnot. But to think about these action maps, it's really, you can sort of view it as these sort of tree diagrams. O of n tells you possible ways that you can encode n-ary operations. So for example, if I had an O of 2 right here, and if I had an O of 3 and say an O of 4, I can plug the outputs of those 3 and 4-ary operations into the inputs of the 2-ary operations. And together, that should give me one particular example of a 7-ary operation. So if k1 is 3 and k2 is 4 and n is 2, I should have a map into O of 7 that says, like, well, some of the 7-ary operations are those that come from composing a 2, 3, and 4-ary operation in certain ways. And this is supposed to play nicely with respect to the equivalence on these particular inputs, meaning, like, I permeate around the leaves of these trees, things should sort of behave as we expect them to. Actually, writing all of that down is somewhat of a task, really. So it's useful to think of operads as being monoids with respect to this composition product of symmetric sequences. So this, I'm certainly not going to talk too much about what this is. It's a sort of complicated product structure. But we think of an operad as being sort of a monoid, meaning there's an associative and a unital map from O circle O down to itself, and that this circle product has a unit, which I'm just denoting by I. And I is the symmetric sequence whose sort of first level is the unit, and it's trivial everywhere else. So, right, I should say maybe one thing here, a useful way to sort of interpret this composition product is that if I give myself sort of two formal power series, those are sort of determined by their sequences of coefficients, if I ask what the nth coefficient of the composition of those two guys would be, that's sort of the structure that underlies this composition product. I think of these O of ns as sort of being the coefficients of some formal power series, but in a sort of categorical framework. So we care about operads really because we are interested in describing extra structure on their algebras. So an algebra over an operad in this category of symmetric spectra, so I have some strictly commutative product, just think of it as tensor, I'll write that in, just think of this in your head if this is unfamiliar as z modules tensor and then z. That's a very reasonable thing to sort of compare this to. So an algebra over an operad O in spectra is a spectrum together with action maps from the nth level of O smashed with the n-fold smashed product of x down into x, taken sigma n co-invariance for all n subject to associativity and unitality conditions. So maps of this type, we really think of as parameterizing the possible n-ary operations on my spectrum x in that what is a two-ary operation? You give me a pair of elements and you tell me all the possible things that they can map to. And this is saying, well, if you give me a pair of elements and then you permute them, that permutation should also agree with the permutation action that happens inside the operad itself. So as an example, if my operad at level n is, well, let's just start by saying if it is a point, so if O of n is the unit, then this map would just be, so we'd get that we have a map that fits into this operatic algebra from the n-fold smashed product taken sigma n orbits to itself. This means any way of rearranging all the possible inputs, all the possible things you want to multiply together have to go to the same output. So this would be something that is a strictly commutative monoid in spectra. If I relax that condition and say that my operad should only be contractable or equivalent, homotopy equivalent to S at each level and some condition on the symmetric group actions, then we think of this as describing monoids which are commutative only up to some coherent choice of homotopies and often these are referred to as E infinity algebras. I want a spotlight. There we go. So we'll just set alge sub o to be the category of such algebras together with structure preserving maps and just note that an algebra is equivalently an algebra over the associated monad. So anytime I have a monad meaning a monoid in the category of endofunctors on a category, I have sort of rich categorical structure on the category of algebras over this monad and the parts that go into this co-product are exactly the maps that you get from this algebras structure up here. Any questions so far? Well, I take another sip of coffee. So far I've said a little bit about functor calculus. I've said a little bit about working with operads. Now the goal is to sort of mesh those two together and when we did functor calculus we sort of started off with this idea that the identity functor is its Taylor tower is strongly related to understanding stabilization of that functor of that category of spaces. So in fact to understand functor calculus in any sort of suitable category it's sort of a prerequisite to understand what the stabilization of that category should be. So for the rest of this talk we're going to fix an operad in O in spectrum with O of 0 being a point. So this just means there's sort of no zero airy operations on my space. I can't sort of create something from nothing. This is sort of an equivalent statement is saying that the algebras over this operad are sort of non-unital in the same way that a ring can be non-unital. For simplicity we'll