 Hello and thanks for watching this video on bifibration of polycategories and the relation to classical denology. I'm Nicolas Blanco and this is joint work with Noam Zalbego. You can find the draft paper for this talk on my website at VSUile and I will also post the slide on my website. So before going into the details of polycategories, let's first recall some context and especially models of multiplicative linear logic. So to model interesting multiplicative linear logic we can use monidel closed category and for classical multiplicative linear logic we can use start and most category that were introduced by bar or linearly distributive category introduced by croquet and city where a linearly distributive category with joules is equivalent to start and most category and in both case the idea is that you will ask for two models structure a new category that will interpret the tensor and the power and a monoidal duality between those that will interpret the negation. Another way of interpreting logic in you know categorical setting is by using multi-categories. So multi-categories have been introduced by Lambeck so it's some knowledge by Lambeck to represent second calculus to interpret the synestically second calculus. So in a multi-category instead of having morphism that have one input and one input you have multi-maps whereas the domain can have many objects. So using this multi-categories to represent second calculus is inspired on one way on one hand so by the notion of multi-linear map in linear algebra and on the other hand by the interesting second calculus. And so using this interpretation we can define the connective by universal property instead of defining them by structure. This can be extended to classical second calculus by considering polycategories. This has been done and introduced by Zabo which was one of Lambeck's students and the idea is that in a polycategory you have polymap where now you have when up both the domain and co-domain have multiple objects and the cut the sorry the composition is done is done along one object to mimic the cut rule from second calculus. So this makes it different from poor or poor for example where you can compose along multiple objects at the same time. So polycategories have been used by Coquette and Silly to model a multiplicative linear logic in what they called two terms of polycategories with duals and we can also find in the literature the terminology representable star polycategory which is the one that we will use in this talk. And finally I want to mention another line of work by Cloggio Amida. So in the at the beginning of the millennium Amida observed that there are a lot of analogies between multi-category theory on one hand and five-part category theory. So if you go to his nice paper there you will see a table table that recap all these analogies and to make these rigors you introduce the notion of multi-categorical vibration. It's more precisely of co-variant vibration of multi-categories. I did that in 2004 and something that is quite interesting is that multi-categories that are fibered over one over the terminal multi-category are in correspondence with monoidal categories. So this makes this relation precise between representable multi-category on one hand and multi-category fibered over one on the other hand. He also used this theory of vibration of multi-categories to study the algebra for an upgrade. He proved that an algebra for an upgrade is the same thing as a discrete vibration over this operator considered the one object multi-category. After that people have been interested in introduce contra-variant vibration and then bifibration of multi-categories. So it has been used for example recently by Likata Shulman and Wadi to represent some logical structure. This is closely related to the notion of vibration of close categories. So if you take a multi-category that is bifibered over one then it is the same thing as a monoidal close category. If your multi-category is co-variantly fibered over one you get just a monoidal category and to get closeness you need to also ask for it to be contra-variantly fibered over one. And there is an important asymmetry event that is that the pullback so for contra-variant bifibration has to be parametrized by a choice of inputs. For the push forward you don't have to do that because you just have one object in the co-domain but in the domain you have possibly multiple objects so you have to choose one along which will perform this pullback. So in our paper we have several contributions. The first is to define a notion of universal polymap that will be parametrized either by an input or by an output. And using this notion of universal polymap we can get all the connective of mll by saying that they are characterized by the existence of certain universal polymap. So with this analogy in mind we introduce star-representable polycategory which will be polycategory that has all possible universal polymap. And then we prove the following theorem which is our first contribution that this notion of star-representable polycategory is equivalent to the notion of representable star-polycategory that has been studied by Coquette, Silly and others to represent multiplicative linear logic. After that we define the notion of vibration of polycategory that extends the notion of vibration of polycategory. So like in the contra-variant case for polycategories, in the vibration of polycategories the notion of Cartesian polymap will be parametrized by a specific input or specific output. And we prove that the polycategory is bifibred over one if and only if