Appalachian Wireless has a plan to make your life simpler. It's called forward pay. No contract, no credit check, no problem. Plans start at $19.99 a month and include unlimited talk and text or ticket to the max with unlimited data plus for only $89.99, which has 50 gig of high speed data because we are you. We are Appalachian Wireless. The Beta Club at Pike Central will be hosting a Stay Warm donation drive January 14th. All donations received will go to the East Kentucky Dream Center and will be distributed to residents throughout the community. Our January project, we're partnering with the East Kentucky Dream Center and we're taking donations, coats, blankets, anything to help keep people warm in the cold times. We take care of people around us. We live in a really big county. So there should be a lot of people that are in need and they take care of the help. Well, we take care of our school. We always try and clean the lunch room the best we can. And then we tutor the younger kids in the feeder schools. The school will be accepting donations of coats, gloves, socks, blankets, and similar items. As first year teachers, we were kind of in the rut of how else can we give back to our students? How else can we give back to our school? And I had a student approach me that asked if we could bring Beta Club back. So I was like, okay, yeah. And then we've been best friends since like, well, at fifth grade. So we graduated together from here. We were graduating middle school together. And I was like, well, I'll ask Miss Case to be my co-sponsor and we'll get this Club back started and we'll do it right. And one of the main reasons I went into teaching was to give back to my students and give back to my community. So Beta Club is all the reasons that I love teaching. It's because I get to be able to be with my kids. I get to give back to the community. I get to help people and we love doing that. Donations will be accepted from 11 to 5 p.m. at Pike Central. All donations must be new or in good condition. Reporting for Mountaintop News, I'm Brianna Robinson.


I am great and very happy to see you all again So let's continue learning our numerals. We've already learned about numeral 1. This is numeral 1 and after 1 we learn about numeral 2 1 and 2. What comes after 2? 1, 2, 3. This is numeral 3 1, 2, 3. How many circles do I have here? 1, 2 and 3 circles. To learn more about numeral 3, let's enter into my kitchen and make some healthy salad. To make the salad the first thing that I need is some plates. So I have 1 2 and 3 3 plates and now we will start putting vegetables in them. So the first vegetable that I am going to put is tomatoes. I have 3 tomatoes. 1, 2 and 3, 3 tomatoes and then I am going to add some onions to it. So I have 1, 2 and 3 onions over here. After onions what do I put? It's summer season and I love to eat cucumbers at this time. So I have 3 cucumbers. 1, 2 and 3 cucumbers. Now my vegetables are ready in the plate. To make this salad more yummy and tasty I am going to add your favorite thing. Guess what it is? It is cheese. I have some cheese cubes over here. 1 2 and 3. So I am going to make salad for 3 people and I have 3 plates. 1, 2 and 3 plates to make salad for 3 people. Now, do you know how to write this numeral 3? Let's quickly learn how to write numeral 3. Let's do a quick a rhyme. A rounder tree, a rounder tree and that makes numeral 3. Let's start writing saying this rhyme again. A rounder tree, a rounder tree and that makes numeral 3. Wasn't that easy? Let's try one more time. A rounder tree, a rounder tree and that makes numeral 3. One more easy way is to say one half circle go back and make another half circle. Let's do it again. One half circle go back and make one more half circle. So this is the way we write numeral 3. I hope children you enjoy learning about numeral 3 with me. Thank you.


They made a camera that's so small. It's the size of a grain of salt world's smallest camera. What? It's the size of a grain of salt. I just see how does it even possible to make some images that look like a full size camera? How little do the people have to be that make that? I don't believe it. A little Kevlar elves like making me say it doesn't even make sense. Like, how do you capture those images? That's the part trips me out of my mind. I can't wrap my brain around. I still don't understand how regular cameras work. This is what it says here. So on this grain of salt size camera are a million eyes, roughly the size of an HIV virus covering the camera. I like how they matched it to it. Why HIV is a virus? They're just to show you how big this is a virus. I don't know why they use that virus. Why would you use that one as an example? We're going to put this inside you. Yeah, right. It looks a lot like. Well, first off, I think cool.


