So I am going to review some basic results on Topozis. So of course Topozis, the Nation of Topos, was introduced by Rotendik in the late 50s or around 1960. And they were first introduced as generalized categories of sheaves on so-called sites, which means categories endowed with topology. But it was discovered a little later that Topozis can also be presented in a very different way as classifying objects for first-order theories in the sense of logic. So it is a very different point of view on exactly the same objects. So the original presentation of Topozis by sites can be seen as a geometric way to present Topozis. And here what I'm going to talk about is the linguistic way to present Topozis from a mathematical theory, linguistically presented in some way. So my lectures will consist in three parts. So first I will introduce a sheer notion of classifying Topozis. I will immediately state the theorem of existence of classifying Topozis. And then I will spend some time explaining the meaning of this theorem. The second part of the lectures will sketch the proof of this theorem. And I shall especially present so-called syntactic categories endowed with their syntactic topology, which allowed to construct classifying Topozis as Topozis of sheaves on some sites associated to the theories we consider. So these two parts of the lectures date back to the 1950s. I will immediately say the names of the people who were involved in this theorem and its proof. And the last part of the lecture will also consist in basic results, but which are more recent. So this is taken from the Piaget thesis of Olivia Caramello 12 years ago, and her book Theories, Sites, Topozis. So these basics results allow to begin to make the theory of classifying Topozis efficient for establishing concrete results in mathematics. Okay, so by now I begin with the notion of classifying Topozis. And I immediately state the following theorem. So this theorem, I think, was proved in full generality in 1975. And quite many people contributed to this theorem. So here I have written the names of MacGyne and Reyes, because in fact this theorem was stated on proof for the first time in a book by MacGyne and Reyes. But this work certainly depended on some very important earlier work by Lawyer. And I think some other mathematicians or categoricians of the same circle played a role in establishing this theorem. So I also wrote the name of Joial, André Joial. And I should add that this theorem systematizes or generalizes some ideas which are already present in particular cases in the Piaget thesis of Monique Akim, who was a student of Grotendi. So this means that at least part of the stop, in fact, was certainly already present in the mind of Grotendi. So here is a statement, but maybe at that moment you will not fully understand the statement of the theorem. And the purpose of the first part of this lecture will be to explain, to fully explain the meaning of this theorem. So, yeah, you consider the so-called first order theory, which is geometric. So geometric is some kind of technical condition which will be explained later. But in fact it is not restrictive, which means that for any first order theory, in fact, there is a way to make it geometric without changing the set theoretic models of the theory. So this technical condition is not restrictive, is not very restrictive. And this is the first thing we can have in mind about this notion. Then, for any such theory, there exists a topos. So I shall explain a little later what is a topos. A topos, endowed with a universal model of the theory, such that for any other topos, there is a natural equivalence of categories. Between the category of models of the theory in E, and the category of morphism, of toposis morphism, from E to the classifying topos ET. So this statement looks familiar for anybody who knows the notion of classifying object, of representable from top. Here, what we are saying is that any theory, any first order geometric theory, defines a functor of so-called models, and this functor is representable by a particular topos. So this theorem is extremely general. It applies to any first order geometric theory, and it allows to associate to any such theory, so which means to any linguistic presentation of a first order theory. It allows to associate to it a topos, which, in fact, is a geometric object. The toposes were invented by Gotendic as the most general notion of space, of geometric object, which could exist in mathematics. This is really what Gotendic thought. Okay, so even if you don't fully understand the statement of this theorem at the present moment, it is not important because we shall explain that with more details in the coming hours. Okay, so before I begin with explanation, I want to make a few remarks. So the first remark is that, given such a theory, is classifying topos, endowed with a universal model of T in this topos, is uniquely determined up to equivalence. The second remark is that T belongs to the world of logic. So here, logic should not be understood at some very particular part of mathematics. Here what it really means is formalization of mathematics, linguistic presentation of mathematics. Anytime we do mathematics, we begin by presenting one or several theories. And even if we don't think about it, we are using the language of logic to present any theory we want to study. So this is really how things have to be seen as respect with this theorem. Okay, so on one side of the theorem, you have formalized mathematics, a linguistic presentation of some theory. And on the other side of the theorem, you have an associated topos, the so-called classifying topos of T, which belongs to the world of geometry and topology. So for growth and dig, I repeat, the notion of topos was the most general notion of space. Yet another very important remark is the particular case of this theorem, when we take for E, the simplest non-trivial topos, which is a category of sets. And if we just specialize the theorem to this particular case, we get an equivalence between the category of points of the classifying topos and the category of set theoretic models of the theory. So this particular case is especially important, because in general, when we study theory in mathematics, we are looking for its set theoretic models. For instance, if we think about the theory of growth, what do we mean by your group? We mean a set and doves with a structure of group. What do we mean by your ring? We mean a set and doves with a structure of ring. So this means we are considering set theoretic models of the theory of groups or set theoretic models of the theory of rings, and so on. And here, what the theorem says is that there is a geometric interpretation for that. The set theoretic models of any such first order geometric theory can be seen as the points of some associated geometric objects, the classifying topos of this theory. Okay, so here this is, so this means that in some sense, this theory of classifying topos is could also be called a functorial theory of models. By now remark five, which is also extremely important. It is a fact that, of course, as we said, for any first order geometric theory, there is an associated classifying topos, which is uniquely determined up to equivalence. But in the other direction, given any topos, there are infinitely many theories with the same classifying topos up to equivalence. And this remark gives rise to the theory introduced by Olivia Caramello of toposis as bridges, because what Olivia has begun to use in a systematic way is a relation between theories, which is defined by the property to have equivalent classifying toposis. So this can be seen as a theory as a general theory of translations, because, as I said, theories have to be understood as linguistic presentation of mathematics. Any theory is a choice of a language and a grammar to present some mathematical content. And the associated classifying topos is a geometric object, which incarnates the meaning of that theory. So when there are two theories with the same associated topos, it is exactly as having two languages, possibly very different, which have exactly the same mathematical content, which means which exactly tells the same story. And as we already remarked, for any given content, which means for any given topos, there are infinitely many theories, or if you prefer infinitely many languages, which allow to present this content. So this remark is the beginning of the possibility to use toposis as bridges between theories, which possibly could be completely different. Okay, so by now I want to explain the different ingredients in the theorem. And so the first ingredient, of course, is the word topos. So I want to explain a little or at least to recall what is the topos. So in order to have an idea of what is the topos, we need to begin with examples. The first example is consistent toposis associated to topological spaces. In fact, this is really like that, that toposis were born. So starting from any topological space, it is possible to associate to it the category of sheaths. And here I mean sheaths of sets on the topological space X. So I will not recall here what is a sheath. In fact, Olivia in her own lectures this afternoon will do it. But immediately I state the following proposition, which tells us that it is almost the same thing to know a topological space or to know the category of sheaths on this topological space, which means to know the associated topos. First, there is a general notion of points for topos and the first basic fact is the fact that any point of the topological space X we consider is a usual sense in uses in use points of the associated topos. And this map is one to one, if X verifies an elementary properties, a property to be sober, which in fact is verified in most cases, it is verified by a whole or topological concrete topological spaces. And in the same way, open subset of X correspond to open sub-toposis of the topos EX. So I don't define here the notion of open sub-topos or the notion of point of a topos. But what I stress is the fact that in the world of toposis, there are categorical notions, such as the notion of points and the notion of open sub-topos, which allow to recover not only the underlying set X, but also its topology, so the full topological space. And so this means that the notion of topos generalizes the notion of topological space. And it also means that this notion realizes an embedding of topology into category theory. So this is really amazing because it means that in this way, we go from one world to another world, from the world of topology to the world of category. The world of topology is a world of continuous structures, whereas the world of categories is a discrete world. So it is very surprising that such an embedding exists. And the sheer fact to go from one world to such a different world is already extremely deep in itself. And not only it is deep, but by going from topology to category theory through the construction of the associated toposis. In fact, we enlarge a lot the world of topology. And this also is very important. This becomes immediately clear when we come to the second type of example of toposis, which is given by groups. So for any group, we can consider the category of sets endowed with an action of the group. And this also is a topos. In fact, here we have the two basic examples of toposis. On the one hand, categories of shifts on a topological space. On the other hand, categories of sets endowed with an action of a given group. And the notion of topos is pertinent, in particular, because these categories of shifts on a space or of sets endowed with an action of the group, have very similar categorical properties. And as Grotendick had remarked. Okay, and the example of groups can immediately be generalized in the following way. If we start with any small category, then we can consider the category of pressures on this small category C. This is by definition the category of contravariant factors from C to the category of sets. So this is a generalization of representations of groups. And this again is an example of a topos. So all these categories, which I have mentioned, shifts on a space, actions of a group on a set, or pressures, which means contravariant factors from a small category to sets. All these categories have similar categorical properties. And in fact, I will come to that a little later. What are the main general properties of topos? So when we come to these categories of pressures, it is very important to state the following proposition, which of course is very well known. So start from a small category C, then consider the category of pressures. And the content of the proposition is that this category of pressures C at can really be seen as a completion of C. So the first point is that C is embedded in C at. For any object of X, you consider the associated