it's star-representable. And so this gives another a correspondence between star-polycategory and polycategory bifibred over one. And so one of first motivation to study that was to study the structure of the star-polycategory and the category of finite-dimensional banana spaces and the contractive map. So something that is really interesting about this this category is that it's a category that comes from the category f of vector spaces. So it has a forgetful function in it. And it is also star-autonomous but it's not compact close contrary to the category of finite-dimensional vector spaces. So there are two norms that you can put that you will put on your tensor project that will give you the tensor and the par all-code projective and injective cross-norm. And they are known in the community of functional analysis. For example, because they have this universal property that they are extremal on the set of norm that you can put on the tensor project. So if you look at any norm that you can put on the tensor project and that is well-behaved in some sense, will lies between those two norms. And we will see that this property of being extremal come from the fact that they are defined using Cartesian polymaps. And finally there is a third contribution that we won't have time to get into the details in this paper, in this talk, sorry, that is polycategorical growth and the correspondences. So we give a collection of different growth and the correspondences in the polycategorical case where, for example, we give correspondences, one-to-one correspondences between one-on-one and bi-fibration of polycategories and on the other hand, pseudo-functors that goes into the two polycategories of multivariable adjunctions. So using this fact, we recover an observation by Max Schumann, but start on this category of Frobenus pulmonary in this two polycategories. So now let's get into it and first define polycategories. So our polycategories will be taken to be planar, which means that they are not necessarily symmetric, basically because it's possible to define that and symmetry doesn't doesn't really play any picture in this story. So you could take symmetric polycategories and do everything that we are doing here with only slightly, slight modification. So we thought it was better to do the more general case. This is in line with the work by Croquet and Cilly, where they consider linearly distributed category to be non-necessarily symmetric. So in particular, the notion of start-on-mouse category that we have is also non-symmetric. So it's planar start-on-mouse category. And in particular for jewels, we have both right and left jewels. That being said, a polycategory is a structure analogous to a category. It has a collection of objects, except that it has a set of polymap from any finite list of objects to any finite list of objects. So the domain and the finite list of objects. These lists can be anti. It should have identities, the notion of composition that are unital associative, that have an intention flow, but will you explain later? So the fact that our polycategories are planar and not necessarily symmetric puts some restriction on the composition, but I will explain just now. By looking at how to represent polymaps graphically. So the idea is that we will represent will represent our string diagrams that goes from left to right. And a polymap will be represented by a box that takes some inputs. So they're gamma and spit out some outputs. So delta so delta one, delta two, and A. Then we will represent composition by plugging the wire from one output of F into one input of G. So we can see that that composition happens only along one wire, one object. And so the planarity condition, which is also known as the no-crossing condition, state that wires should not cross. And so in particular it will, so what it will exactly mean is that either this list or this list is empty, and either this list or this list is empty. Finally, the intention flow say that taking polymap H, and if I want to composite with two polymaps F and G along two different inputs of H, then it shouldn't matter in which order I'm doing it. So I can start by plugging F and then G or the other one. And of course, there is something similar for outputs. Finally, I want to give you some examples that will be important in this talk. First, there is a terminal polycategory. It has only one object and one polymap for each arity. So given two arities M and N, there is a polymap that goes from M copy of star into N copy of star. So this terminal polycategory is trivial. It's not really hard to define, but it will be really important to know the theory. Then given a linear redistributive category, as introduced by Croquet and Silly, or if you would prefer a star autonomous category, which is a special case of linear redistributive category, you can define a polycategory where the polymap goes from the tensor product of the input into the part of the outputs. And in particular, any monodal category can be considered as a linear redistributive category where tensor product and part are taken to be both to coincide. And for star autonomous category, it will give the notion of compact close category. And in this case, of course, we get a polycategory where the polymap are given by the map that goes from the tensor product of the input into the tensor product of the output. So just for information, I'm just recalling the second calculus rules for multiplicatively analogic. If you are not really familiar with it, I invite you to pause the video. And using that, we'll define in a polycategory, which is a