So it's really a great pleasure for me to introduce Olivia Caramello, who is in University of Insubria in Como, and all the Gelfand chair in the EHOS. And I mean, who contributed so much, you know, to the theory of toposis. And she will talk about introducing, if you want, the geometric theory of toposis, and the new theory of relative toposis, which with respect to the logical aspect of toposis theory, will surely play a fundamental role in extending the theory to higher order logics. So Olivia, it's your turn. Thank you so much for this excellent introduction. So yes, as Alain said, this course is going to be a geometric introduction to topos theory by using the language of the shifts and stacks in relation with relative topos theory. And by this, I mean, doing the topos theory over an arbitrary base topos. So authorizing a change of the base topos in a similar way as Grottenbeek used to do in algebraic geometry with things. Relativity techniques for schemes play a central role in his re-foundation of algebraic geometry and we are trying to develop a similar formalism for toposis. So the plan of the talk is this. So I shall start this course by reviewing the classical theory of shifts on topological space. In fact, it is quite important to take this as a starting point also with the purpose of developing relative topos theory because we wanted to keep the geometric foundations at the center. And so we infect one of the central ingredients of our approach to relative topos theory will be an adjunction, which provides a wide generalization of the very classical pre-shift bundle adjunction. For topological space. So it is important also to understand these new developments to start from the very classical theory of shifts on topological space and pre-shift more generally. Then I shall proceed to recalling the basic theory of shifts on a site. So we shall get to the definition of the Grottenbeek topos as any category equivalent to the category of shifts on a small site. Then I will make a methodological interlude on the technique of toposis. Because it will be applied both in this course to derive results and also in other lectures or talks at this conference. Especially to derive concrete results in different mathematical contexts by exploiting the possibility of presenting a given topos in multiple ways. So this is a sort of the basic technique that can be used for extracting concrete knowledge from toposis or more precisely from equivalences between toposis presented in different ways or morphisms between the opposite again presented in different ways. So as when we do relative topos theory actually we are concerned with the study of morphisms of toposis because what is a relative topos? Well, it's just a topos which we decided to consider over another topos via a morphism connecting with it. So basically doing relative topos theory essentially amounts to studying morphisms between toposis. And so we shall investigate morphisms between toposis from the point of view of site presentations of toposis. So we shall describe how one can induce morphisms between toposis starting from functions satisfying the suitable properties. We shall see that there are two main classes of functions which induce in a contra variant or in a covariant way morphisms between the associated topos. These are the so-called morphisms and comorphisms of sites. In fact, the morphisms of sites represent an algebraic point of view on morphisms of toposis while comorphisms represent a geometric viewpoint. And in fact, we shall see that they are in a sense dual to each other. Then I shall also review a classical notion already introduced by Grotendick of the function of between sites, the notion of a continuous function between sites, which in fact plays an important role in the context of the vibrations, which are very important for developing relative topos theory. We shall see that whenever one has a morphism of the vibrations, this gives rise to a comorphism of sites in a canonical way, which is more of a continuous. So continuous comorphisms of sites, as we shall see, induce a special kind of morphism between toposis, the so-called essential morphisms, which satisfy pleasant features. And so in preparation for the last part of the course, we shall describe these continuous functions also in relation with the vibration. And then, as I said, the last part of the course will give an introduction to our working progress with Riccardo Zanza on developing relative topos theory by using the language of stacks. So we shall take, as a starting point, Giro's paper, classifying topos, where the notion of classifying topos of stack was originally introduced. We shall extend many of its results. And then we shall also turn our attention to relative shift topos, so we shall introduce a notion of relative site. And we shall also compare our approach with the usual more classical one that has been pursued by category theories, which is based on the notion of internal category and internal site. So we shall see that this formalism of stacks and relative sites is much more flexible than internal categories and internal sites. It allows parametric reasoning. And it will also pave the way, as Alan has mentioned, to an extension of geometric logic, which is, as Loran explained in his course, the logic underlying groten-dictopos is overset to some higher-order geometric logic provided by the fact that we can change the base topos. And so this will allow us to quantify over parameters essentially coming from the base topos in a certain way, which is still to be made precise, but these are developments that we are pursuing at the moment. And so I will present you some of the results we have already obtained in this connection on relative toposes. For the moment, we have mostly focused on the geometric side of the subject. But then, once all the geometric aspects will be completely clarified, we shall also introduce a higher-order relative parametric geometric logic corresponding to relative. Okay, so this is the plan, and so now we can start with recalling the theory of pre-shifts and shifts on etopological space. Okay, so the etopological space, what is a pre-shift on the space? A pre-shift is simply a way of functorially assigning an open to any open set of the space, a set. In a contravariant way, so in a functorial contravariant way, so we want that to any inclusion of open sets corresponds to a function going in the opposite direction. In addition, in fact, traditionally is called a restriction map because the idea one has when the basic example of a pre-shift one has in mind is that of all continuous functions on some open set of a given space. You see that if you pass from an open set to a smaller open set, you can restrict continuous functions on that open set to the smaller open set. And so you see that in this way to an inclusion of open sets corresponds a map going in the other direction from continuous maps on the big open sets to continuous maps on the smaller one given just by restriction. Of course, in general, you can have pre-shifts or even shifts which do not look at all like this, but this example of continuous functions or other kind of functions on a space has been one of the motivating examples in the development of the theory. And so these maps are still called frequently in the literature restriction maps. Okay, so we have said what the