contravariant representable hash if. So which consists in all rows from an arbitrary object to X. And so this function, which any object of C associated the function it represents, so this function is fully faithful. So we have a fully faithful embedding of C into C at. And on the other hand, any object of C at, so any pressure, can be written as a core limit, or if you prefer an inductive limit of such representable factors. So here, in order to realize that the indexing category is the so called category of elements of the pressure if you consider. So this means that the objects of this category of elements consist in object X of C. Endowed with an element of the image of X by the pressure P you consider. So this is an indexing category, but the first important factor is that any object in the category C at is an inductive limit of representable object. So this is why I say that C at can be really be considered as a completion of C. Okay, by now, I propose a general definition of topazis. But once again, only if you are with an impact point this afternoon. So here I define a topazis as a category, which is both a quotient on the subcategory of some category of pressures. So a topazis is a category such that for some small categories C at. There exists a pair of adjoint functors from C at to C on from E from, excuse me, from C at to E on from E to C at such that so first this pair of functors is adjoint. So this more precisely, the functor from C at to E is left adjoint to the functor from E to C at. Secondly, the right adjoint functor from E to C at is fully faithful. So it is a fully faithful embelling. It can be seen as through this functor as a subcategory of C at. And as this pair of functors is adjoint, the fact that this functor is an embelling is the equivalent to say that the composite functor of the embelling by compose with its left adjoint. Is isomorphic to the identity functor of E. So E is a subcategory of C at. It is also a quotient of C at through this left adjoint functor, which is called the shiftification functor. And the composite of these two functors is naturally isomorphic to the identity functor of E. And the last point in the definition is that we want the left adjoint functor to respect finite limits, finite projective limits. Okay, so this you see is a very categorical definition. Then we can make a few remarks. So first, the fact that the G lower star component from E to C at respects arbitrary limits. This is because it is right adjoint to some functor. And the J upper star component from C at to E, the shiftification functor respects arbitrary co limits arbitrary inductive limits. This is because it is left adjoint to some functor. As a consequence of this remark is the fact that if we consider the canonical functor from C to E, which is a composite of the Yoneda embedding into C at with a shiftification functor. So this composite can be denoted L. Then we get that any object of E can be written as a co limit of images of objects of C by this canonical functor. So this is just a consequence of the fact that this is already true for C at and the functor G upper star respects the limit. Okay. By now, I want to recall that such presented in a very abstract way can be defined in a more concrete way by a so-called grottendic topology. So here we can define topologies in the following way, consider an arbitrary object X of C, and let's consider sub-object of the representable pressure associated to X. So what is such a sub-object in the category C at it is just a family of morphism from arbitrary object X prime to X, which is stable under any composition with any morphism from some X prime prime to X prime. It is stable under right composition. So such a family of morphism stable under right composition and which all go to X is called a sieve of X. And by now, if we have a topos presented from Seattle as in the previous page, a sieve on X, which means a sub-object of the representable pressure associated to X, is called covering if this embedding from S to Y of X becomes an isomorphism after Shifififififififififififif. Okay, so here we have a notion of covering sieve for any object X, and we can denote GX, the family of all covering sieves of such an object X, and then we can consider not only one X but all possible objects X of C. Also, we have an index family of covering sieves, and this is called a grotendic topology on sieve. Okay, and by now it is easily checked that such topology defined in this way verifies some axioms. So in fact, there are three axioms, which are very easy to verify. There's a maximality axiom, which says that when we consider an arbitrary object X, then the full sieve of X consisting of all rows to X is always covering. Secondly, the stability axiom, if we consider any row from X prime to X in the category C, and any sieve S on X, which is covering, then its spread back through the morphism we are considering is still a covering sieve of X prime. And lastly, there is a slightly more subtle transitivity axioms, which says that if we are considering a covering sieve S of some object X, and a sieve S prime of X, would spread back by all elements of S are coverings, then it is a covering sieve of X. So this is a little more subtle. The important point is that it is not difficult to get that any topology defined as in the previous page verifies these three properties. And now the second part of the theorem is the fact that conversely, any index family of sieves which verifies these three axioms defines a topos endowed with two factors, which allow to recover the topology jet. So to consider a topos presented from Seattle through these two factors is the same thing as to consider a growth endic topology on X. Okay. Okay. And by now, I want to state the main categorical properties, which are verified by two poses. So two poses are particular types of categories. They are categories, which can be constructed from categories of precious by the type of construction we presented in the previous pages, but it happens that because they are deduced from categories of precious in this way. They inherit from categories of precious some very special properties. So first, any topos is a locally small category. So these things that given two objects in a topos, the morphism between these two objects make up a set. Secondly, in an arbitrary topos, there are in any topos, there are arbitrary co limits on arbitrary limits. Thirdly, for any morphism in a topos, the associated pullback factor defined by considering the fiber product over this morphism, not only respects arbitrary limits, but it also respects arbitrary co limits. Another important properties is that in a topos, sums are designed. So these things that, for instance, if we consider two objects on the topos and we make up their sum, the limit of the diagram consisting in these two objects without any morphism. Then, of course, the sum contains as sub-objects the two objects we started with. And the intersection of these two objects inside the sum is equal to the initial object of the category, or if you prefer to the empty object of the category. So this, of course, is familiar to us in the case of sets, but here we say that it is also verified in an arbitrary topos. Another extremely important property, which is also familiar to us in the case of sets, but which is verified in any topos, is the fact that to consider a quotient object of sum object E, which means to consider an epimorphism from E to sum object Q, is the same thing as to consider an equivalence relation on E, which means a sum object of E cross E, verifying the usual axioms of equivalence relations. Contains a diagonal, it is symmetric, and it is transitive. And the link between these two presentations is the fact that the equivalence relation is just the fiber product of E with itself over the quotient. And the quotient is recovered from the equivalence relation by considering the co-limit of the diagram consisting of R, the equivalence relation, E, the ambient object, together with the two morphisms, the two projections from R to E. Okay, another important property is that in a topos, the sub-object of any object and the quotient object of any object make up sets. And the last property I want to state is the fact that in a topos, a morphism which is both a monomorphism and an epimorphism is an isomorphism. Of course, this is also familiar to us in the case of sets. If you have a map which is both injective and onto, then it is objective. In other words, it is invertible. But this is true, this is still true in an arbitrary topos. And here, there is a very remarkable theorem of Giro. So Giro was a student of Autendic. The theorem tells us that these properties characterize toposis. For this, we need an extra condition, which is the fact that the category we are considering is not too big. So here is a precise statement. We consider a category E, verifying all properties of the previous page. And then we suppose that in this category E, it is possible to identify a small full sub-category, such that for any object of the E, there is an epimorphic family of morphism from objects of C to X. So in some sense, this means that E can be considered as generated by C. And then, for such a C, there is an induced jet. You decide that a sieve of an object X of C, which means a sub-object of Y of X in SEAT, is covering if it contains an epimorphic family. So here epimorphic refers to the categorical structure of E. So this is a definition. And then the statement is that the category E is the coefficient of SEAT defined by the topology G. In other words, it is a category of sieves on the category C, endowed with a Grottendig topology J. So this means that by now we already have a double characterization of toposites. So the first characterization was constructive. Toposites were categories which could be constructed in some way. On the earth, thanks to this theorem, we have a second characterization. Toposites are categories which verify the properties of the previous patch, plus the existence of such a generated small subcategory. Another remark we can make here, which is very important, is the fact that this theorem means that a given toposite can be presented in infinitely many ways from small categories C, endowed with topology. Because here we can take in this theorem, we can choose for C any small category, any small subcategory, full subcategory of E, which is big enough to generate E. And there are infinitely many choices for that. And this also is part of these extremely interesting properties of toposites that they can be presented in infinitely many ways. So I already mentioned that they can be presented in infinitely many ways from theories, as we shall see later with more detail. But here, thanks to this theorem, we also know that a toposite can be presented in infinitely many ways from categories endowed with topologies. So this presentation, of course, is more geometric, whereas the other one is linguistic. And so here I mentioned again that the full theory of that Olivia Caramello has proposed to use these basic facts to make toposites bridges between theories and also between categories endowed with topologies. So I will come back to that later because in fact we shall make use of several such bridges to establish present. Okay, so by now I can just recall the basic examples I had already talked about. So the first basic example is the example of topological spaces. So we said that for any topological space, there is an induced toposite, the toposite achieves this topological space. But of course, when we talk about topological spaces, we also have to consider continuous maps. So topological spaces make up a category. And here, the basic fact is that whenever we have a continuous map between two topological spaces, then this map induces a pair of functors between the associated toposites of sheaves on X and Y. So a pair of functors which are right joined to one to use the other. So these things that they go in the two reverse directions. So there is a so-called push forward functor from which transforms sheaves on X into sheaves on Y. And there is a so-called pullback functor which goes in the reverse direction. It transforms sheaves on Y into sheaves on X. And it is a fact that these two functors are adjoint. So the pullback component is left adjoint to the push forward component. So this means the push forward component respects arbitrary limits, the pullback component respects arbitrary co-limits. And as a matter of fact, the pullback component also respects finite limits. Okay. By now, we remark that if we have a topos which is defined by a topology J on some categories C, then the topos C-hat and the topos C-hat and J are related by a pair of functors. And for us, it was a definition in the way I presented toposes. So they are related by a pair of functors which verify exactly the same properties as the functors