tensor object. So a tensor object will be taken to be an object that comes with a binary map that goes from A B into A tensor B. So you can think of vector spaces here. And so it's a billionaire map that goes from two vector spaces into that tensor product. So it should have a universal property that say that any map that has A B in the input should factor uniquely through this A, through this M, this billionaire map. So this is the property in vector space that tensor projects linearize billionaire maps. And we choose this notation because this is closely related to other rules from the second calculus. So this corresponds to the tensor left rule. So given something that has A and B in the input, I get something that has A tensor B in the input. And this M is closely related to the right rule for the tensor product. We get it by looking at the right rule and taking the special case where you have identities. And so by an identity from A, A and B, I should get this map that goes into A tensor B. And this will induce a natural isomorphism that goes between A B and between maps that goes from A B and map polymer that goes from A tensor B. So in one direction, we get this isomorphism by composing with M and on the other direction by factorizing. So this natural isomorphism is represented by this correspondence between these two maps. We have something similar for the power, but where we exchange the notion of input and output. So we'd get a co-binary map, a map that goes out of the power into A B, but as a factorization property where this binary map is now post-composed. So everything factors through the post-composition by W. And natural isomorphism where now we have A Bs in the output and A that gave A power B in the output. Finally, we can define the star in a way that is really similar to the way that it's defined in compact class category by saying that we want two maps that we'll call the cup and the cup, but as this, this typing. So the cup goes from the anti-list into A A star and the cup goes from A star A into the anti-list. So that they satisfied what is called the snake identity. So it's going to say snake identity because if you draw it, so you will get a snake like that. And taking the snake identities is yanking the snake. And so taking the snake and making it an identity. So we'll have both snake identities and this will induce a natural isomorphism between a map with A in the input and map with A star in the output. So using all of this notion, we get the notion of representable star polycategory that has been defined by your eco-cat and series by saying that it is a polycategory that has all times A power left and right joules. So now that we introduce that, we want to give a slightly different definition that use the notion of universality. So the idea is to get one notion or two notion of universal and universal that encompass all of the properties of the connective. So we'll say that the map is universal in one of its inputs or one of its outputs if it has this kind of factorization property. So for example, U is universal if in this output, if any H that has the right kind of typing but will make it a potential composition with you will be indeed a composite. So there will be a unique way to factor H through H, there for you sorry, and very very run for inputs. So it's a same kind of factorization property but now both composing by the universal map. So in particular for there if we take U to be the binary map that goes from the answer from the from A and B into the answer project, then this would be exactly the property of the universal property of the answer project. It will say if I have A, B in the input of a map, then I can factor it to get a map that has A times B in the input. Using that we say that a poly category is star representable. If it has all universal object, what we mean by that is that if you give me some object, some list of objects, that will specify all the input and output of a map except from one. So we will leave one input empty. Then I can give you this input and a poly map that is universal in A and of course the same for outputs. So if everything is specified for the poly map except from one output, I can find this output in a universal way. So such that it comes with a universal map. And if a poly category is star representable, we prove that it is equivalent to the poly category being representable star poly category. So it's equivalent to to be star representable and to have all the answer parts under yours. To do that one way is mostly by definition. So we get by definition that the answer and the power are universal objects. So we get them from star representability. What is maybe a little more surprising is that we can also get duals by the same kind of the same scheme of universal property. So we get for example that the dual, the right dual can be defined as a universal object with respect to a cup map. And the universal property of this cup map will let us define what is a cup and it will have built in the fact that it is a snake and entity. But we could choose the other way around and say, okay, we will define duals by the universal property of the cup. And using this universal property, we can derive a cup that will have the snake equation. Also, we should note that cup and cup are also universal in A. And we can use this fact, for example, to prove that the B dual is equivalent to the object that we started with. And the other way, if you give me representable star poly category, so something that has all the answer parts from duals, I can build star, I can prove that it is a star representable