 Hello, I'm Niccolò and this is joint work with Tharmo from the University of Reykjavík and Anzal Berger from the University, from the Corporate Technique. And the starting point for our work was the definition of the notion of closed-category, which was introduced by Alembar Ekelin in 1966. So these are categories which come with a unit object, typically denoted by i here, and an internal hole, a-lolly-b here. So we use this notation from renal logic to indicate the internal home or the internal function space. I'll use this logic notation. This is an ocean that appears in a lot of examples. So, for example, in categories of structure sets, such as the category of monoids and the category of posets, for example, for monoids, the unit would be just a singleton set with a three-bit mono instructor, and the internal home made V would just be the monoint of the mono and homomorphisms between A and B. And another class of examples is given by the categories that underlie inductive systems. The main example here, perhaps, is the single type from the calculus. So now i, the unit object, would be the unit type, and the internal home is just the type of functions from A to B. And in many cases, the internal home actually comes from an assumption with a tensor product of another category. But in the original definition of algebraic and Kelly, this is not required. There is no requirement for the existence of another structure. But if such a structure is present, so if both a tensor and a lolly are present, then when they are, and they are related in S way, in the right way, then actually the closed structure comes. One can prove that a closed structure comes from the mono instructor. And it can be specified in terms of the mono instructor. So this is the theorem that can be found in the original island of Kelly paper, which states that if you are a category with a unit i and two functors, so a tensor and a lolly, now related by disjunction, that is, that's natural in B, then the pair i tensor is a mono instructor on C. If and only if i lolly is a closed structure on C, and remember this assumption here holds internally. So, as I said before, there is the monodal structure in the presence of such an assumption, the monodal structure completely characterizes the closed structure. But if you want to characterize the monodal structure from the closed structure, then there is this additional requirement of the internal adjunction. What does it mean? It means that the following natural transformation here is required to be invertible. And this just states that it's just the fact that this is an isomorphism is the same. It's just an internal variant of saying this adjunction. Yeah, the fact that this is just an internal variant of this adjunction here. And so if one checks the details of the proof, we'll find out that this requirement is used for defining an inverse for the associated alpha. And in fact, in the definition of closed category, there's nothing that is matching the virtuality of alpha. So that's why this is the internal adjunction requirement. So, but raw street started looking at this mismatch, let's say let's call it a mismatch, the fact that there's something in the definition of closed monodal category that doesn't match. That is not matched by anything in the definition of closed category. So raw street started looking into this mismatch and notice that it is possible to fix it if one considers categories with less structure. So in the definition of monodal or closed categories, there's too much structure. If you remove part of it, then you get a good match. And the weaker variants of monodal and closed categories that he considers are these two. So the notion of less Q monodal categories, which was introduced by this Hungarian Algebraist Slakani in 2012. And he introduced also his own variant, like less Q variant of closed categories in his paper of 2013 to match the one that matches correctly, the definition of Slakani. And he formulated this theorem that says that now if you have a category that is equipped to put a tensor in the lolly and they're related by the same adjunction, then the pair i tensor is a less Q monodal structure on C if and only if i lolly is a less Q closed structure on C. And there is no internal adjunction requirements. So there is in the sense there is a preferred match between the two definitions. And before discussing our contribution to this, let's also have a look at the definitions of less Q monodal categories. And let's close categories. So this is the definition of a less Q monodal category. So this is just a category which comes with the unit and tensor and now three natural transformations, lambda and alpha, which have this type, these three types here. And the notable difference with the definition of monodal categories is that these are not required to be invertible. But only natural transformations in this direction that it's written down here in the slide. And this has to satisfy all the five McLean equations. So there is no way, like in the definition of monodal category to dispose of three equations and drive them from only two of them. So there's a called pentagon and triangle equation of monodal categories. In here you mean all five equations of McLean. And then there is also a notion of a less Q monodal category. This is from Street. And so again, it's a category C. We now have a unit, a loading, and we also have these three natural transformations, JINL, which have this type. And they have to satisfy five equations. So now these are, these JINL are also required to... There are structural laws that appear also in the definition of closed category by Alembert and Kelly in the original definition. But in the original definition, there are some additional requirements. So first of all, I is required to be an isomorphism. So there is an isomorphism between any object and A and I only A. And there is another requirement. So clearly here it doesn't make sense to ask for J and L to be invertible. So J in principle