induced by a continuous map between topological spaces. So this pair of functors are adjoint. And the component J upper star respects not only co-limits, but also finite limits. Find out whether we know other situations with such pairs of functors. So in fact, there is one which is very important. So let's consider an arbitrary functor between two small categories. So of course, this functor defines an associated functor in the reverse direction between the associated categories of pressures, because a pressure on D is a contravariant functor from D to set. So if you composite with row, you get of course a contravariant functor from C to set. In this way, you define a functor which goes from D-hat to C-hat. It transforms pressures on D into pressures on C. And here, a very important fact is that this functor, this functor of composition with row has two adjoints, one left adjoint and one right adjoint. The right adjoint is denoted row, lower star. And the right adjoint is denoted in this way with such a right of dot of mark. And as the functor of composition row, upper star has two adjoints, of course, it means that not only it respects arbitrary limits, but also arbitrary limits. Also, we have an adjoint pair, row, upper star, row, lower star, such that the left adjoint component, row, upper star, respects in fact not only finite limits, but arbitrary limits. Okay, so this example, some, especially the first one, lead to the following definition. Morphism of toposys by definition is a pair of adjoint functors, the so-called pullback functor and the so-called push forward functor, such that the pullback component, which means the left adjoint component also respects finite limits. So here it is just a definition. Okay. And here, of course, we remark that as we are talking about functors or pair of functors, they naturally make up a category. Given two toposys, E prime on E, and two morphisms of toposys from E prime to E, so this morphism we call F and G. And by definition, a transform of such morphism of toposys is a transform of functors between the pullback components. So by definition, a morphism from FG is a morphism of you if you prefer a natural transformation from F upper star to G upper star. And because they are adjoint, this is the same thing as a morphism from G lower star to F lower star. So here you have to be careful that taking the adjoint, the arrow has to go in the other direction. So to give a morphism from F upper star to G upper star is the same thing as to give a morphism from G lower star to F lower star. So in particular, the category of points of a toposys is by definition the category of morphism of toposys from set to E. So of course, the category of set is a toposys, it is just the category of pressures on the category consisting in just one object and one morphism. Or if you prefer, it is also the category of shields on the space of the topological space with one element. So a point of a topos by definition is a morphism of toposys from set to E and points defined in this way make up a category. It is not only a set, it is a category. In fact, in general, it will not be a set because it can be bigger, but it is a set in quite many cases, but not always. And the last part of the definition is a notion of embedding of toposys. If you prefer the notion of subtopos. So what it is, it is a morphism of toposys consisting in a pair of adjoint factors, J upper star, J lower star, such that the push forward component J upper star is fully faithful. And this is equivalent to saying, because of adjectness, that J upper star composed with J upper star is isomorphic to the identity factor of E prep. This is the definition of subtopos. So you see that in this definition, everything is categorical. Everything is phrased in the language of categories, which is very natural because toposys are just particular categories. But we see that in the language of categories, we have expressed geometrical notions, especially the notion of point on the notion of sub space, the notion of subtopos. And also the notion of a relative toposys, which means a toposys considered over another topos through a morphism. And in fact, this will be studied much more in Olivia's lecture. So, of course, this definition was given in order to have the following examples. So first, any topological map map, any continuous map between topological spaces in uses a morphism of toposys. And in fact, this is one to one is why is a sober topological space. So this means that if why is sober, to consider a continuous map from some topological space X to Y is the same thing as to consider a toposmorphism from the topos associated to X so EX to the topos associated to Y. So this of course applies in particular, if X is a topological space with only one point. And so this means in particular that the points of why identify with the points of the associated topos, if why is a sober topological space. Another remark is that if we have topology J on a small category C, then the topos defined by J is a sub-topos of the topos of species of C. And the last example is that any factor between two small categories in uses a morphism of toposys from C-at to D-at, so in the same direction. Okay, so by now I have given a review of the theory of toposys. I hope I have given to you an idea of what a topos can be. In fact, I have given full definitions together with the basic most important examples. And by now I want to explain the other alpha of the ingredients necessary to understand the theorem of existence of classifying toposys, which is a linguistic presentation of mathematics. Here we have to use some language which is familiar to logicians and just as a warning we should not be afraid by this language because it is just a way to make ourselves aware of what we are doing all the time when we write mathematics. So the first order theory consists by definition in language, in a language plus grammar rules. So if you prefer in a vocabulary in a list of names on the one hand and on the other on grammar rules. And the grammar rules in mathematics are called axioms. So first we begin with the language. So the first order language consists in three types of names. First there are names of objects, but which logicians call sorts. So, for instance, if we want to consider group theory, here there is only one name of object, which is just the name group, or if you prefer, the letter g. If we consider the theory of rings, there is only one name of object, which is the name ring, or if you prefer the letter r. If we consider the theory of fields, there is only one name, k, or one letter, k, the name field, the letter k. If we consider the theory of vector fields over vector spaces over fields. So then there are two names, the name vector space and the name field. And we can choose letter for that. So for example v for vector space and k for field. If we consider the theory of modules over a ring, and here there are two names, modules and rings. Or we can write them as letters, m and r. Okay, this is just the names of objects we are considering. Then the second type of names are names of morphisms, or names of functions, or function symbols, or they are called. The function symbol consists in a letter f, going from a collection of, from a finite family of names of objects to a name of object. So for instance, if we want to talk about addition, so addition is certainly a name of morphism. So addition is a binary operation. So this thing, it is a function symbol from, so in the case for example ring, from f, f, because addition is on two variables, both in f, from f to f. And we can consider outer multiplication in a module over a ring. So this is the function from r, m to m, because one variable has to be in r, one variable has to be in m, and the function goes to m. And lastly, there are families of, there is a family of names of relations. So for instance, if we want to talk about the theory of equivalence relation, so what is an equivalence relation? It is a sub-object in the product of some object. So here, the equivalence relation will be some sub-object r of, for instance, e, where e, for instance, can be a set, or it can be an object of a topos. You see you have a name of object, e, and an equivalence relation, r, which has to be a sub-object of e, e. Here an important remark is that it is possible in this definition to take n equals 0. So a function symbol from 0 to b is called a constant symbol. And the relation inside the empty family of names is called a proposition symbol. So let's immediately take examples. So for instance, the language of the theory of rules. So in the theory of rules, you have one name of object, which is g, and there are three function symbols. So first for multiplication from g, g to g, then going to the inverse, this is to g from g to g. And lastly, there is a constant one, which goes from the empty family of names to the name g. It corresponds to choosing one element of g, one constant in g. And in that case, there are no relation symbols. If we want to talk about the language of the theory of equivalence relations, we have to choose one sort e, which will be the underlying object, no function symbol, and the relation symbol are inside e, because we want to consider a sub-object of the product of an object with itself. Okay, so this is the language of a theory. And by now, I want to spell out what it means to interpret such a language. So technologicians don't use the name language, they use the name signature. So for such a signature, which means a first order language, and for any topos, or more generally for any category with finite products, including a terminal object, then we can talk about interpretations of the language of the signature sigma. So what is a sigma structure m in such a topos e, or in such a category e? It is a map. A map where it shows the associates to any sort, which means to any name of object a, an object, which is called m a. And it has to associate to any function symbol f going to a1 a n to b, a morphism, which goes from the product m a1 cross m a n to m b. And here we are, of course, when we write this m a1 cross m a n, this is a product of the object m a1 m a n in the category e, and m b is also an object of e. So it has a meaning to talk about morphism from m1 cross cross cross cross m n to m b. In this case, n equals zero. So we have a constant symbol, and then we structure, sigma structure consists in associating to such a constant symbol. A map going from the morphism going from the terminal object to the object of name b, which is m b. Any relation symbol are inside a1 a n.  The sigma structure is a way to associate to any name of object, an object of e, to any name of morphism, morphism in e, between the corresponding objects. For any relation symbol, a corresponding sub-object of the product of objects associated to the names of objects, which are the context of the relation symbol. And by now, it is also possible to talk about morphism between two such sigma structures. So let's consider two such sigma structures m and n, which means two maps, which are associated, I repeat, to any name of object, an object with the name of function morphism, to any name of relation sub-object. Also, a morphism of sigma structure is a family map, which associates to any name of object a, to any sort a, a morphism in the category e, from m a to n a. So morphism between the corresponding objects with the name a, associated to a, through the two sigma structures m and n. And they have to verify two properties. So first, they have to be compatible with the function symbols. So this means that for any function symbol f, from a1 a n to b, the associated square, which is drawn on the edge, is commutative. And for any relation symbol r, inside some a1 a n, there is an associated commutative square as at the bottom of the page. Okay. And so this means by now that given a signature, which means given first order language, we can associate to any topos, or more generally to any category with finite products. The category of the sigma structures in this category e are not only that, but if we have a functor between two such categories, which respect finite limits, or more generally which respect finite products on monomorphism, there is an in use factor between the associated categories of sigma structure. So in particular, for any morphism of toposis consisting of a pair of adjoint factors, f lower star and f upper star, there are two induced factors, which in fact are adjoint between the categories of sigma structures on e on e prime. So the pullback from top and the push forward from top. They are well defined because both f upper star and f lower star respect finite limits. So in particular, they respect finite products, they respect the terminal objects, and they respect sub-objects. Okay, so I just look, yeah, so by now I am going to Excuse me, this is what you wanted to say.