category. So I can build any universal poly map. And how does this work? This work by, so if you give me the typing of the map, except for one object, I want to derive this object and our universal map poly map in this. So the way that we do that is that we use a cup to take all the outputs and put them into inputs by starving them. And then we take all the inputs and we transfer them to get a final object. And so this object will come with this universal poly map. We can do something similar for universal by flipping. So exchanging input and outputs and replacing all the transfer by parts. So the cup will be flipped and we just give cap instead. So this gives us a first contribution, which was to give an alternative definition of what it means for poly categories to be representable. And now we can go and define bifibration of poly categories. So first some terminology will say that we will call the front of poly category, the poly refinement system. This terminology is inspired by the study of the type refinement system as a factor of category. So we will draw them graphically in this way, vertically, where the diagram on the top will be in the domain and the diagram on the bottom will be in the pre domain. And in such a way that things that lie over another one are sent to it by the poly refinement system. So for example, r is sent into a, phi is sent into f. And so this precise diagram is expressing the preservation of composition because it says that a composite is sent into a composite. With this graphical representation, we can give a concise definition of Cartesian poly map. So both in and out Cartesian poly map, which correspond to, for the unit correspond to Cartesian map and the out correspond to Cartesian map. So they are defined for example for out Cartesian in this way by saying that phi is out Cartesian in this output. So it's still parameterized by one, one specific output. If any map that lies over a composition by the image of f will factor for f. So if it's lie over a composition, then it's a composition, basically. And the same is true for in Cartesian with respect to one specific input. So if something lies over a composition by the image of this input, so along the image of this input, then it will be a composition along this input. So something that is interesting here and that I will come back to later, but I want to explain it graphically. It's to give you some sense of white work is that if I take the base category, the base category to be terminal, the terminal category, then all this condition will be trivial. So everything will be trivialized on the base and we can just forget about the base. And we see that this factorization property then becomes just the factorization property of universal map. So this explains why your polycategory that is bifibre over one is the same thing as the star-representable polycategory, because then Cartesianness corresponds to universality. And in the similar vein, if the bottom diagram is a universal diagram, then the top diagram will be universal. And so we can think of Cartesian morphism to be a way to lift universality. So universal map from the bottom polycategory into the top polycategory. Talking of lifting, we called Cartesian lifting any map obtained like that. So given a polymap in the base and if I have the typing of every, so if I have a refinement, if I have an object over every object of the domain and codomain of f, except for one, I should be able to find one object over a in such a way that it comes with a Cartesian polymap that lies over f. And so this object that lies over a, we called the pullback of sigma along f in context pi1 and pi2. We often just refer to it as a pullback. And the map will be called the Cartesian lifting of f. There is something, of course, similar for output. And we called the dual notion of a pullback a push forward. We will also talk about Cartesian lifting. And so finally, a poly refinement system is called a pullfibration or just a fibration. If it has all in Cartesian lifting, it will be called a pushfibration or upfibration. If it has all out Cartesian lifting and a bifibration if both are true. So if it has all Cartesian lifting. And as I said before, a category, a polycategory that is bifibred over one corresponds to star-representable polycategory. And this is just because the notion of Cartesian map will be parametrized by some base polymap that is trivial. And so this will correspond to universal map. And as a corollary, we get that star-autonomous are just represented by polycategories fibred over the terminal polycategory. We can use this to lift the star-autonomous structure. So as I said, if you take the Cartesian lifting of a universal polymap, you get a universal polymap. So Cartesian is a way to lift universal polymap. And so using this fact, if I have a poly refinement system between a base category that is star-representable, so it will have all universal map. And in such a way that this poly refinement system has all Cartesian lifting of universal polymap, then the top category will be star-representable. So this is a way to lift star-autonomous structure, because we will lift all the connectives by just taking the Cartesian lifting. So we can use that as an example that I mentioned before of banana spaces. So first we can define the polycategory of finite-dimensional vector spaces by just using the fact that the category of finite-dimensional vector spaces is compact close. So we get poly-linear maps as a polycategory that are defined by a map that goes from the tensor into the tensor. And then we can define a polycategory