 Well, thanks a lot for this invitation. I'm happy to be here. So the title and abstracts were important. I think that belongs to Damien, just FYI. So everything I talk about today will be joint work with John Francis, and parts will be joint with Nick Rosenbloom and Aaron Maselge. So my goal for this talk is to talk about this, I'll call it theorem. We're not quite finished with the proof. It's been a long time. So is to talk about this, how it's related to other things you might know, and indicate some of the aspects of its proof. And please speak up. I'm happy to tailor the talk to your interests. So this result says the following. Factorization homology with adjoints, that's a term which will be described in a bit, defines a fully faithful functor from an infinity category of pointed infinity n categories with adjoints to space valued functors on a category of solidly framed stratified spaces. Now that's a lot of jargon. And again, much of the talk will be unpacking some of that jargon. And it's not any old functor. It's one that has the following values. So if you take a pointed n category, the value on a closed k-dimensional disk will be the space of k-morphisms in the n category. And the value on rk will be the space of k-endomorphisms of the distinguished object. So here the white part is repeating what I said in words. And I will indicate some parts of the right-hand side before actually defining it more thoroughly. So again, the domain of this functor is an infinity category of infinity n categories with adjoints. If you're wondering what it's meant by saying with adjoints, that's to say this, each k-morphism in it has both a left and a right adjoint for k between 0 and n. And again, the right-hand side, the co-domain of this, is space valued functors on some very peculiar category of solidly n-framed stratified spaces. So before saying giving a specific definition of that, let me indicate what an object is. So an object in that MFD category is a stratified space. Strictly speaking, it's what we call conically smooth. That's just a type of regularity that ensures that every stratum is a smooth manifold. And furthermore, the links between two strata have some regularity to them. Think of Whitney stratified space. So an object is a stratified space, but not just a stratified space alone. It's solidly n-framed. And what's meant by that is that it's equipped with an injection of its tangent constructible bundle into the trivial rank n bundle. So an example of an object, then, is just a smooth n manifold equipped with a parallelization. Another example of an object is, for instance, a point equipped with the unique injection of its tangent bundle, which is the zero vector bundle, to the trivial bundle. So that's in just a moment. I'll compare this result to other things you might be familiar with. I'll say right off the bat that one of the reasons for that conjecture altogether is that essentially an immediate corollary is a proof of the Tangle hypothesis, as well as the Cobortism hypothesis. To see the logic for how that assertion implies this is posted on the archive. There is a related conjecture to that one. It's like the k-linear version of that one. I'll just, I think I'll let you read this. What's that? Is every stratified space as an injection of tangent space Tx to y? It's equipped with an injection. So the data, what's that? It may not happen always. That's right. Yeah, thank you. So in fact, if the stratified space has dimension greater than n, then there is no such injection. So this structure of there being an injection requires the ambient dimension of the stratified space to be no bigger than n. Yeah, that's right. So there is a k-linear version of that, which we haven't given substantial thought to, but expect for it to be true. So I'll just state it as a conjecture. This conjecture is proved in dimension one in joint work with all those names up there. And I'll just leave this here for you to read as you wish. So I'll now try to connect this result to other things you might know. And I'll do it in three passes. First, I'll talk about factorization homology in maybe a more familiar guise. And then I'll talk about other manifold invariants that are captured by this. So remark this, meaning that, extends factorization homology e n algebras in the following sense. So e n algebras count as examples of n categories with adjoints, in fact, pointed ones. Namely, given an e n algebra, you can construct an n category from it that has a unique object, a unique one morphism, et cetera, a unique n minus one morphism. And then that e n algebra worth of n morphisms. That construction is a d loop construction, n times, so b n. This is, in fact, fully faithful. Then there's the functor of the result to co-apprecives on this category of solidly framed stratified space. And the sense in which this factorization homology is compatible with the one already that you might be familiar with for e n algebras is that if you take an e n algebra, it's value on at least, if you evaluate on not any old stratified space, but a stratified space that's just a stratification of a smooth n manifold. Smooth means conically smooth, the smooth manifold. No, but here it is the condition is conically smooth. If you adjoint this. Yeah, that's right. So a general object of M, MFD, is a conically smooth stratified space