 First of all, I would like to thank organizers for giving me a chance to talk in this conference. I'm gonna talk about two-dimensional extent homotopy field theories. More precisely, I will tell what they are and what we know about their classification. Let me start with two-dimensional topological field theories. If you remember Claudia Salk, she mentioned two ways of studying topological field theories. One is as a functorial field theories and the second one is factorization algebras. This is completely just factorial study of topological field theories. Let me remind you what is a two-dimensional topological field theory. It's just a symmetric monoidal functor from coborism category to any other symmetric monoidal category. Let's take this to be categorized back to space and linear transformations. And what is an object of this category is just oriented circles. Morphisms are just oriented from a class of co-bordisms between circles. Like an example of co-bordism or morphism in these categories. Something like this. The first thing is to realize is that this category is not hard to understand. Basically, objects are just disjointed of circles. The co-bordisms are just like this and just writing, it is not very hard to write generators and relations for the co-bordisms of this category. And then it is under disjointed and it is it is really well understood and they are classified by commutative robin solgibus. Like 2DTFT's Abram classified down 96 by commutative robin solgibus. And we want to understand homotopy field theories which are introduced by Tryev in 1999 which are basically just Tryev applied the axioms of topological field theory to metaphors and co-bordisms which equip with maps to some fixed target space. Therefore to define HFT, homotopy field theory, we need to fix a target space. We call it x and for this talk it is always be a KG1 space which is the homotopy type of KG1 space. Basically, we want to understand those two-dimensional TFT's with principle G bundles for the script group G and how do we do this? We just take homotopy class of maps to this KG1 space, which is a pointed space and now we define this new category x-cope 2 whose objects are pointed oriented circles with homotopy class of pointed maps to this space and co-bord-morphisms are co-bordisms equipped with homotopy class of maps to like this pointed homotopy class of maps to this target space. And if we can think of just taking the universal cover of this by pulling back along this homotopy class, we have a principle G bundle over this thing and try to classify such TFT's by crossed Robbins algebras to the HFT's with target x, which is the KG1 space, by crossed Robbins algebras. What is cross Robbins G-algebras? What is cross? Cross replaces the commutativity, but I mean we have a G-algebra. When we take two elements, AB is not necessarily BA unless G is commutative, but if G is not commutative, we have an automorphism of algebra, which replaces commutativity. This is the cross structure. Yes, it's a G-graded algebra with inner product, non-degenerative product, and there's an automorphism of where rho is here. Rho is an automorphism of A, and it's like conjugation type. We also have extended field theories, which are symmetric monoidal two-functors, two-dimensional extended field theories, symmetric monoidal two-functors from extended borderism category to some symmetric monoidal bi-category. Let's take the bi-category of algebras by modules and by module maps. Christian Mopris classified them in terms of some separable symmetric Frobenes algebras, and he introduced this bi-category. Let me tell what it is. Objects, monomorphisms, two morphisms. What are the objects of this board? They're just oriented points, like this, and objects of this bi-category just K algebras. And monomorphisms are just broadisms between points. Like this. And here monomorphisms are by modules. And here two morphisms are certain type of surfaces with faces. What do I mean by certain type? Make it more precise. What I mean by certain type is this vertical boundaries are always trivial, or product type. There is no vertical boundary like this. They're always product of the zero manifolds and boundaries at the corners. And here two morphisms are by module maps. And Christian Mopris classified such symmetric monodil two functors in terms of specific K algebras, which are separable symmetric Frobenes algebras. Let me write it here. To the extended TFTs separable symmetric Frobenes algebras. Well, what does this line segment means is a natural question. This term is also called zero one two theories. Say because we also, the data that this gives us is assigned some some algebraic data to points, monomorphisms, and bordems, and bordems between bordems. What this line segment means is this, we can think of HFTs as generalization of this because for each homotopy field theory, by just taking trivial homotopy classes, we just get a topological field theory. And similarly, if it's in a similar way, if you start with an extended field theory and we restrict