pre-shift is. Then of course there is a natural notion of morphism between pre-shifts, which is simply a collection of maps between the sets corresponding to the open sets, which is compatible with respect to the restriction map. Now, the way I have presented pre-shifts so far is very concrete and explicit, but in fact, categorically speaking, one can simply say that the pre-shift is just a factor defined on the opposite of the category of open sets of the space with values. So O of X is a post-set category whose objects are the open sets of the space and whose arrows are just the inclusions between them. So using this language, we can rephrase what the morphism of free shifts is as just a natural transformation between the corresponding functions. So we have a category of pre-shifts on X, which is denoted as written in the slides. So it is basically a category of set valued sum. Okay, so now we are going to define shifts as particular kinds of pre-shifts. So a shift is a pre-shift which satisfies some gluing conditions. So what do I mean by gluing conditions? The idea is that one should be able to define in a unique canonical way a certain global data starting from a set of local data that are compatible with each other. So this idea of gluing from compatible local data is expressed formally by these two conditions in the definition of. So formally the condition refers to coverings of a given open set of the space by a family of open subset. So the gluing condition is formulated with respect to such covering families. So for each such covering family, one requires that whenever one has a set of elements of the pre-shift indexed by open sets in the family, which is compatible, which satisfies these compatibility conditions. You see this is expressed by the fact that their restrictions on the intersections of the two open sets are equal. Then there should be a unique amalgamation of this local data, a unique global data, which restricts to each of these local data. The uniqueness of such a global data is ensured by condition one. So condition one ensures the uniqueness, condition two ensures the existence. So together we have uniqueness and existence of a global amalgamation of a set of locally compatible data. Okay, so now as we did for pre-shifts, we might wonder if it is possible to categorify the notion of shift on a topological space. Now there are a few remarks that we need to make in order to arrive at such a categorical generalization. First of all, we remarked that the shift condition is expressed as we said we respect to covering families of open sets of the space by families of open sets. So if we want to replace the category of open sets of our space with an arbitrary category, and if we want to be able to formulate a shift condition, we need to have a collection of families of arrows going to a given object which should provide a replacement for coverings of an open set by a family of sub-open sets. This, as we shall see, will be provided by the notion of a growth in the topology on a category. This is the first ingredient. Then there is another element which requires some thought in order to arrive at the categorical generalization. It is what you see in condition to concerning the compatibility relation you see for the local data which involves the topological setting considering intersections of open sets. So of course in general in an arbitrary category you will not be able to have an analogue of that. But by using the categorical notion of a sieve we can get around this problem and we will be able to define shifts on any category equipped with a growth in the topology by using this device. I shall give the details in a moment. But for the moment I just wanted to remark that basically three shifts can be just defined as contravariant functors on a category with a losing set. So we don't need any additional data on the category. On the other hand to define shifts we need to specify a collection of covering families and this will be provided by a notion of growth in the topology. Okay, so just a few remarks before going to the categorical generalization. So categorically speaking we can reformulate the shift condition as an equalizer condition because we can consider, so our pre-shifts have values in sets. We can consider arbitrary products and so we have canonical maps as written on the slides. And so you realize immediately that you can formulate the shift condition for a pre-shift as the condition that the canonical map going from f of u to the product of the f of ui where the ui form a covering of u. This canonical map should be the equalizer of the two canonical maps between these two products. And yeah, so it's important to remark that in fact the shift condition is actually a limit kind of condition in the category of sets. And secondly in preparation for the categorical generalization it's important to remark that by using the technical device of shifts we can avoid referring to intersections for formulating the gluing condition. So how does one do this? So given a covering family f of open subset ui of an open set u, you generate a sieve starting from that which in this case will be the collection of all the open sets of u which are contained in some ui. And so by doing this you can basically rephrase the compatibility condition by requiring instead of a family of elements indexed by the open sets in the covering family, you take a family of elements indexed by all the open sets in the sieve. And the compatibility relation becomes the relation that whenever you have an open set w prime included into w, you should get the equality between the value at w restricted at w prime and the value at w prime. So you see that in this way you have eliminated the reference to intersections and so you can see that by using this idea of sieves of taking everything which is below everything which is essentially generated by composition on the right from a certain family of arrows because you can regard this inclusion as arrows in the category of open sets of x. You can avoid referring to intersections and so you can already understand why it is possible to define sieves on an arbitrary category equipped with an essentially arbitrary notion of covering families in it as it will be provided by a so-called growth index. So now we shall go in more details about all of this, but before this I would like to give some examples of the sieves. So I have already mentioned the main motivating example provided by continuous functions on the topological space. Of course, sieves are used in many other areas of mathematics in the context of the differential geometry, analysis, differential algebraic geometry, etc. So for instance, you have sieves of regular functions on a variety of differentiable functions and the differentiable manifolds of allomorphic functions on a complex manifold, etc. In mathematics, sieves arise in many different contexts. Very frequently, sieves appear as endowed with more structure than just the set theoretical one. In fact, for instance, in algebraic geometry, one has a sheaths of modules or sheaths of rings, even sheaths of local rings, etc. So far we have talked about the sheaths of sets and in fact there is a good reason for taking as starting point the sheaths of sets rather than sheaths of more complicated type of structures or even more general structures. In fact, Grotendick himself