intuitively is just a function on the intuitive level that picks out the identity, the internal identity from A to A. And L, it's something that just performs post-composition. So take something of a function from B to C and the function from A to B and then compose them. So it doesn't, there is no reason why this two should be J and L should be invertible. And in fact, they're not required to be. But in the original definition, one also takes something that can be derived from J to be invertible, which is this J hat, which now takes the morphism from A to B in C and returns some of it from I into A to B. So somehow it's a way of turning an external morphism into an internal morphism. And it's defined using J in this way. So we're playing first J and then was composing with the identity of F. And this is still required to be invertible as well in the original definition. Sorry, in off-close category. But while in here, neither I nor J hat are required to be invertible. So right now with the definition in place, we can continue and talk about our contribution. So first contribution in the following. So it's talking about trying to basically extend, prove or find variants of the street, street theorem, including normal conditions. So what is the normality condition? So, for example, in the, in a less Q monoidal case, we call the normality condition the invertibility of a structural law. So either raw lambda or alpha or more or multiple those. And notice that one obtained the definition of a usual definition of monoidal category by just requiring, requiring all the normality condition to hold. So all three structural laws are invertible. And then as I said before, we also managed, so we can prove a street, a version of street's theorem of the Allen Berkeley theorem, including normality condition because we managed to find the, what does it mean? What are the analog, the analog, the normality condition of the scumular categories, less scumular categories in the less Q cross setting. So, and more precisely, we have this theorem. So now in the presence of an adjunction as before. Now, don't, not only their system isomorphism between less Q monoidal and less constructors, but, but now the scumular and the scumular normality condition also correspond, is a one to one correspondence in those. So, let's be more precise, let's see the precise correspondence. So this is in this, in the less Q monoidal setting, okay, for the less Q monoidal structure. So this is the conditions for the, the normality conditions for the, for the left street closed structure. So the invertibility of raw corresponds to the invertibility of y. And the invertibility of lambda corresponds to the invertibility of j hat. So these are the two conditions of, that are usually also required in the definition of closed category. Then we managed to also find a normality condition associated to the invertibility of alpha in the, in the less Q cross setting, which is the invertibility of this L hat. Where L hat now is this, is this the final rule, is this the rival morphism, right, going from this coin here into this other concept. And, and again, of course, the, the, the invertibility of alpha is also equivalent to the invertibility of P of, so to the fact that there is an internal adjunction. So, but this, the ability of L hat, this was the missing piece somehow in what created the MIS match. So if we ask, for example, for a, if you add to the definition of closed category, also the requirement that L hat must be vertebral, then one gets a perfect match. So, with the, with the, with the definition of monolithic category. So this new definition of closed category would match precisely with the definition of monolithic category in the Allenberg, Allenberg theory. So, and notice, so how do this L hat and J hat things arise. So, these are actually interdefineable with J and L respectively. So, we saw before that J hat can be defined in terms of J, but it's also true, the opposite. So if we take J hat is primitive. And this is the scales for L and L hat that are also interdefineable. So one in principle can define, can, can have a definition and equivalent definition of left skew, closed category, not in terms of I, J and L, but in terms of I, J hat and L hat. And then talk about normality condition asking for any of these three structural law to be invertible. So, what will, so maybe I can give some intuition and what does it mean to be, for example, for the inverse of L hat, what does the inverse of the hat do. So, it takes a morphism from A into B, only C, only D, and it returns an element of this, on this co-end. So, it's an object X and two morphisms here, two morphisms here. So, the idea here is that X should be a formal weak tensor of B and C. Right, because then you have a morphism from A into X, only D. So if you think of some B tensor C here, then this is what one would expect. And again, and here, there's a way of embedding B and C into V tensor C. So it's some kind of the existence of a weak tensor that the invertibility of L hat correspond to the existence of a tensor of B and C. But interestingly, if you look at the first morphism here, this one set here, we will, we are actually allowed to eliminate the tensor only to D, not in any other object. So, in this sense, this is a weak tensor. And so, yeah, so as I said before, we can replace, we can substitute the internal adjunct requirements in the original algebraic theory and with the invertibility of L hat. And then we will have a perfect match. So if we ask for this invertibility of L hat, also in the original definition of closed-calibre. Interestingly, this condition was already identified in the past by Dey and La Plaza, and it appears in a couple of places. But then it did, we didn't manage to actually find it after the 70s in the literature. So it was kind of lost. And so the second contribution of our work is the fact that, well, nothing forbids us in principle to just swap the orientation of all the structural laws in the