also assume that the first level of the operad is just the unit for my for my monoidal structure on spectra. And this is not too much of an ask. It's the statement that says the only way that you know the the only one airy operation is just the identity. I don't have any interesting operations which take a space and then do some stuff to it and then give me out of space in a way that relates to the operatic structure that that guy has to begin with. Okay so given an operad that satisfies this we'll define the nth truncation denoted tau n of O as follows. It just kills off anything that is of order higher than n. So if I wanted to say multiply more than n things in tau n of O that that composition would be forced to be trivial. So there would be it's essentially like I just if I want to multiply five things in my ring too bad you only you only get zero out. It's no good above that level. So associated to this each truncation I have a map from tau n into tau n minus one. This is just induced by collapsing the the highest level of this truncation and there's a tower of operads sort of receiving a map from O where O just maps into these guys by just destroying everything above level n. The bottom map of this tower induces a change of operad's adjunction. Before I say what a change of operad's adjunction is I'll just sort of bring it back to this world of algebra. If I give you a map of rings there is always an adjunction between r modules and z modules. The left adjoint in this case just takes an r module tensor is over with r with s on the other side. And this guy down here just sort of forgets about the s structure and only cares about the r structure. And this this construction is exactly what the change of operad's construction is really doing because we have adjunctions of this type and let's get rid of that. So the receiving guy goes on the outside. I think of this as some sort of tensor but tensor in this more complicated category of operad's but you can sort of think of them as encoding the sort of module structure on their algebras. It looks like so. Really though what I want to do is consider a factorization. It's not super important that you understand this step. This is just for sort of nice homotopical properties. I want this forgetful functor to preserve cofiring objects. I will say only that about this. And we're going to define the left adjoint of this red guy to be q and the right adjoint to be u. So q is the thing that sort of kills off up to homotopy all of the two area and higher operations and you just sort of says well if I have such something like that I can just sort of trivially give it structure over this operad as an algebra where I just you know I only care about the one area operations on this guy that started over here. So we'll set tq to be the left derived functor of q. Often this is called the topological quill in homology. I'll say more about this in a second related to classical sort of Andre quill in homology of a ring. But the thing I want to remark on right now is that it's a theorem due to Bastira Mandel that q and u is really equivalent to the stabilization adjunction for algebras. So understanding stabilization in terms of algebras boils down to sort of understanding what happens with respect to this adjunction right here. In particular tq of o algebra is thought of as the suspension spectrum associated to this o algebra. Suspension spectrum in the sense of Balsu and Friedlander where you use the simplicial tensoring to sort of just abstractly define what it means to be a spectrum in this category. So let's continue. Functor calculus in algeo continued understanding the Taylor tower of the identity. It's a Harper Hess and also Pereira show that the Taylor tower of the identity in algeo takes the following form. And yeah, so there it is. What does this kind of mean? I will sort of draw an analogy here. So if x is a nonunital communicative ring, then the equivalent version of this tower in this setting is I can take x and I can mod out by all of the things that are products of two elements. This is often called the nv composables. And you know, tau two, it kills off everything at level three and above. So that's equivalently modding out by all the things that look like composites of three or more elements and so on and so forth up the tower. There are maps like so and taking the limit of this tower is often called the x-addict completion in the classic sense of the commutative algebra book, which name I'm forgetting at this moment. But the indie composables, this is often if I suitably understood to be related to the Andre Quillen homology of this ring. So it's reasonable then to name this comparison topological analog of this Andre Quillen homology because this guy really does look like the sort of derived indie composables of the algebra over by operad in this sense. So let's see. Okay, it'll stay. There's enough room. So in particular, looking at this tower here, if I want to understand what the derivatives of the identity are, well, I should first understand what the homogeneous layers at level N are. Harper Hess showed that the derivative, the nth homogeneous layer of the identity is equivalent to such. Remember TQ is like suspension spectrum and U is sort of playing the role of loops infinity. So this is fitting into the same paradigm we knew before from Goodwill's theorem. And