ourselves, they just forget the points, but circles and more circles and co-bordems between circles, we again get non-extended TFT. Once we have this diagram, it's an interesting question. I believe is to understand what are two-dimensional extended HFTs. And Golo, this talks to define these things, these gadgets, and classify what they are. Any questions so far? Can you vary the target category? No, to define HFT, we just fixed point to target. Yes, yes, yes. There is a Christian Maupers has a classification for any symmetric model by category. We have a similar it is not as clear as this case, but it is given by some data in this category, symmetric model by category and we have a similar result generalizing this thing. And definition of two-dimension extended homotopy field theory is fairly straightforward. We do the same thing as we did here. What we do? We fix this KG1 space, point it, and consider pointed homotopy classes. So what do we have? We can, since these are all point to homotopy classes, we can just associate, we can canonically assign some group elements to these things. But we want specifically for two morphisms to be these vertical faces, boundaries to be identity. Where E is identity element. Yes. We always want X to be KG1. HFTs drive the finds for arbitrary CW complex, but for general arbitrary CW complex there is not a classification theory. But once we take KG as a KG1 space, we have good classification theory. And that is the kind of where this homotopy name comes from. When we take space X as any arbitrary CW complex, we just consider maps of these things, not homotopy class of maps. But once we take KG1, there is no higher K invariance. Maps can be replaced by homotopy class of pointed maps. This is the definition of, we define this symmetric monoid by category of this type. Could you, could you replace X instead of by just a KG1 by a KG1 sort of in a, sort of a Posnokop tower to level 2 or something? There are, there are not in this way, but there are works for such things like KG2 and non-extended case. But in this case, we just stick to the KG1. Yes, this is the definition of two-dimension extended homotopy field theory. And now what is classification? Theorem is this. There is an equivalence of by categories, which is, let me tell what these by categories are. Objects of this by, this by category are just extended TFTs, extended HFTs. One morphisms are just symmetric monoidal transformations between them. And two morphisms are symmetric monoidal modifications. And these by category objects are what we call quasi-biangular G-algebras. And one morphisms are compatible G graded mortar context. And two morphisms are equivalences of G graded mortar context. Let me tell you what is this classifying object quasi-biangular G-algebra. Its name is a bit strange, but it is just some sort of some sort of Frobenz G-algebra. We call this quasi-biangular algebras because it's generalized by angular algebras introduced by Triam in the study of lattice HFTs. Their definition is just not that complicated. Quasi-biangular G-algebra is a Frobenz G-algebra. That means a G-algebra with a non-degenerator product such that the principal component, the identity component, Ae is separable. And each component Ag is both left and right rank one Ae modules. In short, in here we have quasi-biangular algebras G-algebras. What they are, they are basically just copy of this for each component. And here again we have to go here and go here. Here we just restrict to those maps with trivial maps. It gives here, which just restricts to identity, this principal component. And here we just restrict to just circles and cobaltism between them. A G-grade mortar context. This is G-grade mortar context is defined by Boisan in 94. It's just a it's a generalization of usual mortar context in the way that modules become G-graded. What does it mean? Basically G-graded mortar context between G-algebras is a quadruple is a unit and coordinate of the injunction that the usual things by module map. And these both are invertible. And this M and N are G-graded modules. Like what do I mean by this? Like BG X1MH MGH and similarly for N. So does this mean that you have a 2-1 category there? That 2-0 category? Yes. Yes it is. It is 2-0. Yeah these are all invertible. Any coolness of great mortar context is just a straightforward thing. If you have another great mortar context it's just a map of by-module maps which commutes with these things. Virtual by-module maps. Any other question? And the main corollary of the main corollary is the structured co-board is my part this is in a very special case. The corollary of the Vennk Let me take this algebra K Vennk is algebraically closed field of characteristics zero GSO2 structured GSO2 structured co-board is my part this holds for 2D SO2 structured algebra K valued Extended TFTs holds true. The proof of this is basically just comparison with Ordovidovich GSO2 fixed homotopy GSO2 fixed point computation where she has just semi-simple G graded algebras overcome like this field and each