In our previous video, we were able to get ourselves from the Bronze Age and into the Iron Age by smelting raw iron ore we had collected in a bloomery and producing a bloom containing metal iron. This was us at the transitional point in history when iron began to emerge and become the dominant metal, supplanting the previous main metal of choice, bronze. One of the main drivers of change was warfare and the ability to manufacture weapons from a more plentiful ore source. In today's video, we're exploring what that early transitional period might have been like when bronze and iron technologies would inevitably face off against each other in the battlefield. We'll be exploring the unique and different manufacturing methods required for both bronze and iron and then put them to the test against each other. Will one hold the significant advantage of the other? Let's find out. My original goal was to try and make something out of the iron bloom at the end of the video over actually working with the iron bloom proved to be difficult. Only having a vague clue what I was doing, I enlisted some help from some more experienced Blacksmiths to see if I could forge anything out of my iron. Hi, I'm Joe Marcello. I've been a professional sword maker, weapon maker, historical reproducer for the last nine years. I have a lot of various passions, most of them involve metalworking of various sorts. Happy to be here helping, making some how to make everything. I'm Adrian Noland. I'm a bladesmith, not quite as experienced as Joe and I am super pumped to be here helping out making a sword. First off, the spark test to see what is in my bloom. So we'll just go down the row and see how they spark. Anytime we got the sparks on there is showing where the carbon content is. So you've got a pretty low carbon bloom, but there's a little bit of steel kind of hidden in there. So I think once that's all consolidated, you'll have kind of a nice chunk of bloomery iron. The difference between iron and steel wasn't known until later in history, but both compounds are actually alloys of the elements of iron and carbon. The different percentages of carbon are what make a large difference to the properties of the metals and are what separate wrought iron, steel, and cast iron. How are you telling the difference between the iron and the steel?  So the wrought has a little bit of carbon in it. This has .8% carbon.  Steel, you get a little bit of more of a straw color that kind of sprays. None of this was quite to the high carbon steel level, but there's a couple little sparks in there that told me steel. The bloom looked to be more, much more on the iron side of things. So a couple interesting sparks in there. I'm debating if we want to start hacking some of these apart. Yeah, I'd say crush them up, see if you get some nice hard pieces. Any of the ones that have a lot of ferocity too to see if we can separate them. This looks like iron, right? Yeah, this isn't breaking it off. This chunk looks reasonably easy to just start flattening it out. I'd place it right in on that block of stuff in there. Forge welding is weirdly easy.  There's a lot involved, but the nature of the process is actually pretty simple. Get it hot, eventually it'll stick together when you hit it hard enough. And it's amazingly gentle. I love forge welding, like the initial welds, just lightly by hand. It's very cathartic.  Until then, it's pretty nice. I'm gonna just very gently try to... So it's supposed to happen? When it crumbles into a mass, it's definitely not, not the easiest to work with. We'll start with that piece. So that's bad. That's good. But out of that chunk, we basically got that usable. So, we're... It's interesting because there's iron in there. It's like that. There we go. Like that. So it was a successful smell that's too small to actually be able to really get anything out of. So I mean like right here, it's a nice little chunk. You have iron, like it's there. It's in a very, very porous stage. Definitely chunks all around that are going to be usable. Getting all of the other stuff removed from it. This one contained a little more. So so far, this is what we're getting of usable material. In a perfect world, you get a much larger, consolidated chunk of iron in your bloom. And so then instead of having to break it apart like this, you can kind of actually just work it down into a single piece. With this, there's so much around it. It definitely worked. It just not efficiently. The other thing I have as a recommendation is you could look up aerostable furnaces. They're little. Like that might be a good option for like redoing this down into a usable bloom to then kind of approach this later. While successful in producing iron, it'll take some extra work in the near future to make it usable. Still wanting to explore the differences in these two historic metals, we substituted my bloom with a comparable piece of iron. But first, let's cast one last Bronze Age sword with the help of Greg, the sword casting guy. For the sword we'll be making, I made a pattern of the now two, one of the most important and longest lasting sword designs. The design lasted for seven centuries, spanning from the Bronze Age and well into the Iron Age. I'm back with Greg, the sword casting guy. And previously, we cast a few different swords, including this one out of pennies. We did get a lot of comments that that is a knife and not a sword, but ancient Bronze sword experts get to decide like what's a knife and what's a sword and they've decided if it is longer than a qubit, it's considered a true sword. If it's shorter than a qubit, it's still considered a dagger. What in the world is a qubit? A qubit is the length of your forearm. That's considered a sword kind of just barely, right? Yeah. A lot of the swords were that size because swords were a secondary weapon in the Bronze Age. They were kind of a side piece, second to the spear that they were using. One we're going to be casting today is really on the big side. That would be really pushing the limits of the technology because you imagine hot bronze being poured into this is cooling off the whole time and it has to get all the way to the point before it solidifies and that wouldn't be a certainty with something as long as this.  So most Bronze Age swords would be smaller than this, but I'm excited to see if we can do it. So I had to actually make this just for this. Yeah. And you even had to enlarge your box. That's right, kind of Frankenstein it. I'm excited. Good to actually fit. This one is quite a big one. This one is bigger than most swords of this type would have been. It's definitely a sword. Nice shape to it. And it is not like ridiculously heavy. Like for a sword that long, that's doable. That is doable. And you can kind of feel its flexibility a little bit. I mean, bronze is pretty pliable. So in a sword this long, you can imagine in battle, it's going to get bent. Like that'll happen. You'll get some bending it. Now you can harden the edges and you can also harden the middle of it kind of they bend them back and forth sometimes and just like get some of the bending out of the way like a paperclip. It's harder to bend. You're doing that with the sword too. Lastly, to grind and polish off the surface and before getting too far, hammering the hell out of the edges in a process called work hard name, which can greatly increase the strength of the bronze. So what's happening there is you're causing little defects in the crystal structure. And so those defects amount to being harder because you're kind of limiting some of the ways I can bend. As we move firmly from the bronze age to the iron age, this will likely be the last video featuring the help of Greg. If you want to learn more yourself about how to do this historical method of bronze age sword casting, be sure to check out his classes. He travels around the country teaching how to cast swords and he can likely make a trip out to you. Check out his website for more information. And so now we have our first sword made of cast bronze. Bronze was traditionally cast because the bronze alloy has a relatively low melting temperature that makes it feasible to fully melt and pour. Iron however has a much higher melting point in reaching temperatures to cast iron was out of the reach for some time in the iron age. So in contrast to bronze, iron is forged by being repeatedly heated above its recrystallization temperature and manually shaped with a hammer into its desired form. This is similar to some extent to the process of cold working I previously explored with native copper. When surviving to periodically anneal the metal by heating it to above its recrystallization temperature, the iron is just continuously worked while above that temperature. Forging oftentimes produces a stronger result than casting. Let's explore the process in Forge the Second Sword. I will guide you with where I'm striking with my hammer to show you where I want it to be. Unfortunately working on a tight production deadline, we have to speed things up with the help of a modern trip hammer. It's amazing how quick that thing works. Long get that up. That