in particular, it shows that the nth derivative is just going to be equivalent to O of N. And these guys come with a naturally occurring action by this nth symmetric group. So these are equivalent as sigma N objects in spectra. In particular, they're then equivalent as symmetric sequences. And sort of that's compelling because it has been longstanding conjecture that these two symmetric sequences actually should be the same as operads. Now, we said underlying an operad is sort of, I need to know that it's a symmetric sequence plus extra structure. And in this case, a weak equivalence of operads is going to be detected as a weak equivalence of underlying symmetric sequences. But I need to know that there is a comparison between these guys as operads, meaning as monoids with respect to this composition product for symmetric sequences. And the hard part of this conjecture is that it's not quite so obvious how to describe a monoidal structure on the derivatives of the identity, especially one that you can compare with the operad O. So the argument that Ching gave for the identity in spaces doesn't carry over so nicely into other categories. In particular, it doesn't necessarily tell you anything about the derivatives of the identity having a particular structure in Aljo. And this is where my work comes in now, because theorem, the derivatives of the identity possess an intrinsic homotopy coherent operad structure with respect to which it recovers O. So you can sort of suitably forget about the homotopy coherence and say that O is a homotopy coherent guy where the coherence is sort of trivial. And essentially, this is resolving that conjecture just in a slightly reduced, relaxed setting of homotopy coherent operads. So if operads are already confusing, homotopy coherent operads probably sound like a real mouthful, but I'll sort of unravel where this homotopy coherence comes from. It's really just sort of an artifact of the proof of this theorem. So rather than just jump right into the proof, I'd like to give sort of an idea of this proof, which is relating to something that I think we all probably know. So let me just read this off first. Our method is then to adapt a technique of McLaren-Smith that if I have a cosimplitial space, which is a monoid with respect to the box product due to batonin, then when I hit this guy with tote, it's an A infinity monoid in spaces. So there's kind of a lot that goes into here. Well, it's a cosimplitial space. A simplitial space can always be realized. So like a simplitial set can be realized to give you say a CW complex. Tote is sort of the dual version of that for cosimplitial spaces. So you can suitably think of this as like, all right, this is telling me something about the space is built out of pieces that sort of fit together in a certain way. And when I glue them all together, the thing that I get out is an A infinity monoid. I won't give a definition of this box product because it's kind of opaque at first. But sort of an idea for where it comes from is this box product is the same thing that will give you the cut product on cohomology in terms of a pairing on the level of co chains. So cut product tells you that well, cohomology has this nice sort of pairing on it's on the sort of graded in a certain way. And you can see this as a result of an underlying pairing on the co chains, which by sort of doled con is equivalent to some sort of cosimplitial object in a suitable category. So here's an example. The end goal here will be to show that if I take loops on a space X, so the space of loops on X is an A infinity monoid. This isn't necessarily something new, right? I know that if I have two loops, I can concatenate them. I can take a loop. I can do another loop. If I do a third loop, well, now I have to sort of choose different ways of which ones that I do first. And we don't really ever think of this as a drawback. It's just that there's not really a way of choosing that. And our monoid structure on loops is only determined up to some homotopy that says shift around where you chose to sort of base your paths. So one way we can see that is as follows. If I start with a base space, its diagonal map will turn it into a co-monoid in spaces. With respect to this co-monoid, I can build sort of the co-bar resolution. So what are the maps that go into here? Well, X is based. So each of the sort of outside co-face maps is just going to be the inclusion of a point. And then the inside maps here are what's coming from the diagonal. So diagonal applied to X maps me into X squared. I can choose to apply the diagonal on the left factor or the right factor and map into the associated left two or right two factors of X, the threefold product of X, and so on. This gives me a naturally occurring cosimplicial object. With respect to this guy, there is a pairing. A box product pairing is sort of like a tensor pairing. It's sort of saying that I need to know how to take the p, the p-th level and the q-th level of this guy and map it into the p-th plus q-th level in some sort of relative way that agrees with the co-face maps. In particular, in this case, the structure is just induced by the natural isomorphisms. At level n, I have just X times n, and there's isomorphisms of this type. In particular, once I hit this guy with tote, once I sort of realize it, well, the thing that I get out is equivalent