component of the thing is just rank one principal component but the same simple separable algebra is just the same as semi-simple that are in under this condition characteristics zero fields. Maybe I can remind to people what is the structured co-board is my part is this briefly for a group gamma gamma structured co-board is my part is this is due to Lurie that is stated as follows symmetric monoidal functors to any symmetric monoidal N category answers I need to explain what these are this is an equivalence of symmetric monoidal infinity categories for any symmetric monoidal infinity category C this board N is fully extended board is in category with structure gamma which means we have a principal G bundles over all manifolds principal gamma bundles over manifolds and these are all TFTs symmetric monoidal the infinity category of symmetric monoidal symmetric monoidal infinity functors and here we have this full subcategory of infinity and subcategory of C consisting of all the divisible objects this is once we take the subcategory we take the underlying infinity group point and we take the homotopy fixed points where this is basically and the specific case we have the correspondence there's W Dovich compute homotopyx points and classify such extended HFTs with the structures and we have our proof does not use any cobalt as my part of this and we get the same result in this in this case not exactly I can tell a couple of words about the proof of this direct basically proof in a sentence we just generalize some of the techniques Schubert-Briss introduced in his thesis to this manifolds equipped with maps to this KG1 space how do they go that is this way he has a planar decomposition theorem and we make it into G planar decomposition theorem what it is it is basically replacing those objects with diagrams with some combinatorial data and we encode what what I did is just encode this homotopyx class of maps or we just just just label links into his diagrams consistently and to get the G planar decomposition theorem like what does these diagrams in dimension one that is basically just Morse theory it's just consist of diagram consists of critical values of Morse function just in this case projection and some some nice open cover here we have some G label and we encode this adding this G data and here with some additional data on this diagram in dimension two he stratifies this jet bundles for a surface sigma and he gets similar data in some diagram in R3 in R2 and in in for and since we take equal deformation class of surfaces he also do this is like two dimensional Morse theory and then but since we have deformation classes he also stratifies this jet spaces like two dimensional set theory how these diagrams are related understand and we do similar things in in his diagrams we add some additional data which just encodes this this characteristic maps of the this maps this characteristics maps of the manifolds and we get the G planar decomposition theorem and the his theorem and why these are useful then once we have this G planar decomposition theorem we basically replace this category with a coolant category category P lanar diagrams this category just consists of diagrams and this is freely generated and Schumer-Pries has a coherence theorem for symmetric monoidal two functions out of freely generated by categories like any it says that means what what do I mean by freely generated there is a list of generators for objects one morphisms two morphisms and relations within two morphisms and the coherence theorem says symmetric monoidal two functions out of such freely generated by categories are precisely determined up the coolant up the coolant to where generators cause subject to relations and that gives the classification exactly what he did and what the additional data that we encode in this diagram does not change anything in this algebraic methods yeah that's can you modify your proof to also get like the framed version or unoriented version of something like this framed version is done by Piotr and unoriented we have the same thing I have unoriented case just generalizing the Schumer-Pries unoriented case similar way we have kind of G still algebraous again up to G graded more technical ones more context like a general statement for general yeah for for we have a result for any target just as in this which just consists of some data which is the image of the generators and relations in here but for any like group mapping to O2 or something like could you do spin case spin case hopefully next project there are some work not complete the work on by someone else the related spin two-dimension spin field theories with homotopy field theories yeah I'll try to do there's no work yet spin case but yeah possibly favorite example favorite example of G algebra just matrix algebra's works for this one matrix algebra's are works and our group links also work our cost to normalize two cos cycles works are the examples or just taking any anything with this and copies of G copies of this it's not boring is the speaker again