moment between heats where you actually get a chance to like kind of think things through. Yeah I'm crazy. I wear shorts when I'm working. I'm gonna get the length vaguely drawn out then we can work on flaring that out and necking that down. Just want to get it drawn down to something usable. We'll do a little hand hammering just to kind of re-profile everything. Yeah once we start pulling it back work on those fins and the thing. There's certain like refining things you just cannot do with a trip hammer. I need to draw about another inch out then we can start making all the fun shapes happen. It's amazing how much work it goes into a sword. Definitely getting closer. Yeah we can do a little bit of hand forging now and drawing the tip out will be good because then we can get the full length the rest of the way. Exit pass to help flatten out the blade. This section we've hit with the flatter it's taking this undulation out. Yeah it's a little chunky towards the tip still. Give me one strike right there. We're getting close. Only rules donate me in the face. One strike there. One strike there. One strike there. Perfect. We're getting a lot of that weird washboard pattern out. Let's turn it into a sword. It's a beauty of steel. We can kind of just move it around. So I'm excited to get more into steel bronze. You cast it like it either worked or didn't and there's not a whole lot you can do about it. Whereas steel definitely has a little bit more tweaking that you can do in the process for sure. Steel's very fun. I enjoy playing with it a lot. What do you think? Start drawing the fins out? At least draw them a little bit and then draw the rest of the tang out. That fin section is not easy to forge. Definitely easier shape to cast than to forge. That's getting there. So we got that shape started for sure. We will succeed on having made a sword today. Bit of a bend. Nothing too major. Probably from anvil height weirdness. It's just a tiny bit of twist. So right in this nice cherry red heat they can take it and kind of bend it where we want and that's about it. You can definitely see the similarities here. The cross section is just a rectangle. Forge down. What we're gonna do is we're gonna take it and we're gonna convex each side into that shape rather than forging in our bevels and having a flat in the center. A convex edge is going to hold much better at lower content. It'll hold this edge longer. It'll stay a little sharper. Which would have been done with files but we have a belt grinder and that's all there is to it. Rough ground at this point starting to feel like a sword. You can definitely feel that it has an edge now. I mean that is unmistakably a sword. It's weighs just a little bit less than the bronze one I think which is pretty typical for the period of iron swords and see that cross section now it's like a double teardrop. Both sides are sharp so it's just an arc. For something we kind of just whipped out. Yeah. Yeah for one day that's more than I've ever done in a day. Now there are two swords. Lastly just need to halve the two blades. We can try them out. Thanks to Greg we now have the last bronze sword we'll be making in contrast to some of our other swords such as the Mycenaean one we did which is primarily for thrusting and stabbing or the co-pesh which is mostly for slashing. I kind of learned the hard way. It's not great at stabbing all the time. In contrast the Nautu is kind of a general purpose sword and is able to both stab and slash. That kind of allowed it to remain popular through the ages into the Iron Age where they basically just replicated the castings. This one kind of dwarfs all the other ones. It comes at about 23 inches closer to 23 and a half and thanks to Joe and Audrey I have the iron one that was forged. It comes in slightly shorter about 23 inches weight-wise. Once I polished down the bronze one it was actually pretty close. It's only like three or four ounces more than the iron one which is a little bit shorter and a little bit thinner. I had a little trouble forging out the little wings on the handle. It's easy to cast that shape a little bit harder to actually forge it. The bronze one I was able to edge hard out which kind of give it a little bit of a wobbly look after polishing out all the hammer marks and the iron is basically just as it is. Later on other techniques of hardening it through quenching and other methods were discovered that eventually made iron superior to bronze but at this first stage it was probably pretty comparable. So we're going to put that to the test and see how true it is and see how well they fare against cutting and chopping a variety of different objects and then against each other and see what happens when they strike. Gather round children, I will tell you a story of the Great War. Many lives were lost that day. I saw my friend die. He got hit right in the melon. Rumor has it the same enemy has made a new weapon so you guys got to keep your eyes peeled and if they come back you have to avenge me. Yoink! We've got some volunteers here to try out our now twos on so we now have two now twos. Two fruits, two swords. See if there's any noticeable difference and then move on to some harder stuff. Let's see how they compare. He's watching you bro. You're next. I got it in my cup. You got the other one. The real test, coconut. Wish you're this cool. Got a stick. See how well it can chop it. Probably do some damage. We're in the middle here. Pretty sized little cut. It looks like a slight dulling where it hit. But overall, not too much damage. Maybe we should roll on it a couple times. Probably a little bit dull, slightly. But overall, actually pretty impressed. Didn't bend. Still pretty sharp. Pretty good. Got maybe a quarter inch. I think the first hit went the deepest. Pretty good. Let's see what steel does. No noticeable damage. Definitely went a little bit deeper. Got a little bit more penetrating power. Cool. That was less destructive than I thought. Now we're going to do probably the most realistic kind of trial for if someone with a bronze sword faced off against someone with an iron sword, and that's going to be strikes on each other. Back when we did our video with Daman, talked about the co-passion such. He learned some bronze age fighting skills. He talked about how you don't really parry with bronze because it's going to bend. It's going to fold your edges and such. So you're going to try and void as much as possible either by dodging or using a shield. And if you have to, you can use a flat side of it. We're going to do a few experiments of trying to parry with the flat side against a blade. See how well that holds up. And then we'll do blade to blade and see how much damage that does. I have faith in Lauren that she's not going to kill me. We'll see. I don't see a mark on here. Did you do anything to the blade? Oh yeah. I got dull, flattened. Yeah. That's a decent amount of force. I'm impressed. I don't, I don't see, I don't even see a marking. Oh, right there. There's a little bit of bronze in there. Oh wow. Yeah. Barely anything. So it didn't scratch the iron, but left pieces of bronze on it. Wow. So let's trade and try the other way around. That is a gas. Oh, okay. Yep. Anything on that? There's a little bit of bronze on the tip. Did it lose any of the edge? It doesn't look like it. It looks pretty sharp. Wow. That is a lot better than I thought it would. We put a nick in there. I don't even want to look at yours. Bailey knows that we're going to snap their sword. Wow. Yeah. It didn't put the dent in this at all. Really? Yeah. Wow. It really didn't. That's just two hits. Yep. This one's just fine. When it blocked with the flat side, it did put a little bit more of a bend into it. Not nearly as much as this one. Yeah. It's hardly noticeable. That one looks like it's wiggling. Wow. All right. So the end result was not quite expected. I think I read this said that these guys were, should have been pretty comparable. They had a similar penetration and sharpness. Everything we were able to cut through, they did pretty similar with. So it was really when they went up against each other that the main differences came out.  Some pretty major nicks and bends. And if you were up against an iron sword and they were just wailing on you and you're trying to protect yourself, it would not have been hard for them to just break all the way through your sword and then your toast. If someone with bronze age technology went up against someone with iron technology, it does seem that the iron had a very definitive advantage and could potentially make quick work of anybody with bronze. So welcome to the iron age and thanks for watching. Thank you to all of our supporters on Patreon. Be sure to check out our live stream at 1pm this Saturday and be sure to bring a snack. Thanks for watching. If you enjoyed this video, be sure to subscribe and check out other content we have covering a wide variety of topics. Also, if you've enjoyed these series, consider supporting us on Patreon. We are largely a fan-funded channel and depend on the support of our viewers in order to keep our series going. Thanks for watching.