realized about the importance of first talking about the sheaths of sets in order to have better categorical properties when you consider the whole category of sheaths of sets on an on a topological space or more generally on a side. Because if you replace sets with another category, in many cases you lose some pleasant categorical properties. So it is good to define categories of sheaths of sets. And then try to understand sheaths of more complicated structures as relative to these sheaths of set valued structures. So formally the way this is done, at least for geometric theories in the sense which has been explained by Locang in his lectures is you look for instance at a sheath of models of a certain geometric theory. In particular it could be sheaths of modules or sheaths of rings, sheaths of local rings, etc. You regard this as a model of the theory of such structures, a model of the theory of modules of rings of local rings formalized within geometric logic inside the category of sheaths of sets on the space or the side. So this is the way we can naturally deal with these sheaths of more complicated structures. So yes, this is an important remark to make from a formal view point. Okay, now let's proceed to talk about this fundamental adjunction between pre-sheaths and bundles on a topological space which I mentioned in the introduction of my talk. So after this we shall describe sheaths on a side. Okay, so how does this very classical adjunction work in the topological setting? So first we define given a topological space X a bundle over S simply as a continuous map towards X. We have of course a category of bundles which is simply the slice category. So the top here denotes the category of topological spaces and continuous maps and here I am taking the slice category over X. So then the objects are arrows going to X continuous maps towards X and the arrows are commutative triangles. Okay, now why is it interesting to consider bundles in relation with sheaths? Well, because there is a very nice construction which allows us to build a sheath from an arbitrary bundle through the consideration of the so-called cross sections of the bundle. So given an open set of our space X, a cross section over this open set of a bundle is simply a continuous map defined on that open set going to the domain of the bundle such that when it is composed with the bundle map it gives the inclusion of the open set into the space. So of course you can collect all the cross sections over a given open set in a set and this gives a set, the set of all cross sections over the given open set U. And if you think a minute about this you realize that this operation is factorial in a contra variant way in the open set because if you switch from an open set U to an open set V contained in U, you get the restriction operation because you can restrict a cross section over U to a cross section over V. This is completely clear. So in this way you actually get a factor, a contra variant factor on the category of open sets of X with values in set, namely a pre-sheath. And it is not hard to show that this pre-sheath is actually a sheath and this is called the sheath of cross sections of the bundle T. So given the fact that we can build ships from bundles it is natural to wonder if one can go in the other direction as well. And this is possible through the construction of the so-called bundle of germs of a pre-sheath. So suppose that you started with a pre-sheath on a space X, you can build the bundle out of this by considering germs of sections of the pre-sheaths at the points of the space. So first we have to define what a germ is at a given point of the space. So a germ is an equivalence class of sections defined over open neighbors of the point. And the equivalence relation is what I have written in these slides. So two sections are equivalent in an open neighborhood hub. Of course the open neighborhoods are different so they are considered equivalent if there is some open neighborhood of the point contained both in U and in V on which their restrictions agree. So of course this is an equivalence relation so we can take equivalence classes and we can do this for each point of the space. And so if we fix the point and we take the collection of all germs at this point we get what is called the stalk of the pre-sheath at the given point. And of course we can consider the disjoint union of all the stalks. So it is a union indexed by the points of X. And of course we have a projection map to X which is defined in the obvious way so it takes just the point on which the germs are defined. And we can topologize the domain of this projection map in such a way that it becomes a local homeomorphism. And so what we get is actually what is called the bundle of germs of the given pre-sheath. Okay so now we have two constructions one going in one direction the other one going in the conversed direction and in fact what happens is that so we have these two functors gamma which is the functor of the cross sections and lambda which is the functor going giving the bundle of germs. And in fact these two functors form an adjoint pair so lambda is adjoint on the right. On the left sorry and gamma is actually a global section, well it's a section functor. So in particular if you can apply it to the space itself in this case you get the global sections of the bundle. And so in fact a key result is that this adjunction actually restricts to an equivalence of categories where you have on the one hand sheaths so the pre-sheaths which satisfy the sheath condition and on the other hand some particular bundles which can be characterized as being the etal bundles or also called local homeomorphism. So in fact the restriction of this adjunction is what you get by restricting to the fixed points, fixed points of the adjunction. So in general whenever you have an adjunction you can restricted to an equivalence of categories by restricting to the fixed points. So this is a general process and if you apply it in this case this is what you get. So this is very nice because it allows us to geometrically think about sheaths as particular kinds of bundles, namely the etal bundles. This has several pleasant consequences concerning the geometric understanding of the number of constructions on sheaths and pre-sheaths. So in particular I would like to point out two of these nice insights that such an adjunction brings out. So first of all the sheathification process. If you take most books on Topos Fury you will see that the sheathification is described by using the plus-plus construction which is a technical means of constructing this, but not necessarily very geometrical intuitive. So thanks to this adjunction one has a more geometric understanding of the sheathification process. Because in fact we can think, we can describe the sheathification of a given pre-sheath as simply the result of applying successively the two functions of forming the adjunction. So basically the sheathification of a pre-sheath is simply given by the sheath of cross sections of the bundle of germs of the pre-sheath.  