to the space of loops in X, and moreover, the a-infinity monoidal structure on this tote is equivalent to the a-infinity monoidal structure on the space of loops. So these are equivalent as monoids with respect to this monoidal pairing, homotopy coherent monoidal pairing. And yeah, so just recall that space of loops is a monoid with respect to composition. Any questions so far? So let me just now sort of briefly sketch the outline of this proof. I won't say necessarily too much. I think what there's 15 minutes left in this talk, and I'll like to leave maybe 10 or 5 minutes for questions. So if you've gotten this far, this is pretty much what I would have hoped that you would get out of this talk, understanding sort of what functor calculus is, sort of understanding where operads fit into the derivatives and then how to sort of apply functor calculus in the setting of algebras over an operad. The idea behind the proof is that if I can say that the n-th derivative is something like the tote of a cosimplitial object, which is a monoid with respect to the box product, then that is essentially giving me the structure that I'd like. And the idea is that this cosimplitial structure is coming from the Bausfield-Kahn cosimplitial resolution of the identity functor with respect to stabilization. So that is it's coming from this cosimplitial resolution. These co-face maps are the ones that just include UQ into either of the two factors and so on and so forth. And using our model for U and Q, this guy is equivalent as objects to sort of the iterates of this circle over construction, where I kill off all the higher area operations and then sort of forget back and then just continue this out. So we'd like to understand sort of how the n-th derivative fits into this cosimplitial structure comparison, right? Identity sort of co-augments this cosimplitial object over here. The theorem due to Blomquist is that the maps into the truncated holms, so if I chop this guy off at a certain level and just look at the comparison between identity and that chopped-off part, are sufficiently connected to induce equivalences of this type, where if I apply the n-th derivative, in particular, I can move that inside the holm. And once I look at the comparison into the homotopy limit over delta, it's a weak equivalent. So up to homotopy, the n-th derivative is equivalent to sort of gluing together in a coherent way all of the derivatives of the composites of U and Q. So understanding then how to sort of analyze this cosimplitial diagram, we're led to looking at the algeosnace plate. So this is kind of an idea due to Arone and Cancanrita that I can analyze these guys via this sort of equivalence. And I think I'm just going to kind of skip this for now, because I'd like to say something about applications before I run too much out of time. Yeah, the idea is that applying sort of the co-unit of stabilization means that it splits strongly in sort of this divided power algebra with certain coefficients. And in particular, this allows me to get a model for the n-th derivative of some composite of U and Q in terms of sort of the cosimplitial object I had up here. This is essentially just stripping off this circle over O with blank part at the end, so that we get a model for the derivatives of the identity as homotopy limit of this cobar-like resolution, which looks like so. So C of O has this form, co-face maps are induced by sort of inserting the map O down to J at the I position, so I just sort of put in another O circle over O and then map that guy down and code degeneracy maps are given similarly. Right, so sort of three parts that go into this this proof then are, well, first I need a suitable fibrant replacement, just going to ignore that. This guy, C of O, is a monoid with respect to the box product. I have this word oplex because the box product with respect to the composition product is not nearly as nice, it's just saying that it sort of only goes in one direction and that direction it goes in is the direction we'd like it to go in, so everything works out nicely. In particular, this C of O looks like a cobar resolution in the same way that this L did that gave us loops. This isn't exactly right because this guy right here is not necessarily a co-monoid, it's a very, very weak co-monoid in that you get some ugly zigzag of equivalences which doesn't necessarily tell you it's rigid enough to build this cobar resolution, but you can sort of think of it this way. Alright, once I have this cobar resolution and I have this pairing, well, I know then that I can apply totes and I will get an a-infinity monoid structure on the derivatives via this sort of relation up here and also this guy C of O had a co-augmentation from the operad that we started with and in particular this co-augmentation carries through and gives you an equivalence of sort of homotopy coherent operads and thus proving the theorem and resolving the conjecture that had been open. So this is really the main technique and it's really just using sort of extending this McClure-Smith idea into this much worse category of dealing with symmetric sequences and their composition product. Trying to describe what a-infinity monoids with respect to the composition product are, you're forced to move into this world of colored operads and it's just, it's sort of a nightmare to really get out what