Hi, this is your host of the Bhartiya here at Open Source Summit in Dublin. And today we have with us John Fontanage, your head of engineering at GINA AI. First of all, it's great to have you on the show. Thank you very much for having me. Now let's talk about GINA AI. First of all, tell us a bit about the company itself. So our company was founded in 2020 and we are working. We are trying to build an ecosystem to help developers to build neural search and multimodal machine learning applications and bring them to the cloud. This is our aim and this is what we are working for. Now let's talk about the open source aspect, you know, and you're involved with the Linux Foundation as well. So our, we have many open source projects, but the main one and the first one that we have, and it's where it's named after our company is GINA. And this is an ML Ops framework to build neural search and multimodal machine learning applications. And this is the main project from which all of our other projects depend and work upon. Can you talk about the scope of GINA Project? Because Linux Foundation, they have a lot of other projects. So what GINA does is in this world of neural search and multimodal machine learning applications is quite new. We don't develop directly the models or we are not in the research part of these models. We are trying to, at every touch point of the developer, from building their own local application until bringing them to the cloud, until production and to have tools that help them and empower them to build these applications easily and saving time and headaches for them. What are the scope of GINA you see in the market still? Because we kind of live in a world where more and more things are getting driven by AI ML technologies.  Analytics is done to extract value from the data, not only to make their services more appealing to customers, because in a lot of other cases, it's just to improve the service itself. And sometimes you're searching the customer directly because that service impacts how customer uses that. So talk about where it is being used. So it has been used mostly on these neural search applications and the scope of GINA is to empower not only the big players to make use of these applications or teams with maybe less resources so that only with a focus on your business logic to be able to have an impact and to scale your product and to bring valuable products to your customers. Can you give example of some of those applications? So with neural search, you can do, for instance, to search images given text, to search audio given text as well. So this is what GINA focuses on, on building these multicross model applications. Can you also, as you said, you know, there are big players, small, can you give, if you can give names that would be great, if you cannot give the names, if you can just give example, these are the industries that are using GINA. In many industries, I know even the type of applications that is powered and that you can, where you can use neural search is very high. So from video games industry to tech industry. So everywhere you may have some part of your applications might be based on these technologies. So maybe it's not the main core of it, but part of the application or the value that you bring to the customer may involve these kinds of applications and these kinds of interactions between data. So in any of these cases, GINA can help. Can you talk about for how long you have been involved with the Linux Foundation? So I worked in GINA since almost the start, so since 2020. And I think GINA, since its birth, is involved in the Linux Foundation. We were founded in February 2020. And GINA is an open source project. Talk a bit about the community itself. Of course, your company is involved. What are the other players who are involved with the project? Who contribute the code? So we are mainly driven by the community, not only in terms of code contribution, but through our Slack and through our GitHub. Our community is reporting bugs, is asking for features and so on. So we are kind of steered by the community. And this is helpful for them because they get what they desire for us because it's a very easy and fast way to get feedback. So that's the benefit of open source for a business like us. Since we are here at the Open Source Summit, what is the presence of GINA here? Talks, booths, sessions, panels, what are you folks doing here? So we are here to do a workshop on one of our projects that is focused on generative AI. That is, we are trying to show how we can use GINA to create awesome art driven by AI. And also another presentation to present our Open Source ML Ops framework GINA to the Summit. Let's talk about commercialization aspect of Open Source because Open Source itself will not succeed if there is no commercial angle or because that's what supports it, that's what sustains it. So talk about what kind of business model GINA AI has. So we are developing, so we have, our core is Open Source but we are developing some products around our Open Source project that can add value and that this is some way to productization. And then another part of our production strategy is to offer these open source and bring them to the cloud and as it can be in our cloud so we can monetize from that side or so that customers can use it on their cloud in an easier way so that we handle the provisioning and all the challenges that come with that. John, thank you so much for taking time out today and talk about the project, the company, your presence at the event and this is the first time we are meeting but I look forward to talking to you again in the future. Thank you. Thank you very much for having me.


One thing that I love about SCU. One thing I love about SCU. What I love most about SCU. That I love about SCU. What I love about SCU. The thing I love about SCU is the faculty. Here's my chiropractic procedure classes because all the doctors are willing to go out of their way to make sure we do things right and get the hands on experience. There's one thing specifically is my cohort and as cheesy as that sounds we all help each other and at the end we know that we're gonna be great chiropractors over here. It's that they offer asynchronous classes which allows to take classes from the comfort of our own home. It's when they host really fun events on campus. It kind of gives us a moment to just break from all the classes we've been taking all day and it also gives us a chance to meet other students from different programs. It's the cohort-based system and the camaraderie that comes with it. As you develop your inter and interpersonal skills with your cohort and people you meet, they're there for you on the good days, and more so on the bad days. We make sure that you're on track and you keep progressing actually.


A Bemidji City Councilman has been fined $750 for campaign finance violations. Michael Beard was fined $500 for accepting a $2,400 in-kind contribution from a committee and $250 for not having a proper disclaimer listing the name and address of the committee which provided the funding for the Bemidji first YouTube video and website which Beard was included on. The issue boiled down to four individuals who paid $7,200 to Pinnacle Marketing Group to create the Bemidji first website and YouTube video for Mayoral Candidate Jill Venny and City Council Candidates Don Heiningen and Mike Beard. The individuals who made the contributions were Dean Thompson and Mary Ann Thompson who are a husband and wife and Steve Hill and Jill Hill who are also a husband and wife. The financial reports for each candidate listed $600 in-kind contributions from each of the four individuals. The Minnesota Office of Administrative Hearings ruled the Hills and Thompson's acted as a committee and not as individuals. If you enjoyed this segment of Lakeland News, please consider making a tax-deductible contribution to Lakeland PBS.


We see that leverage may not affect the vague for a pump because low cost of debt may exactly be offset by higher cost of equity and as a result there is no net benefit for the pump and the vague may remain constant for the pump Now how to compute vague for multiple securities For this we have an example where we have equity of $440 debt of $500 and warrants of $60 only The cost of debt is 5% and the weighted cost of warrants or the return on warrants is 75% whereas the return of levered equity is equal to 18.18% To determine the return on levered equity we have an additional computation where we have to calculate first the expected value of payoffs on the equity and using these payoffs we have determined the return on equity which is 18.18% Now we have to determine the proportional weights of the equity debt and the warrants Then we are multiplying the individual securities return with the securities computed weight and this will give us the weighted cost of debt particular security as then in the end we sum up the weighted cost of the three securities and the resulting cost is termed as the vague and that vague for these three securities is equal to 15% Now what is levered and its relation with unlevered betas We know that a firm's unlevered beta or its assets beta is the weighted average of its equity beta and its debt beta This means that beta of unlevered firm is the sum of its proportional equity beta and its proportional debt beta Now this unlevered beta measures the market risk and this unlevered beta can also be used as proxy for other firms A change in capital structure without changing the investment pools does not change the unlevered beta of the firm because its equity beta will change to reflect the change of capital structure on its risk This means that the beta of equity will be the sum of unlevered beta plus the additional beta due to the leverage effect and this means that the beta of equity in this particular case or the levered beta may also increase with the leverage in the firm We have an example to understand this unlevered and levered betas We have equity beta of 0.8 debt equity ratio of 0.1 We have debt beta equal to 0 So how the asset beta would be We have the model of unlevered beta which is equal to the proportional equity beta and the proportional debt beta using this value In this model we have the asset or unlevered beta equal to 0.73 Now assume that we have debt equity ratio of 0.5 Increase from the earlier of 0.1 So what will be the equity beta Now we have a little change to our unlevered beta Now the equity beta or the levered beta will be equal to the unlevered beta plus the additional beta due to the debt as multiplied by the debt equity ratio Now when we are putting the values into this model the levered beta is equal to 1.09 So there is an increase in the unlevered beta due to the leverage effect This means that the levered beta de-cost of equity and the debt beta will be increased as the firm is increasing its leverage So is there any effect of cash on the firm's cost of capital Now cash and other marketable securities are treated as a risk pre-assert for the firm and because they reduces the required risk premium on the firm's assets This means that excess cash holding has an opposite effect of the leverage on risk and return on the firm's securities This means that the additional cash holding and the marketable securities are viewed as a negative debt for the particular firm If we are to determine the firm's enterprise value we can determine its enterprise value by deducting its cash holding and marketable securities from its overall value or the total value This means that we have to take in fact the net debt of the firm to determine its enterprise value and to determine this net debt we need to deduct its excess cash holding and other marketable securities from the firm's total debt Now we have an example to understand this phenomenon of cost of capital and the excess cash holding We have market capitalization of 140 billion debt 25.4 billion cash and other short-term investments by 60.4 billion equity beta 1.09 debt beta we assume 0 risk period is 2% market risk premium we assume 5% So we need to determine the enterprise value and the return of the unlevered firm To determine the enterprise value we need to deduct the net debt from the market value of the firm Market value of the firm is 140 billion Now to determine the net debt we need to deduct the cash investment of 60.4 billion from its debt value of 25.4 billion The resulting enterprise value is 105 million Now the other computation is the unlevered cost of capital for the unlevered equity cost Now we put the given values into the model of the unlevered cost of equity The resulting value is 9.25% So this means that as we are decreasing our debt then the unlevered cost of debt has also decreased