So it's not just a formal portion construction but you have a geometric realization of such elements. And we shall come back to that because in fact we shall be able to get a generalization of this in the context of an arbitrary site and even in the context of the stock. So just keep in mind for the moment these features because we shall then provide the generalization in the categorical and stock theoretic setting. Okay, so we have talked about the sheathification geometrically understood. There are also other advantages of this point of view of sheaths as a tail bundle. For instance, suppose you have a continuous map between topological states and suppose you want to describe the effect on sheaths of such a continuous map. Of course, there is a direct image of sheaths which is defined in a striped forward way basically just composed with the action of the continuous map on open sets of the two spaces. So this is completely striped forward but suppose you want to understand the inverse image of sheaths along this continuous map. Well, this is not completely striped forward if you want to do it in the language of sheaths. We shall see that you can do it by using a can extension for instance. But here by using the identification between sheaths and the tail bundles you have a very nice simple description because taking inverse images of sheaths on Y along F corresponds precisely to take the tool back. Along F of the etal bundles corresponding to these sheaths. So in fact, one can show that taking the pullback of an etal bundle still gives an etal bundle. And so this is how the inverse image operational sheaths actually works. So you see that really this adjunction brings some very nice geometric intuition in the picture. So it has been a question for several years whether one could find a good analog of this working for arbitrary sites or even possibly extending to stats. And in fact, in our joint work with Ricardo Zanfa we have indeed provided such a generalization, not just for pre-sheaths but more generally for indexed categories. So this will be described in the last part of the quiz. Okay, so now that we have talked about this fundamental adjunction, we can go to the categorification of sheaths. So how to define sheaths on an arbitrary site. So for this, as I already anticipated, it is necessary to talk about sheaths because you see sheaths were fundamental for giving a notion of compatible family of local data without requiring the intersection property. You remember about that. So it is a technical device that is essential for defining sheaths in the general categorical setting. So formally, what is a sieve in a category? Well, given a category and an object of this category, a sieve in the category on that object is simply a collection of arrows in the category towards that object, which is closed under composition on the right. So the condition is that whenever an arrow is in the sieve, the composite of this arrow with an arbitrary arrow should again be in the sieve. So you see this is a categorical generalization of the condition we had for topological spaces. We wanted the sieve to contain any other smaller open set, you see, and here we get the condition that it should be closed under composition on the right, which is just the categorification of it. Okay, so sieves are very nice objects, in fact, because you can make a lot of operations on sieves. In fact, you can understand the sieves more abstractly as the sub-objects of the corresponding representable functor in the category of pre-sheaves. So this is an abstract understanding of sieves as sub-functors of representable functors, but you can avoid that point of view. You can reason about the sieves in a perfectly concrete way. So in particular, you can compute pullbacks of sieves, which give rise again to sieves. So here is the operation. So if you have a sieve on an object and an arrow going to that object, you can pull the sieve back along this arrow to get another sieve. So these sieves consist of all the arrows which, when composed with the given arrow, belong to the sieve. So it's a very naturally defined operation, and in fact it really corresponds to taking a pullback in the corresponding pre-sheave topos. But as you see, you have a perfectly concrete description of this operation without involving pre-sheaves. Okay, so now that we have talked about sieves, we can introduce the fundamental notion of a grotendic topology on a category, which will be the basic setting for us to define sheaves. So a grotendic topology is a way of assigning two objects of a category, a collection of sieves on those objects in such a way that some natural conditions are satisfied. So the first condition is called the maximality axiom. So it requires the maximal sieve on each object to be in the topology. So of course, this is quite intuitive. So of course, the maximal thing should be covering. It's quite natural. And we have a second axiom, which is quite important. It's called the stability axiom. So it requires that the pullback of any covering sieve should be covering. Again, this is quite intuitive. In fact, in the topological setting, it corresponds to the fact that if you have a covering of a certain open set by a family of open subsets, when you pass from that open set to a smaller open set, and you intersect each of the open sets in the family with that one, you still get a covering of the smaller open set. So this is stability. So you see that in the topological setting it is satisfied. So it is natural to require it as well on an arbitrary category. Then you have the so-called transitivity axiom, which says that whenever you have a sieve such that the pullbacks of this sieve along all the arrows of the covering sieve is covering, then the sieve itself should be covering. Okay, so this axiom is a bit less important than the other ones, especially the maximality axiom you can always make it hold without any problems. But the stability axiom is really the crucial axiom to have. In fact, for defining sieves, basically you just need maximality and stability. And because then you can generate the groten-dictopology starting from those, and it will have exactly the same sieves. So basically not all the axioms of groten-dictopology have the same statutes. In fact, the most important one is really the stability axiom. In any case, one often works with the basis for groten-dictopologies or some smaller presentations for groten-dictopologies. And in general, it is important to dispose of techniques for computing groten-dictopologies presented by some families of sieves. There are formulas for this. In my book, for instance, you can find a formula for computing the groten-dictopology generated by an arbitrary collection of sieves and some other techniques for computing topologies starting from basis satisfying certain properties. So in general, it is an important theme that being able to compute a groten-dictopology starting from certain sets of data. From a logical point of view, in fact, as Loran will explain in the last part of this course, being able to generate a groten-dictopology starting from a collection of sieves corresponds to deriving theorems within geometric logic starting from certain axes. So as you can see, it's something quite significant, especially when you can really achieve a full description of the groten-dictopology. It means you have a classification of all the geometric logic theorems that are provable in a given theory. Okay, in any case, so the sieves which belong to the topology are called j-covering, where j is the topology. And then a site is defined simply as a pair consisting of a category and a groten-dictopology. So which kind of