together with this tangential structure. But here I'm just saying to say a sense in which this new factorization homology extends the one you might already know for e n algebras, I'm just saying what its values are on stratifications of a smooth n manifold, like a triangulation of a n manifold or something. I'm not in this description saying what it, how it evaluates on a general stratified space. So here I actually mean stratification of a smooth n manifold. We still have a list? Yes, thank you. A smooth frame parallelized in manifold. And the answer is just delete all of the positive co-dimensional strata. The result is a smooth n manifold. And evaluate factorization homology, maybe call it alpha for just the e n version. So this extends the factorization homology for e n algebras that you might already be aware of. Let me just speak to this in case you're not aware of this factorization homology alpha. And what I want to emphasize in the next comment is that this version of factorization homology for e n algebras I think of as codifying the observables of a perturbative topological QFT. And the intention behind this, actually really the K-linear version, abstracts the observables of a not necessarily perturbative TQFT. So let me say give a comment that reflects that sense in which this is abstracting observables of a perturbative QFT. And this is inspired by work of those books. So for example, if you fix a field, let's just say characteristic 0 and a point in a derived K scheme, then there's a couple of things you can do with this. First off, you can construct a formal scheme by just taking the completion of this scheme at that point. So this constructs a formal scheme, denote it like that. And it also constructs a Lie algebra, namely the tangent complex at that point. So this is a Lie algebra. And these casual duality articulates a sense in which these two objects determine each other. Given a Lie algebra, there's a formal construction. This is present in a paper of Ben Knutson that constructs, I'll put an e n on the top of that. There's an e n tangent complex version of this, which is an e n universal enveloping algebra of that Lie algebra. The result is this is an e n algebra. And therefore, it defines the input to this composite construction. So I'm going to say what this is when that's closed, parallelized. And the result here is some K module. And this is, if you're wondering what this K module is, well, it's the functions on maps from the manifold into this form of completion. So number n on the right? Yeah, that's the feature. Well, that's the dimension of the manifold. So the functions on this, the observables of this, well anyway. What is m? m is smooth over complex numbers or over this K? m is a closed, parallelized, smooth, n-manifold. Smooth, it's infinity. See, infinity n-manifold? And maybe you're asking that to know what is the meaning of this, because the. The field you specified, K, with characteristic p. And m is over characteristic 0. I'm sorry, in this step, in the way that I use casual duality, I don't know it without the characteristic 0 assumption. I forgot I indicated what some morphisms in this are. I'll just leave that there, though I won't. I'll come back to it later. As another remark, I'll indicate another value of factorization homology. So up there, that previous remark said that it agrees with factorization homology for en-algebra. And for this remark, this is the intention behind this is to indicate that this version of factorization homology potentially captures something non-perturbative. So suppose you're given a sequence of maps of spaces. So I'll just call that z-bullet maps between spaces, like homotopy types. So from this, we can construct a couple things. One is c sub z-bullet. This is an n category with adjoints. Let me say what I'll describe it just by describing what the space of k-morphisms is with k-morphisms with a k-morphism. Well, it's a map from this flag of spaces to that flag of spaces, where what's the domain here? It's the flag of spaces given by the various skeleta of the closed end disk regarded, say, as a CW complex by the hemispherical stratification, so the hemispherical CW structure on the closed end disk. So in other words, a zero-morphism or an object is simply a point in z-zero. A one-morphism is a pair of points in z-zero, together with a path between them, between their images in z-one, et cetera. And this organizes nicely into an n category. So the statement is that for m closed parallelized smooth n manifold, this construction applied to this n category. That space is familiar. It's a mapping space just into the last term. So this is an enhancement of non-Abelian point of the duality. Oh, I should have said there's always a map. And it's an equivalence provided the map from the zi to zj for each i less than or equal to j is i connected. So here, the right-hand side is a mapping space. The left-hand side, as I hope we'll see, is some version of a state sum. And this is the pattern we expect, in general, is that this factorization of homology, and actually most interesting is the enriched version, is a way of giving state sum models to observables for non-perturbative signal models. So the next thing I'd like to talk about is how that result is related to three possibly also familiar constructions of TQFTs. And I'll just reiterate