Welcome to Saul Cast, where we're talking about Season 5, Episode 4 of Better Call Saul titled Namaste. This is our instant take. We just watched the episode, so these will be our raw immediate reactions to having seen it, but make sure to check back later in the week where we'll have our full recap and analysis. By the way, if you're listening to this as a podcast, check out the link in the show notes where you can watch this as a YouTube video on our channel One Take. And if you're watching this on YouTube, subscribe to the podcast to get that full recap later in the week. Without further ado, let's jump into tonight's episode. I'm Gill, I'm here with my brothers Alun, Yo, and Adam. Hello. Let's get into it. So a few things to talk about tonight's episode. Let's start with the character we ended on, Mike. He went looking for trouble. Again, found those same hoodlums and got himself beat up and stabbed. So what is he doing here? Why was he looking to get into another fight? And where do you think he is at the end there? Are they hoodlums or are they punks? Because I wrote punks in my notes. You called them hoodlums last time. That's why I used the word hoodlum. I got that from you. I had a change of heart. Last time with Gill. Are you sure you should call them punks? I think they might be hoodlums. It's complicated. But what's even more, what's less complicated are the reasons that he even went there. I think without even being trusted to have the responsibility of watching over Kaylee babysitting her, he's got nothing left. He's depressed for him. It felt good to get in that altercation last time. So he just kind of went out looking for trouble because he didn't know what else to do. Also, if it's anything like last time, he's been drinking a lot and so probably not making the best decisions. He might have seen a picture of the Sydney Opera House. Yeah, had set him off. Well, when he was told that he wasn't babysitting that night and he walked away, I literally wrote down in my notes, Mike looks like he's looking for trouble. He had a look on his face that said he wants to hit somebody. Yeah, in addition to all of that, it's possible he also feels he deserves that. Maybe he felt like he hasn't been punished for the wrongdoings he's committed. Maybe this was a way of feeling like he is getting what he deserves. That was one of my questions. Do you think he went into that fight hoping to win, hoping to lose, or did he just generally not care? I think he didn't care. Yeah, I mean, I think honestly, I think he wanted to take out some frustration, but I think he also did kind of want to get beat up a little bit himself. And then afterwards, he wakes up in a hospital of some kind, not exactly a hospital, I guess, but it's almost like a farm of some sort out in the desert. Who do you think rescued him? Was he rescued? How did he end up there? I have one theory, and that's, I think, given how thorough Gus is, there's a good chance that Mike is under surveillance most of the time. And Gus also seems to like Mike and see him as an asset. I'm not backing this theory 100%, but there's a chance that one of Gus's people saw this happen and took him somewhere to convalesce. Yeah, my first thought was maybe they stabbed him, one of the hoodlums or punks realized that he was dying, and they thought, we don't want to get involved in a murder right now, so let's take this guy back to my place and care for his wounds. Then I realized, that doesn't seem very likely. My next thought is exactly what you said, is that Gus is monitoring him, saw that he was being threatened, and not Gus himself, but one of Gus's people stepped in and saved Mike. But that leads to a question of, how much is Gus going to put up with? We know he's going to continue to work with Mike, but he already had the Werner screw up, and now he appears to be borderline suicidal, putting himself into this compromising dangerous position. It'll be interesting to see if Gus really is the one who intervened, how we go from this to Gus saying, yeah, I want to keep working with this unpredictable guy who keeps going out looking for danger. All right, I was going to say, if Gus was involved in this, it could be, you know, Gus still wants to work with him, and maybe he will be offering Mike a purpose. Mike clearly needs something in his life to change at this point. And this is very similar, if Gus is involved in this, it's very similar to what happened, I think in season four of Breaking Bad, when, and Mike was involved in this, he basically, Jesse was downtrodden, and Mike kind of helped Jesse get a spirit back by, I think actually might have been the first half of season five, I can't remember anymore. But he had like these two guys corner them in an alley or something like that, and Jesse basically helped them escape, and it turned out to be kind of staged, but it was a way of getting Jesse to feel more confident, and Gus was kind of part of that planning. Do you guys remember that? Vaguely, vaguely, but another example I remember, and I think that this is, I think you're spot on that Gus is going to use this to his advantage as a way of not exactly manipulating Mike, but using what he's seeing in Mike's behavior to guide him to what Gus wants. And we saw similar behavior when Gus appealed to Walt, and Walt's desire to be a man and support his family. So we've seen in the past Gus finding someone who needs a purpose, and giving them a purpose. That happens to align with exactly what Gus wants. Yeah, Gus, despite his abnormal psychology, he understands the psychology of others. That's right. So speaking of abnormal psychology, we saw Jimmy throw some bowling balls, and he was trying to hit Howard, right? Howard sleeps in a hammock in his driveway. That was before we started recording when I couldn't stop laughing, and you asked me why? I had just thought of that line. That's why I was laughing. And also, I'm going to keep calling him Jimmy, because he did say that he lets his friends call him Jimmy, Sol's just his professional name. Yeah, but then he said, you can call me that too. So it could be in addition to friends. Oh, I didn't catch that. I thought that was an intentional kind of, I thought he was just, it was an intentional kind of poke at him. It could go either way. He could be including him, maybe. I mean, well, considering the next time we see him, he's throwing bowling balls at his car, I think Adam's right on this one. Just a prank amongst friends. Well, so in all seriousness, what is he doing here? To me, this felt like old Jimmy, just super childish, and just lashing out.  They made a point of showing us his whole process of, he was shopping at an antique store for heavy object. He was practicing, kind of tossing them to see. This looked so premeditated. And also he was planning to do it before they even had the lunch. So this wasn't going to be fair. Actually, to be clear, I don't know that we know that. I mean, Breaking Bad has at times shown us. Yeah, we don't know the timing. I suspect that this scene, what my interpretation is that this scene took place later on. And they were just showing us a glimpse of what's ahead. Could be wrong though. Yeah, it's possible. Well, whether it was a reaction to that or it was even more premeditated, in either case, it was still premeditated. It wasn't just an impulsive thing. And I think there's a chance it's just Jimmy doing his usual childish petty things. But he's so clever that maybe there's something more to it. I don't know. It could be both. It could be something childish, but there's just a longer scheme here than just destroying Howard's car. Yeah. Oh, what do you think? Childish prank? Or is there some bigger game here? I think he's going to try to be Howard's lawyer. What? I don't know. I have no idea what he's planning. I'm sure it's more than just getting even with him or something. He's definitely got some bigger plan. I have no clue what he's planning, though. Yeah. I've always been pretty sympathetic toward Howard. I think he's been between Chuck and Jimmy. He was put in a difficult position much of the time. And I think he always treated Jimmy pretty fairly considering he didn't have to. And I never really thought he was he was like a douche until I saw his license plate. Yeah. Did you take that as he's essentially had some kind of religious awakening or found some sort of peace with all the guilt he's been harboring? I got more of like a vibe of like a yoga bro. Well, I think that's sort of what I'm saying. I mean, he also gave Jimmy a very long hug. So I see him as he's been struggling with this guilt and he's in this peaceful, compassionate state of mind. And he's trying to be he's trying to amend his wrong doings. He's reaching out to Jimmy, offering him this job. And I think that Namaste was an over the top way of saying of basically saying exactly that. So at first, I thought that he was just kind of trying to alleviate his own guilt by extending an olive branch to Jimmy.  Now, it could have been him putting on an act, but his face and his tone really sold it to me that I think he actually does want a little the more success Saul has and the more clever things he does, the more people want him. So now the cartel wants him. Kim at the end of the episode wants him for something and she's come to him before for for some craftiness. And now Howard wants a piece. Mm hmm. No, I think that's right. I think both things are sort of true. Man, part of it is that Howard has I mean, am I wrong in saying that he's always seen Jimmy as somebody with skill? Part of the reason he always held back was specifically at Chuck's request. I think since then, he has seen even more capability from Jimmy. So he genuinely wants him to work for him. But I think there he's definitely coming to him with his tail between his legs with a very apologetic tone. But I don't think he's offering the job purely out of guilt. Agreed. I think the actor for Howard, who his name escapes me, did a great job, especially at the end of that scene. Yeah, he's always done such a great job. I mean, we've praised him so many times for managing to be somebody we see as a villain, then becomes this sympathetic character. It's a super, it's super interesting character.  