left-wing model category. And one will take something called the right-skim model category. So now the structural laws are all in dotted other direction. And yeah, there are still, there are some five equations analogous to the one before. But then it's more interesting what happened if you do the same thing with the left-skill closed-calibre. So you just swap all the structural laws. So we start with the definition, taking I, J hat and L hat, and just swap the orientation of those. And one will end up with the notion of a new notion of right-skill closed-calibre, which has this primitive structural laws satisfying also some five equations. And we're again also proved a theorem that it's like a right-skill variant of street theorem. Then now you can connect a joint right-skill model and right-skill closed-calibre on a category. And also there are some similar relationships between the normality conditions. But more interestingly, so we can prove a new theorem that now connects left-skill closed and right-skill closed-calibre structures. So it goes as follows. So if you have a category with a unit and two lollies. So it may indicate the lollies and the lollies are here. And together with the following condition that we call the external lambda condition, which says that there is a bijection sigma that is nothing in A, B, and C of this form. So this takes a morphine from A into B, lowly R of C, into a morphine from B into A, lowly L of C. We call this the external lambda condition. And why is lambda involved in this is because if one looks at the, because this is very much reminiscent of something that is actually of a characteristic feature of lambda syntactic calculus. So something that it actually holds in there. And if you look at what it says, it basically says that A and B can be swapped here. But if you do that, you also have to swap the lollies. And this holds in the lambda calculus where now instead of the lollies, you have the two residuals in that case. And then if you have this condition in place, then a lowly L is less than C, if not if I lowly R is less than C. And again, there are some correspondence between the normality conditions. Now for the left lolly, for the I of the left lolly is invertible. Then this is the equivalent to asking for J R of the right lolly to be invertible. And if J hat is invertible, then this is equivalent to IR being invertible. Then you have also condition for the L, so L hat is invertible if not if LR now is invertible. And moreover, if this is also equivalent to the existence of an internal, I mean something that is kind of analogous to the internal adjunction condition that we saw in the original Halle-Mercalli theorem, which is the fact that the following morphism that is the right one using L hat or using LR is actually an invertible. And if you look at what this is, this is just an internal version of our lambda condition. It's an internal version of this lambda condition. So again, it can swap A and B here, but you have to swap also the L on the left lolly R. Okay, so now let's move to the third contribution, which is examples. So in this paper, we give a very large number of examples because we need to motivate the different multi-conditionals as well. This is also the new notion of right skew close category. So in this talk, we will focus on three classes of examples. So the first is because as follows, you're starting with, for example, the left skew close category. How can you further skew it to the left using a component? Or if you start with the right skew close structure, how you can skew further to the right using a component? Then we'll see how starting from skew close category and amount on it, how to lift the skew close structure on to the class D. And the third example is the non-community greener type lambda calculus. Okay, so here we start with the left skew close category C and I lolly and a common D on it. So we want to generate, come up with a new left skew close structure on C using D. And we can do this if we ask for D to be less close as well, which means that it comes with a map E from I to D I, and then after transformation C from D of B lolly C into D B lolly D C. And it has to be coherent suitably with the data of the left skew close category and the co-logic and co-multiplication of the coordinate. And if we have this left skew, the coordinate is left closed, then you can define another left skew close structure on C where the unit is the same and now the lolly is given as follows. So it's just B lolly C, for this new lolly, it's just D B lolly C, this is the order to apply the common to B. And then we can derive, for example, this is how you derive the new structure of I in this case. So you first apply E here because we will go from I to D I. And then you apply the, you will post-compose it with the, with I applied to A. But this I is the I of the original left skew close structure. And then one can start asking how the normality condition lift to this new structure. So, yeah, so what, for example, if I is invertible, when does the new I is invertible for the new structure is invertible. Well, this is true if, for example, if A is sufficient conditions E, E to be invertible as well. And one can play a similar game, starting from a right skew close category and starting with a monad instead of a commonad and asking for it to be O plus closed. And then one will get a new right skew close structure on C, defined this way, where the lolly is defined this way. So again, now instead of applying D to B, just apply T. Okay, the second class of example, so we started with the left skew close category and the monad. And if the monad, and so now we ask when does this left skew close structure lift to the classic category of T. And this is the case if T is left strong. Alternatively, people call it this, call this internal functorality, which means that there is a natural transformation of this form. From B to C, B lolly C into T B lolly T C. There's also a pseudo-incorherent with the data that