that my goal here is to see how to use this result to capture stuff that's familiar to you, possibly, and also extends your imagination to what else might be captureable. So there's three types of manifold invariance that are easily captured by that result. So there's the, I'll call the first one, the Jones type, because this captures a version of the Jones polynomial. So take a non-empty. What's that? Manifold and smooth manifold. Yeah, so I will always, by manifold, let's just say, I'll mean smooth manifold for this talk. So this will produce, essentially, link invariance. But I'll phrase it quite generally. So say you have any non-empty closed manifold equipped with an embedding of it into Euclidean space of co-dimension k. Together with a trivialization of its Gauss map. By trivialization, I mean a null homotopy of its Gauss map. So I'll take that as the input to this next invariant. In the case where w is a circle and n is 3, then this is, of course, just a knot. And a trivialization of a Gauss map, in that case, is an orientation to the knot as well as a normal framing. That's it. It's the same data. So now let this be a pointed n category with adjoins. You can pair this pointed n category with that higher dimensional knot to get the following. A map from k anamorphisms of that distinguished object to n anamorphisms of that distinguished object as follows. So there is this essential calculation. Recognizes this as the factorization homology on Rk of C. The morphisms that actually I'm glad that I wrote, I did not indicate what morphisms in this manifold category are. And the morphisms in it are designed exactly so that you could do the following maneuvers that I'll do in this Jones type as well as the next two types. And one of those maneuvers is I want morphisms in this manifold category, in particular, to be opposites of surjective fiber bundles. So here's a surjective fiber bundle using that W is non-empty. So the just projection from W cross Rk to Rk is certainly a surjective fiber bundle from that to that. And I'm just asserting, I'm just telling you that that defines a morphism the other way in this manifold category. And therefore, a map this way between these spaces, which are the values of factorization homology, I'm also just telling you that another example of a morphism in this category of manifolds is an open embedding. Using that this map is equipped with a trivialization of its Gauss map, it extends as an open embedding from Rk cross W into Rn. So this is induced by the link together with its framing. And then the calculation up there identifies those as well. So that's what this composite map is. And I call it the Jones type because in this example, it literally is the Jones polynomial. So if you take N to be 3, W to be the circle with a normal framing and orientation to the associated link, and C to be the twofold D loop of the category of finite dimensional representations of quantum SL2. So a result of Drinfeld gives that this category of such representations is abraded my nominal category. Therefore, it can be twice D looped just to be regarded as a two category. In fact, it will be a pointed two category. And the fact that I'm looking at finite dimensional representations guarantees that every object in this category has a dual. And that manifests then as this then pointed three category having adjoints. So we can input all that. And in that case, this arrow is simply a map from the space of objects of this to the space of endomorphisms of the unit, which maps to, at the very least, the ground ring. And this is the colored Jones polynomial. That's why I call it the Jones type. Is that gone forever? No, no, nothing. Is this a key? Ah. Can you say something about any good form, W is good to S3? I can't say anything useful. Yeah, that would be, if you, I would be keen on anything anybody has to say about that. But that is the point, is that these constructions are defined in all dimensions. I mean, finding examples of four categories is not an easy task. Was there? Yeah? There is a restriction on the co-dimension k over n. It can be co-dimension. Yeah, when it's written, there's no restrictions. Right, that's right. Yeah. No, that means how I ask the question because of the knots, it should be, that was written, A and B are the specific written with the knot. So there's another type. I'll call it the Tri-Virotype. And that's as follows. So take the pointed n category to be input, to be, now this requires some explanation. So just an aside, quick notes on some formal category theory is there from when n is equal to 1, I'll say, I hope this has meaning to you. It's the full subcategory of all k modules, just consisting of those generated from k by finite co-limits and then by retracts thereof. So this is perfect k modules. It has a natural symmetric monoidal structure, just given by tensoring two together. As so, you can look at the full subcategory of categories equipped with an action of that symmetric monoidal category. And then you can look at those generated from this itself under, again, finite co-limits. And retracts, et cetera. And that, again, has a symmetric monoidal structure. So this can be repeated, thereby defining per nk inductively as the smallest full subcategory of modules for that symmetric monoidal n minus 1 category, which are generated from it by finite