I think in the previous episode where Jimmy had his run in with Howard and we saw that he basically barely acknowledged him and then walked away, we thought it's in the past. He really doesn't care anymore. This seems to have brought something back to the forefront and it couldn't have been buried that deep if it was so easy to bring it back up to the surface. So I think this is kind of the first crack in the facade that we think Jimmy's got this hardened shell. He doesn't care about Howard. He's moved on. Maybe he hasn't really moved on entirely. Yeah, that would not surprise me. So I'll say in this episode, I think I hated Gus more than I've ever hated him before because the way he treated poor Lyle, making him clean that fryer over and over again. I was waiting to see him stick that kid's like head in the fryer. I think Gus, we've talked about him being crazy before and having psychological issues, but aside from everything else we've seen already, he's got an obsessive disorder of some kind. Yeah, I think so. And I think there's something that triggers it. And at first, I thought he was nervous. He's nervously waiting for the phone to ring. But do you think? Yeah, that's what I thought. Yeah, that's also what I thought. And I was debating is it nervousness or is it anger where he's very confident his plan will work, but he's angry that he's been forced into this position. But I do think it's nervousness. I mean, it's the classic you're waiting for a phone call and it's pro tip, by the way. And I think of one, you'll agree with me on this. This is the equivalent of when a guy or a gal texts somebody and they're waiting for the reply back. You can't just sit there and stare at the phone. You've got to busy yourself. Like Gus had the right idea. He should go and clean something or do something, but he shouldn't have Lyle do it. He should do it.  Poor Lyle. Exactly. You know, honestly, poor Lyle. And so I will say we last episode, Gus was kind of backed into a wall. And our question was, how is it going to get out of this? We know the DEA is going to check out these dead drops. And Gus's people are going to be there. So how does he not alert them that he knows the word got out so that Nacho gets in trouble? So it turns out he let a few arrests happen. He let close to a million dollars get seized. My one question is that that guy at the end or the third guy that Hank and Steve Gomez had to chase, it seemed to me that he was seen on purpose and he wanted that chase to happen. It's unclear to me why that had to happen considering they did get a few arrests already and they were able to seize all that money. Yeah, it's a good question. I'm not sure why given so at first I didn't you know, we didn't know there were any other arrests. And so I thought this was a way to make it look like a close call that it wasn't a stage to drop. Right. Yeah, that's what I thought. Given that given that there were other arrests, it makes me less certain. Yeah, we'll have to see on the rewatch if if there's anything we missed in this first viewing. But yeah, when I first saw it happen, I was like, Oh, that was his plan was just to do the drop, but then have the guy run away. And then immediately afterwards you realize that happened. Yes. But there were actual arrests. They did seize some money. And by the way, how awesome was it where Hank is obviously disappointed, but he's able to turn it on for his team and give that speech. We're all going to O'Brien's made me really want to work for Hank. He's a good leader. He is. And the more we see him here, the more it's so depressing knowing where he ends up.  Yeah, it does. And Kim goes to the courtroom sees Jimmy's antics. And as always, whenever Kim sees Jimmy do some of his scummy lawyer stuff, I always wonder what her reaction is going to be. It's either going to be her admonishing him or her loving it and wanted to get involved. I think the writers know that we're going to be wondering that because immediately after Jimmy does his switcheroo, we see Kim for a second, but then there's a good 30 seconds where the camera is behind her head. And I keep wanting it to pan around. I want to see the expression on her face, see what's going through her head. And then when Jimmy comes back from the chambers, she says she wants his help with the whole Mr. Acker situation. So as we've seen time and time again, Kim gives Jimmy a hard time. But when she needs Saul Goodman, she goes to him anyway. Yeah, that's great. Yeah. There's a lot of Jimmy doing stuff where I wonder what his plan is. So again, here where Kim sends him to Mr. Acker and gets himself hired as his lawyer. What is the plan here? This could really go in any direction. I can't see Saul really screwing him over that badly, given we talked about it last episode. He tends to feel loyal toward his clients. But maybe he's going to, I don't know, maybe he's going to try to get him to basically agree to leave for more money. That sounds way too simple. I don't know. That's not really a scheme. Have Kim's colleagues at Mesa Verde met Saul yet? I can't remember. I don't know. I don't know at that party. Was that through Mesa Verde? I think it was. I think that Kim's boss, the lawyer knows Saul. I don't know if the Mesa Verde folks do. Okay.  Oh, that's right. Yeah. Well, I was just wondering if maybe they could just pretend they're not associated with each other. I mean, I think she would like for Mr. Acker to get to keep his property and for Mesa Verde to go to that other lot. So if they're not aware that Kim and Saul are together, then Saul could be his legitimate lawyer. But I don't know if that makes any sense at all. Well, that's what I was struggling with. So Adam, you were saying you don't know if Saul would screw up Mr. Acker. When I think about it, now that after hearing that, I realize probably they're going to try to get Mesa Verde to go to the other lot. Right. Exactly. They are fighting on Mr. Acker's behalf. But to a lunch point, if Saul represents Mr. Acker and they turn this into a lawyer versus lawyer battle, Mesa Verde, Kim's boss, they're going to recognize Jimmy. They're going to recognize Saul. And you've got to think they're going to look at Kim and say, what the hell's going on here? Yeah, this might be the thing that gets her in some trouble. Yeah, which ultimately may not be such a bad thing for her. It seems like she really doesn't like this job. Yeah, but I hope it doesn't. I don't know if it jeopardizes her license though. That's true. That's true. How awesome, by the way, was it Jimmy's method for winning Mr. Acker over? Yeah. I'm going to start using that. Carry a picture of that around in your wallet. Every meeting I go to, that's the first thing I'm going to show them. I'm just going to put it like projected on it. It shows up in the middle of a PowerPoint and you're like, oh, how did that get in there? Well, now that it's here, let's look at this for a second. Now you notice the man is the one doing the act. The act. They're like, Alon, what is this a metaphor for? And you go, hey, what's a metaphor? This is just a picture. The horse is your clients. Wait, no. There's something here.  You can really say anything. It's funny because I wrote down for that scene how smart Jimmy is. He really knows his audience and I think that there is a very specific audience where this will work. I don't know if there's many people besides Mr. Acker where this would have been the right move. And Alon and I. I like how long Mr. Acker had to look at that photo to process what he was really looking at. He's like, is there, am I missing something here? Yeah, it's absolute genius. Any other thoughts on the episode while it's still fresh? Yeah, I noticed no Lalo this episode. I know. I know that's always a disappointment. I even was paying attention to the opening credits. I saw Tony Dalton and I was like, yes, that's right, Lalo. Yeah. And also when I forget what scene it was, it might have been in, I forget, at some point you hear the sound of cooking. And I thought maybe we're about to see Lalo, but we didn't. And more of his tacos. Yeah. I also wonder the 50% off guys. I wonder if there's more to that or have we pretty much closed out their story at this point? It's so far, it seems me too. I mean, the interaction between Saul and those two boneheads is just great. But I wonder if it's just, A, entertainment, B, it shows you a little bit of the problems caused by the whole 50% off and basically encouraging more crime. I wonder if there's more to it than just that, if we'll see more of them. And if they'll, it's almost like those two skater kids where they ended up being pretty important in terms of getting Saul mixed up with Tuko. I wonder if these 50% off guys will play a similar role or if we've basically seen everything we'll see of them at this point. Yeah, I think the show tends to capitalize on the characters it introduces, even the minor ones. So I think we will see them again. They also, they're good comedic relief. They're like an even poorer man's Badger and Skinny Pete. So like if Badger and Skinny Pete are like a poor man's comedy duo, then these are like a homeless man's Badger and Skinny Pete. No, I guess, I don't know if that's offensive. Well, it's like Chris Farley and David Spade. Then it's Badger and Skinny Pete. Then it's those two guys. Yeah. And then it's Adam and Alon talking about it. No, then it's Wendy and the man that whose head she crushed for the ATM. Oh, god. Breaking Bad. And I don't think it gets a little lower than that. Wait, it wasn't Wendy. It wasn't Wendy. It was a different scale. Gotcha. I thought you were talking about Wendy's for a second. And what's his name, Adam? What's the name of the guy you founded, Wendy's? Oh, I will not tell you that because you're not going to get me. If you want to know what that is, what we're talking about, listen to the full take. That's a little inside joke here. Yeah, we'll explain the whole thing. Jokes are best when they're explained. Yeah, I just remembered something about that. I wanted to mention with the guys that beat up Mike, speaking of hoodlums like these 50% off guys. Those are also hoodlums, so good transition right there. Yeah, it's even better when you explain it. Thanks, man. Yeah. So remember last time I was saying how they were pretty bad friends. They let Mike just break their friend's arm and they just stood there and watched and then let him get away. So I'm glad they were able to get their revenge. That's a lot on everyone. They're the good guys in this show. Yeah, they've redeemed themselves good. I'm glad we got closure on that open thread. Okay, well, I think that wraps it up for tonight's episode. Like we said at the front, make sure to check back later in the week for the full recap and analysis. If you're listening to this or you're watching this on YouTube, check out the description. We'll link to the podcast so you can also hear that deeper analysis of the episode. Anyway, with that, thanks for listening. Thanks for watching and we'll see you on the next saw cast.