of finite-dimensional banana spaces and contractive poly-linear map just by defining what it means to be contractive for a poly-linear map. And so this equation holds that for any object A1 through AM in A1 through AM, and any linear form phi 1 through phi m that goes from any B's into the field, into a by-field, if you want to think of it, object of the duals of the B's. Then when I plug the input into F and I take, so this will give me an output that is in the tensor product of the B's, then I can take this output and give it to the tensor product of all the linear form. This will give me a scalar. And if this scalar is lesser or equal or has an absolute value that is less or equal to the product of the norms of the A's and phi's, then we'll call it contractive. And so in particular if you take F to be a linear map, then this notion will be contractive. So it will be the fact that F is of norm less than 1. So something to notice and that is really important is that we can define this polycategory without mentioning any notion of tensor product in a banana spaces. So we have a tensor product in vector spaces but we don't know how to lift it into banana spaces. And the notion of contractive poly-linear map is, I think, something that is easy to define, especially if you compare it to the notion of tensor product in banana spaces. So from that we get a forgetful function but just forget about the norm. And this forgetful function won't be a bifibration in general but it will have all Cartesian lifting of universal polymap. And we saw that it is enough to lift the star-representable structure. So this will give us a star-representable structure on f-band 1. And we will get, for example, that the tensor product in f-band 1 will be the push forward along the binary map in f-vect, the map that goes from a b to a tensor b. So we take the push forward and this will give us the tensor product. And in the other way around the power will be defined as a pullback along the co-binary map that goes from a tensor b into a b. And if you look, so it's explained in more detail in the paper, that if you look at what are pullbacks and push forward in this category for relatively to this factor when they exist, this gives you that the tensor product is a projective cross-norm and the power is the injective cross-norm. So as I said, I find it really exciting because it's hard to see when you look at this norm that there are indeed the right norm for the tensor and the power, that this thing put a start-almost structure on f-band. But they have this universal property. They have this property of being extremal among all the good norm that you can put on the tensor project. And this reflects exactly the fact that they are defined by a push forward and a pullback. And so this gave us a way to find the start-almost structure on finite dimensional banner spaces by a universal property and just by defining the notion of contractive pullback. Okay, and so to conclude, in this talk I presented to you a vibrational perspective on models of classical multiplicative linear logic by showing to you that start-almost category correspond to poly categories that are bifibred of the one. And using this perspective, I showed you how to lift the start-almost structure along a factor if you have a good notion of poly map on your top category and your factor has enough vibrational property, has enough Cartesian poly map. Something that we also discussed in the paper and that I don't have enough time to go into is a bunch of polycategorical growth and the correspondences. And so in particular, a growth and the correspondence between bifibration of polycategories and pseudo-functor into the two polycategories of multivariable adjunction. And using that, so if you specialize this fact to the base polycategory, being the terminal polycategory, so we know that bifibration over the terminal polycategory are the same thing as start-almost category using the growth and the correspondence. So we can get a pseudo-functor. So this will be equivalent to a pseudo-functor that goes from the terminal polycategory into the two polycategories of multivariable adjunction, which is the same thing. So pseudo-functor out of the terminal polycategory is the same thing as a Frobenus pseudomonade. And so we get that start-almost category and Frobenus pseudomonade are in correspondence. This is something that has been proved and observed by Schullmann recently. So some future work that we'll be interested in is first finding a very interesting example of polycategories of bifibration. Then using polycategories of bifibration to build a polarized model for linear logic. And finally, we will be interested in looking more precisely, specifically, as the case where the base polycategory is compact close, which is, for example, true for vector spaces. And so thanks for listening to this talk. If you have any question, you can contact us on the Slack channel during the conference and feel free to go on my website and check the paper and also check this slide if you want to. I will leave you with this nice table that recap all the models of classical multiplicative linear logic that we talked about in this talk. So first start-almost category where connectives are represented by monodestructure, then representable polycategory where they are represented by universal objects, bifibrate polycategory over one, where they are represented by push forward and pull backs. And finally, for monoid in the two polycategories of multivariable adjunctions, where you get the connective by multiplication and commutification. Thank you.