co-limits and retracts. Now, it is an interesting problem to try to make explicit what objects in per n is for each n. But that is a separate and interesting thing to work out. Nevertheless, this has a fine definition. And so I'm not suggesting that you should easily know what an object of it is, but just know that this exists in order for me to proceed. So if we use the enriched version of factorization homology now, this is for this next type of manifold invariance. Well, factorization homology of the k linear category that's as trivial as possible. It's point, point, point. And then at the very top, it's just k. It's just k, certainly. And this point in there, so this is determined by x can be selected out by amorphism from this very simple. This is the role of point. So a map from this into per nk selects out that curly x. And a merida invariance in this enriched case identifies this with k itself. And here, in case I didn't say, this is a closed, parallelized, n-manifold. The composite number, then, is an element of k or a number. If you're wondering why you might expect this merida invariance, well, there is a key example. And the proof of this is the same premised on a type of excision for that thing. And so I'll just mention that in the case of dimension one, this is nothing other than the Hockschild homology of this k linear category. And by a trace map, for instance, that implements an equivalence between that and k. And, well, this is a conjecture, but we expect for that merida invariance to still be true, as implemented by a higher version of a trace. So premised on that, this defines the manifold invariance given an object in perf n. I call it toriovirotype because when n is equal to 3 and x is, so an object of perf 3 is a certain example of, with some finiteness conditions, a k linear 2 category. And if you take it to be the D loop of r, where r is a fusion category, so a monoidal k linear category with duels, which is semisimple and has finitely many simple objects. So in addition to having r having duels, to be in perf in that particular case amounts to r being semisimple and having finitely many simple objects, as well as a condition about the anamorphism of its unit. So in that case, this is the trioviro invariant. And the third type is the triovirotype is premised on some features of this conjecture, I'll just make that clear. But those are features that, if anything is true, then those should be true. This one I have less confidence about, so I'm mentioning it to more just spark your imagination. So if c is just formally equipped with a certain trace map or an action, again, m is closed, then the partition function with respect to that trace determines another manifold invariance. And I'll just give an example. So if c is finite dimensional is the two-fold d loop of finite dimensional representations over c of the unrolled quantum group for SL2 at an even root of unity, then this notoriously is not a fusion category. However, it has this ideal that's often called proj of c inside of this braided monodal category. I'll just put ret of it. And that ideal determines such an action. And in this way, one gets manifold invariance along the lines of those constructed by Nathan Greer in particular. OK, so that concludes a discussion of how factorization homology is related to other things you might know. What's next is I'd like to describe what the values of this construction are. I'll do it first heuristically. And then I'll give more details. So are there any comments or questions? I have a question about these three invariant types. Yeah. How does one go about showing that these are in fact invariants you already knew? That's a great question. So with all of them, it's premised on a type of excision for factorization homology. If that means something to you, that's great. But so speak up if you want me to elaborate on that, and I'll make some choices on whether or not I do. Question? Yes. Don't some written invariance, do they relate to this formula then? I don't know. That would be interesting to know. I'll bet people in this room have better ideas than I would generate. So let me indicate now the heuristic for what the values of this are. And as I do this, I neglected to emphasize a point, which is that somehow I feel like saying it this way is like what is being emphasized is that this is fully faithful. But even before that, it's just to emphasize that it's even just a functor. Like it even exists. It's defined that there's a way of constructing invariance of manifolds from an n category at all, fully faithful or not. For indeed, the results are not just a way of assigning some value to every manifold given an n category. But that value has an action of the topological group of difthomorphisms of the manifold. It's completely coherent and could continue us more than just being fully faithful or let alone being fully faithful. So the very heuristic is the following. So again, let's just, for discussion's sake, take m to be a closed, smooth, n manifold. And c is a pointed n category. So informally, this is a modular space of c labeled disk stratifications of m. So in other words, an element of this is like a triangulation, for example, a triangulation of m together with a way of labeling every k-dimensional face by a k-morphism in c, but not any random way of making such assignments. But