all the coherence should be, but I managed to write this down in at least a partially succinct way and it ends up looking, I think, about as nice as you could possibly hope, which still might be kind of ugly. Okay, so I'll say maybe just briefly about some applications here and then let the open the floor up for questions if there are any. We can use a similar box product pairing to induce a highly homotopy coherent chain rule. So this idea of there being a chain rule on good willy derivatives, well it's sort of the idea that there's a chain rule on regular derivatives. So if these are sort of mirroring this idea of calculus, you know, maybe these guys should have one too, this is originally a result of Oron and Xing for the category of spaces and what they managed to show is that there's a comparison map that would say take the derivatives of f, compose it with the derivatives of g, and you get a map into the derivatives of the composite. So within the realm of O-algebras, you can induce this pairing in terms of box products on co-simplicial resolutions that give you the derivatives as their totalization. Similarly, let me just get this all out because I think I don't like how I raised it, you can show that there's sort of a pairing that sends an O-algebra x to an algebra over the derivatives of the identity. Now this might not be so remarkable because we just show that the derivatives of the identity and O are equivalent as operands, but it's sort of the technique that goes into this that I think can be useful for doing other things. So I can resolve x by a co-simplicial resolution, take its homotopy limit, that's called the TQ homology. This c of x is the same thing that went into the Baal-Siehl-Kahn co-simplicial resolution with respect to stabilization. It's a theorem to the Ching and Harper that if x is zero connected, it recovers the homotopy type of its TQ completion. But the way I'm interpreting this is that if x is zero connected, then I get a naturally occurring sort of comparison of algebras x over O to the TQ completion over the derivatives of the identity induced by this box product pairing. This sort of goes into the second part down here. So understanding how a model category C, so in particular alge O, is related to the category of algebras over its sequence of good willy derivatives of the identity, there's something implicit here where we had to sort of restrict to a nice category in order to make this precise. And really understanding the relation between C and algebras over the derivatives of the identity, I believe should give us some information about some sort of descent properties on the category C. So I'll finish just by saying that if you are ever interested in reading Ching's proof and you find it kind of difficult, another way of approaching this is to use the similar sort of box product pairing to induce pairing on the resolutions that are known for the derivatives of the identity and spaces, which sort of naturally shows that the derivatives of the identity and spaces should have this homotopy coherent operet structure. And underlying it is sort of the same sort of ideas as what Ching gives in his proof. So I'll finish it up there and here are some references if you're interested in reading any more. So thanks for listening. Thanks Duncan. Any questions? Yeah maybe this was addressed at some point but like there's it's like this tower of derivatives is sort of maybe like superficially similar to like a POSNICOP tower so is there some it is a POSNICOP tower yeah and that's that's actually something that good willy shows. Okay cool. So each of those comparisons is is sort of naturally a principal vibration. Is there a class of these objects that we'd expect to be like completely characterized by this tower or is that part of the result? So that's that's sort of what's going into this this last statement here. Understanding this this category of algebras over the operet structure that you can endow on the derivatives is and this is kind of like just something I've been mulling over for a couple weeks now is it should tell you some sort of descent data on the category you started with. In particular in spaces this operet derivatives of the identity is sort of it's causal dual to the sort of co-commutative co-algebra and so sort of rightfully inherits the name the spectral Lie algebra. So it sort of tells you something about a comparison from spaces to spectral Lie algebras and this comparison it can be made quite strong with respect to some of Quillen's work that says like rational homotopy types you have a comparison of those with sort of DGA Lie algebras and co-commutative DGAs. So yeah there's a lot there and it's sort of all in the process of being worked out at the moment I think the story is clearer if you go to the infinity category world but I'm sort of trying to use these box product pairings to say a stronger statement on a point set level quickly realizing that taking keeping track of the homotopy coherence is actually quite non-trivial that's a good question though. I wonder is there something in that analogy so I guess in the classical setting you have these like K variants for the tower is there something that looks like that in the setting? This I don't know. I don't know if I know how to if I have the right person to ask that question too but I think that's a valid question.