sites in terms of sites are we going to consider? Well, for defining groten-dictoposis, one restricts two small sites. And by this, I mean that the underlying category of the site should be small to have just a set of objects and arrows. But for technical reasons, it is important also to consider larger sites which still can be studied, can be associated with smaller sites in a meaningful way. So these are called small-generated sites. And in fact, a site is said to be small-generated if the underlying category is locally small and admits a small j-dance subcategory in the sense expressed in the slides. So in particular, it will be convenient from a technical viewpoint to consider a topos itself as a site with a topology called the canonical topology. And you see that a topos in general will not be small, but such canonical sites will always be small-generated by the definition of a topos as a category having a small set of generators. So it's important to keep in mind that while a groten-dictoposis is formally defined as a category of sheets on a small site, in fact, one can extend to all small-generated sites without changing the resulting categories of sheets. Okay, but this is just a technical point on which we shall come back later. Okay, now let's give a number of basic examples of groten-dictopologies to get you familiar with the concept. Of course, we can always put the groten-dictopology on any category by taking as covering sheets just the maximum ones. Then there is a very nice interesting topology that one can put on an arbitrary category. It's called the dense topology, and it's defined by taking as covering sheets precisely the stably non-empty ones. So the ones such that they are pulled back along arbitrary arrows is always non-empty. So this simplifies in the situation where the category satisfies the right or condition, namely the property that you see displayed in the slide, the fact that for any pair of arrows that would come in a codomain you can complete them to a commutative square. So under such hypotheses, in fact, the pullback of any non-empty sieve is again non-empty. And therefore, the dense topology specializes to the so-called atomic topology, whose covering sieves are precisely the non-empty ones. The atomic topology is very important for several purposes in toposphere, in particular in connection with the topospheric interpretation of Galois theory and its categorical generalization. Okay, other examples of topologies. Well, of course, the motivating example for us was sheets on topological space. So in fact, we were considering pre-sheeps and sheeps on the category of open sets of the topological space. On such a category, there is a canonical growth and dik topology one can consider. So in fact, we postulate that the families that should be covering exactly those which give covering families in the usual topological sense. That is, the open set should be the union of all the open sets in the family. This, of course, can be generalized to pointless topological spaces, also called frames or complete hiding algebras. So the frame is a complete lattice in which the infinite distributive law of arbitrary joints with respect to finite needs holds. So you can really see this as it were the lattice of open sets of a topological space, even though in general you might have frames which are not of that form. Only the spatial frames come from topological spaces. There are many other frames which are not of this form and which can be studied and which can be interesting in their own right. And of course, on such a frame, one can define a growth and dik topology by using joints in this frame. Thanks to the fact that these arbitrary joints distribute over finite needs, which ensures that the stability axiom for growth and dik topologies is satisfied. So the topology we define in this way on a frame is called the canonical topology on the frame. Okay, now another set of examples of a different nature. So given a small category of topological spaces which is closed under finite limits, typically one suppose is that and undertaking open sub spaces. There is a natural topology one can define on it called the open cover topology because it's covering families are precisely given by families of open embeddings to which cover the given space. In fact, the open cover topology plays an important role in the construction of growth and dik toposes in the topological setting. In fact, with Ricardo Zanfa, we have introduced a higher analog of this open cover topology on the category of toposes itself. And in fact, we have shown that thanks to this one can essentially regard any growth and dik topos as a sort of petty topos associated with a very big topos related to that by a local morphism, by attraction, etc. And so in fact, this idea of the open cover topology is an interesting one. I will not have the time in this course to talk about this result, but if you are interested, you will be able to read about it in our forthcoming work. Okay, another very important example of growth and dik topology is the Zariski topology, which can be defined on the opposite of the category of finitely presented or equivalently fan-generated commutative rings unit. So of course, this topology plays a key role in algebraic geometry, and it admits a very simple intuitive definition. So the covering cosives for this topology, so I talk about cosives because I switch from the opposite of this category to the category itself. These are those which contain the, which contain finite families of localizations of the given ring at elements of the ring. The families which are characterized by the property that the ideal generated by these elements is the wall ring. So equivalently, this collection of this set of elements is not contained in any proper ideal of the ring. So of course, you understand the geometric significance of this definition. You see, if you think of the Zariski spectrum of overring, you can see that this in fact corresponds also to a more intuitive kind of covering relation at the topological level in terms of spectrum. Then finally, Loran has talked about syntactic sites in his course. And of course, these are very important kind of sites that one can build from any kind of per-order geometric theory. In fact, depending on which fragment of geometric logic you consider, you have different versions of syntactic sites. So if the theory is regular, for instance, you have a regular syntactic site. If the theory is coherent, you have a coherent syntactic site. If the theory is geometric, you have geometric syntactic sites. So different versions of syntactic sites, but which will present always the same classifying topics. And so you have different ways of, say, embodying the syntax and the proof theory of a theory in a site which actually presents its classifying purpose. In fact, it's quite interesting also to compare properties related to different fragments in which a given theory can be considered. There are compatibility relations existing between different fragments. And for instance, you can understand them very well by using the bridges because you have just one classifying purpose and different presentations for it provided by these different fragments. And the point is that you can understand several invariants from these different points of view and they will give rise to such a compatibility relation. So