in such a way that, for instance, of k as 1, so we're looking at an edge in this triangulation, that edge inherits nearly a direction from the ambient parallelization of m. It doesn't literally, but you could always tilt that edge as needed so that that edge aligns with the first direction of the ambient parallelization. And then that edge inherits an orientation. And with respect to that orientation, the labeling of the two endpoints of that edge should agree with the source and target of the labeling of that edge. To know that that description makes it seem like there was a choice involved with how I possibly had to tilt that edge. And indeed, that is a choice. But that choice is in the wash exactly because I'm assuming the end category has adjoints. So that's a very heuristic, a little less heuristic. I'm saying this to lead to the construction itself, of course. So let's see. Given a manifold, we can construct a space from it. I'll call it d of m. So this is a moduli space of disk stratifications of m. So it's like a moduli space of all triangulations of m. Also, from c, we can construct, I'll call it f sub c, a co-sheaf on d of m. And what is this co-sheaf? Well, with stock at a point in here, so a disk stratification d of m, I think, on a triangulation of m, is the space of c labelings of d, as described in the heuristic above. So with these ways of speaking, then we can take factorization homology. We could define it to be the co-sheaf homology of f sub c. So that's an idea, and it's an idea toward a definition of factorization homology. And I'll just emphasize two constructions that are starting to appear, then, from this way of talking. One is co-sheaf homology. That's a co-limit construction. Another is the association of this co-sheaf from c. And it's like a maneuver where you know what to evaluate on every disk. Now we're finding a way to evaluate on a disk stratification of a manifold. So this is like the result of extending from a basis. So this is a limit construction. This is shedding light on what will become the actual definition of factorization homology. OK, so now better still. So up there, that first squiggly arrow, d of m, that modulized space of disk stratifications, a moment's thought in dimensions more than one recognizes that modulized space as being really crazy. It's infinite dimensional. Like the modulized space of triangulations of a surface is definitely going to be wildly infinite dimensional because you could wiggle an edge around and stuff. So I don't care to actually say what that modulized space is because whatever I would mean by it in the end, this co-sheaf on it will not be a general co-sheaf. It'll be constructible. So specifically, there's a sense in which that modulized space of disk stratifications is stratified. So two triangulations belong to the same stratum if they're isotopic to each other. And as so, just that way, that heuristic description of f sub c is such that the values of that co-sheaf depend continuously through such isotopies. So instead of concerning myself with defining that modulized space and then constructing a co-sheaf on it, I'll recognize that modulized space, once I would have defined it, it has a stratification to it. And then this co-sheaf would be constructible with respect to that stratification. And there's a classification of constructible sheaves on stratified spaces. It's namely, it's functions from the exit path category. So I'm not even going to define that modulized space. I'll just define its exit path category. And I'm only going to be interested in that so that I could then hopefully organize the data of an infinity n category to construct a functor from it into spaces and then define this to be the co-limit of that functor. So the intention with me talking in this way was to make it seem more plausible to even implement this construction through standard categorical constructions. So in the remainder of the time, I'm going to indicate what this infinity category is. Any comments or questions? So definition. So Mfdnsfr is an infinity category with an object or for which an object is. I set it over there. It's a stratified space, x, together with an injection of its tangent constructible bundle into a trivial rank n bundle. Amorphism is. So I hope, good, I left it. Whatever I'm about to say amorphism is. And almost all of the craftsmanship in designing this category is so that all those become examples of morphisms. So I'm going to say what amorphism is and then I'll say how those give examples. Amorphism is a constructible. I'll say what that is momentarily. Over the one simplex, I set one simplex so that you can start to envision that this is being defined in familiar ways of defining infinity categories as a complete seagull space. So it's a constructible bundle over this stratified space. It's just an interval with two strata. And what does constructible mean? To say constructible bundle, this means the restriction over each stratum. So x restricted to that stratum and x restricted to this stratum are fiber bundles of stratified spaces, of course. So the topology in the fibers of this map doesn't change anywhere except as you jump across from that point to the other stratum. Together with an injection from the fiberwise tangent constructible bundle into a trivial rank n bundle. And I won't even say I'm supposed