we have from the category and from the monad. And now we can find the left skew close structure on the classic of T, the classic category of T, where the unit again is the same. And now the new lolly is given by just applying T on C now. So B lolly C with new lolly is B lolly T C. This is the old lolly. This is how you define J and I for the new close structure, left skew close structure. So I is very simple. You just use I on TA. So the new I on A would be the old I on TA. And for the new J on A, you first apply I, sorry, J, so you go from I to A lolly. And then you apply eta here on this A, and you go from A lolly to A. But now this has to be amorphous in the class D category of T. So of T, so then you have to, this is a pure map. You just use J, which is the funter embedding C into the classic category of T on this morphism. And then you get something that is well typed. And again, one can play the same game by starting with the right skew close category and obtain the right skew close structure on the class D of T. If now T is left, it's left closed. Okay, so the last example is the non comm, is actually it's a fraud. It's a, it's a, it's a restricted variant of the non commutative linear type linear type and the calculus unit. We'll see in the second vice and restricted. So this is how it goes. So the objects of this is the categories or the objects of types is the grammar for types. You have atomic types, unit and volume, which is this function spaces here. Context are around the same as our list of types. And this is the grammar for well from terms. So variable, and then you have unit introduction and elimination and obstruction and application. So you see that this is a linear and non commutative linear variant of simple type of the calculus for you can see, for example, for application because first of all, the resources that we have in context split when how so part of them go to the for as resources for for for the for the function argument for for this T here and the second part go for the argument of the application. And they go in order right so gamma goes to the left and that goes to the right so this is why it's non commutative. But I said before this is actually a more restricted version of the non commutative linear type on the calculus with unit because we all there is an additional restriction on the on the on the unit elimination. And it is that all that now, again, this is a similar restriction that we have here so the gamma. So you have to, we have T of type I and you have type A, but the resources of T, so the context of T must come from the left most position from from the left side of the of the context here so gamma cannot appear any in any position of the context here, which is usually what you have in the usual non commutative type of the calculus. But in here, this is more restrictive. And the definition of a quality of terms is the quality and now so the morphism of this categories and now yeah what type equivalence classes or what that well for terms, model of the type of quality. So for example, this is so and this is a left skill close category. And this is how you drive out for example. So you're given in context, x of type, you only see and you want to drive this. This is just lambda y, which are typed to be lambda z and z type A, and then just apply x to the application of why to say, and this is what types. And this is a nice example because also is is a concrete presentation of the free left skill close category that is generated by the set of atomic types. And interestingly, in this example, there is a normality condition that already holds, which is liberty of J hat. And what does mean in the setting, it means that there is a bijection between close terms of type A and B, so close functions and open terms of type B with one free variable in a. So this, this is what then the disinvertibility of J hat tells us in the setting. And one also may wonder, what about the material divide, for example. Well, this is obtained by relaxing the elimination for for the unit with this more permissive rule, but now yeah the gamma in the context of T cannot appear anywhere in the in the context in the in the in the context of the conclusion here. You see before it was restricted to me on the. So, coming from being the left part of the context but now gamma can be anywhere in the context. And then and if you add this rule. So which gives them the ability to buy and associate the left skill close structure. One gets the full non-community bring up type lambda calculus within it. So what one usually called more community than the calculus within it. Okay, so this concludes my talk so we continued working on. Cleaner variants of unimpercated theories that was something that raw streets started working on. And we prove that there is a relation between less humanoidal structure and less to close structure also with partial conditions with partial normality conditions. And. And also we for example we demonstrated that. That is also interesting to consider. Right skew violence of this of more than a close category in particular close categories so what to do with analysis this notion comes out naturally. And we we justify these notions with with a with a big number of examples. And what we like to do next so first we like to come to come up with a number of examples. That are relevant to the semantics of problem of programing languages. I mean the must be so we show that several examples involving more than a couple of minutes. So we believe that this is a good path for for future work. And we also would like to continue what we've done already. And we presented it. Two years ago at MSP. And which is in where we developed the proof theory of. Of of less Q monodal categories. And we would like to also do similar analysis proof to retic analysis of. Also less Q and or partially not partially normal monoidal and what are also closed monoidal closed by closed and symmetric monodal. Okay, so this concludes my talk. Thank you for listening.