in any case, this is just a remark. So if you want to know more about this, you can take my book and you will find several results, several compatibility results of this kind proved through bridges. Okay, so now we are ready to introduce SHIBs on a site. SHIBs are defined in the obvious way, so simply as contravariant functors with values in sets defined on the given category. Then for defining SHIBs on a site, we have to talk about compatible families of local data indexed by covering families in the Grotendictopology. We define a notion of matching family for a SHIB of elements of a pre-shift. So this is defined as a way of assigning to each arrow in the SHIB, an element of the SHIB, in such a way that this compatibility condition you see is satisfied. And that here the SHIB condition is fundamental because you see that here I am considering the composite of F with G. So I am using the fact that since I had a SHIB and F belongs to the SHIB, also F composed with G belongs to the SHIB and so it makes sense to consider that element because I should have an element for each arrow in the SHIB. And so I can formulate this compatibility condition. And so this is what a matching family is, and then for such a family we can define what should be an amalgamation. So an amalgamation should be a single element of the pre-shift at the given object, which is sent by the pre-shift to all this local data along the arrows of the SHIB. So you see everything is very, very natural, very unsurprising. Okay, so again as in the topological setting we can formulate the SHIB condition in terms of an equalizer. And so by considering all the pre-shifts on a category which are SHIBs, with respect to a given growth and diktology, we get the category of SHIBs. Which is what is denoted like this, so SHIBs on CJ will denote the category of SHIBs on the given site and as arrows the natural transformation between these SHIBs regarded as pre-shifts. Just to remark, the SHIB condition can be expressed categorically in a very nice way. So you see I mentioned that a SHIB can always be considered as a sub-object of the corresponding representable. So you see this vertically in the triangle. And the SHIB condition can be formulated as a sort of extension condition. So a matching family for a given SHIB can be thought as a natural transformation from a given SHIB to the pre-shift. This is quite clear because you see the compatibility condition is amounts precisely to naturality. And so you see the SHIB condition by the Yonida lemma can be formulated by saying that every natural transformation defined on a covering SHIB admits a unique extension as in this diagram. Okay, so finally we can define what Grottenbichtopos is. So Grottenbichtopos is any category which is equivalent to the category of SHIBs on a small site or more generally a small generated site. Okay, so we have got to the central notion of this course. Examples of toposes. Well, here are just three classes of examples. Of course, there are infinitely many examples, but I selected these three just to show you very quickly how general toposes are because you see the first example deals with categories. So whenever you have a category, you have an associated topos, the topos of the SHIBs on that category, which you obtain by keeping the category with the trivial topology. On the other hand, if instead of a category you start with a topological space, you have also a topos associated with that, the topos of SHIBs on the topological space. Also, you can decide to start from a group and to consider the category of actions of the group on discrete sets. So you can also take the group to be topological if you want, and in which case you take the continuous actions of the group on discrete sets. And you can show that in this way you indeed get a group of topos because you can present these topos as the topos of SHIBs on a particular site. So the site whose underlying categories that of non-empty transit actions on the topology is the atomic topology which we introduced here. So you see already these three basic examples show you that topos extend categories, topological spaces and groups. So given the fact that all these concepts play a central role in mathematics, nowadays you can understand why topos have a very big potential to have an impact in essentially across all mathematics. Because they simultaneously generalize all of this. And there is much more to that because of course as we shall see topos can be attached also to other kinds of entities. Loran has talked about how to associate the topos with theories, etc. And there are still many other approaches to the construction of topos that one can introduce.


Hello and welcome to another episode of frightfully forgotten horror movies, but before we get started. What are we drinking today? We're drinking slaughtered lamb English bitter awesome, and damn it is bitter At least to mellow out a little bit drink too many of these y'all turn into a werewolf Crunching up and everything Today we're gonna do something a little different and talk about something that is Current yeah, like brand new and that is our thoughts on all these Horror movies and horror shorts and trailers Jumping on the whole Mickey Mouse steamboat Willie bandwagon Yeah, the copyright for the steamboat Willie version of Mickey Mouse came up so Now anybody can do anything with that character and the first thing that happened Was like right away a trailer dropped for Mickey's Mouse trap a slasher styled horror movie And what's funny is it appears that? They kind of knew that this copyright was gonna come up Yeah, and they already started making it with the trailer ready Right in time for when copyright expired on it. Mm-hmm. So it's kind of smart planning It is but I'm kind of curious like I'm surprised that you could do that because like that means that they actually infringed It like maybe not nothing was officially released.  You knew it You already had the movie done And there was also another trailer that popped up for steamboat Willie the return of steamboat Willie Which is looks to be a completely Computer-animated type thing. It's very vague. Mm-hmm. It just kind of is like this camera going through these cows And then it kind of goes underwater and there's like a boat And they're like the boat's abandoned looks like people have been maybe murdered on it Yeah, and then the camera turns around and it's like this mouse kind of runs towards the camera and that's it Don't you hear the whistling too? Yeah? Yeah? Yeah So what do we think about well first of all Mickey's mouse trap we watch the trailer Yeah, well your thoughts on this whole thing jumping on this bandwagon and the movie in general I think the jumping on the bandwagon thing is fine. Like there's that Winnie the Pooh movie blood and honey Yeah, yeah, so stuff like that right so these characters as they're coming up People are taking advantage of it. I think that's perfectly fine. It's how it's done, right, right?  Yeah, yeah But he's always been stuck in the children Animation genre mm-hmm now we can expand him wherever we want But yeah again, like you said, it's how you go about it in the case of Mickey's mouse trap It's just the mask there. It's not Mickey Mouse.  It just looks it looks bad. Yeah, like it looks kind of shitty It reminds me a lot of that five nights at Freddy's well the same setting. Yeah, sort of deal, right?  