to be defining an infinity category. What's the composition rule? Well, informally it's given by concatenating two such things along their common boundary. A little more precisely, it's given by constructible bundles over two simplices and simplices in general. But I'll just leave it at this. There's two constructions, two examples of such that I'll mention to see how those give examples. So if you have x0 to x1, a refinement between stratified spaces onto its image, which is open, then there's a construction you can do from it. Call it the open mapping cylinder, sil o for open, and it's defined to be x0 across the closed interval, union x1 across the half open interval, over x0 across the half open interval. And this way of talking evidently gives a canonical map down to the closed interval. And that canonical map is a constructible bundle. I won't mention the tangential structure that comes along for the ride. And there's another construction of such a morphism, given itself a proper constructible bundle. There's a reverse mapping cylinder construction. It's very similar. And this, too, is equipped with a natural map to the closed interval, making it a constructible bundle. And there's a fact that these two classes of morphisms in this category MFD, or this manifold category, generate all of them. These types of morphisms generate this infinity category, the morphisms in this infinity category. I wanted to indicate some of the technical results in this. And that's one of them. It amounts to some business about stratified space analysis. And now finally, disk NSFR is the full subcategory, consisting of those. So it's the smallest containing these hemispherically stratified k-discs equipped with the tangential structure given by just drawing them literally in a standard way in Rn and closed under gluing faces together. So then to finish over here, I wanted to say what this infinity category is. And I'll say it now. So this makes available a precise definition of this. And a precise definition is definitely necessary to get anywhere with this. This is disk sub NSFR. It's the slice category over M, but not any morphism from an object here to there. But only those that come from that reverse mapping cylinder construction on refinements, or sorry, the open mapping cylinder construction on refinements. And then just to finish literally in that space, I want to indicate where the hard parts are, because I would like to know that if I saw a talk like this. One hard part, so this is the hard parts. One hard part is matching together essentially the higher categorical adjoints and this tangential data of solid framing. So matching higher adjoints and solid framing. And this amounts to inductive description of the orthogonal group as it's decomposed by the Bruha decomposition. I'll say of GLN of R. I said I would fit it in that little space, but I guess I wasn't able to. The other hard part is in showing that it sounds silly. This is just technical, but a lot comes down to it. That the evident inclusion is final, which is to say that co-limits indexed by this agree with the co-limits computed by this. And the essential technical stuff behind this, much like the Bruha decomposition comes in there, is existence of common disk refinements, even in families. So I think of this as some version of Whitehead's unique existence and essentially uniqueness of PL structures on a smooth manifold. And we had to develop some category theory in order to proceed with this. And so I'll just mention there's a paper called Vibrations of Infinity Categories that allows even a way of recognizing certain co-limits of higher categories to be accessible. Thank you for your attention. You promised also to say a few words about the essential image of this function. Oh, yeah. Yeah, so not part of this conjecture is, but we still expect it to be true, is that the image is as follows. It's characterized as not any space valued functions on this, but those that satisfy two descent type properties. The first is that if you take a stratified space and just break it up as a union of its closures of its strata, then that determines a certain kind of cut-paste type cover of it. And the values in the image here assemble as a limit of the values on those parts. So that's what we call a closed sheaf. The other is excision. And that's a little more complicated state, but it mirrors the notion of tensor excision for E and algebras. And establishing excision is an even more complicated version of this, but we expect it to be true. And I didn't mean for that to be part of what I mean by 90% theorem, but excision is something. That's like a 65% result. So yeah, closed condition and excision is the image. So from this model space of this stratification of M, can you find anything related to maybe Donaldson, Thomas kind of invariant or numerical invariant? Is this possible? Because I understand this is infinite dimension. Maybe part of it contains some cycles that you can integrate out some numerical things. Is it possible? Let's see. So I guess very similar is just, in general, a modular space of stratifications of like a smooth four manifold. In particular, in there, are those stratifications just by an embedded closed surface? Is this the kind of thing that you have in mind? And then Donaldson theory has invariance of such surfaces. I don't know.