Yeah light gloves He was like if you could be like a kind of muscular or something with those like no shirt on with the suspenders and the shorts like Like sweating and like a bit more, you know Medicine looking then just some fat bum in a jersey. Yeah He went for a jog and then decided to put the mask on or whatever.  That's lame it'd be cool if like Right in the opening scene of of the movie not this one But a movie that would be made if you see the guy like you don't see what he looks like or anything But you see this guy putting suiting up. Yeah. Yeah, the stuff on right and he's maybe whistling that steamboat Willy it could be anybody right the way they make it more of a murder mystery kind of heart back to the 80s sort of that Be kind of neat. Yeah, because like the slasher genre isn't dead You know that Thanksgiving movie just came out even though it's full of like the nostalgia type 90s slasher cliches It was a good slasher right and it you know, so you can still make a good slasher by kind of Falling back on those cliches, but it's again It's how you go about it.   So let's see what else these other people come up with For the return of steamboat Willy like again And this appears to be like a completely CGI computer animated movie Mm-hmm, you know we're not big CGI people, but I think that's more when it comes to like in a Movie where there could be practical effects and they use CG instead Yeah, in this case, you know a whole movie which is computer animated could be interesting Yeah, because you can do whatever the fuck you want. You're not being Limited by anything. Mm-hmm, and it's it's neat because Usually like you were saying like in a normal movie a real movie with real people and then you see CGI It looks fake and it takes you out of the movie this whole thing is CGI and it's already based on a cartoon which is Fake which you know so it sort of follows that same line Yeah, it doesn't there's nothing to take you out of that movie, right?  It's a legit mouse You know it's a killer mouse in this world of weird creatures, you know, which could be super interesting Yeah, it also could be the first full-length Animated horror movie. I can't think of another one.   Yeah, like a month later Which is crazy.  It's it's crazy how quickly people jump on that Yeah, well, it's it's kind of interesting too to see how People can switch the narrative too, right because it's like steamboat Willie or Mickey Mouse is so wholesome He's such a wholesome nice family character just to flip that and to make them evil Yeah, it's funny. That's the first thing people did Yeah, you know, that's the first thing people did was take this beloved nice character and turn them into a murderer Yeah, let's not make a comedy out of this.  Yeah, let us know what you guys think Mm-hmm and until next time keep drinking


I'm O'Brien welcome folks this is Tom O'Brien of TFNN we got five days a week we go seven hours a day we go 24 hours a day in the internet at TFNN.com always remember folks what have you think about you bring about whatever you focus on the road so everyone's having a great day safe day make it a great night and a great week folks you are what you believe you are humans are powerful magicians you have the power to make yourself what you are right now but it's not your reasoning mind that controls your power it's what you believe. Let's take a look at it out here we have the Dow industries down 19 Nasdaq off three S&P's up seven and a half gold gold contract up twenty dollars twenty eight dollars and fifty cents at twenty one twenty four ounce you get silver up seventy three cents twenty one dollars ten cents an ounce light sweet crew down a buck twelve seventy eight dollars eighty five cents a barrel notes and bonds ten year note down nine ticks trading one ten twenty four the thirty year down fourteen ticks at one nineteen nineteen and King dollar. King dollars down forty one ticks trading one oh three eight twenty Euro at one oh eight again at one fifty British pound at one twenty six to one U.S. dollar our phone number is eight seven seven nine two seven six six four eight give us a call folks one notes going on in your world and the world of the S&P's let's take a look at them what do you have well bottom line you get a small ABC up in both of them my take is that you know we're going to get a pullback yet because you do have a huge contraction of volume but guess what I think the ABC is going to finish first so right now on the spy word five thirteen and it's check this out yeah it's five twenty six so you get a long way to go here man we'll see how it shakes out but we took that B point out we took it out with volume on Friday so and it's a it's a decent ABC on the way up so five twenty six or at five thirteen we go to the NDX 100 the three cues take a look at the three cues same another ABC up right now they're flat you're at four forty five and this is four fifty four you know you can see Friday Friday was the big day had wide price spread plus you had the accelerated volume takes it out man takes it out with volume so we have an ABC up up market anyway so I suspect window dressing really is going to be closing up maybe tomorrow today tomorrow so the next catalyst here is going to end up being Powell in front of Congress let me see I think that starts Wednesday right I have it this one yeah well I get I get oh we get the beige book two on the sixth yeah Powell in front of Congress going to do it though now check this out we know that the gold contract's been on the run but this is pretty cool look at this okay so this is a GLD I just brought the GLD back twenty years the GLD is breaking all highs okay we've been we are the consolidation that we were in was a three-year consolidation that's four-year consolidation okay now we broke that now this is what's important to understand though right watch this I'll leave that one up just so you can understand how this works because we're actually higher than that because the GLD has a four tenths of one percent fee structure yeah it's four it's four tenths of one percent okay so in four years okay what do we got one point six percent you see what I'm saying so we're really breaking this and we get the juice behind it so bottom line you get to new highs guess what new highs higher higher you can go I'm going to take the generic contract and the reason I'm taking the generic contract right now is that I want to see what we're going up against I'll put this first on a 10 year I might have to expand it to bigger than that yeah I do but this is pretty cool look at this see you broke it now this is what gets interesting yeah do you remember well of course you remember four months ago the bottom line we broke it and failed now we're not at that price point yet but the bottom line we'll see whether you know we can do it but you can see how this is shaking out so put this back 20 yeah same deal so what what happens folks is that you know when you get to new highs you don't have any resistance I mean that's that's that's the bottom line because no one's bought yet it's a big number man it's totally a big number let's get into the silver contract take a look at the silver contract silver contract today really got some juice friday friday wasn't bad but today is really get some juice 81,000 contracts that's big contract volume for the silver yeah you can see it's the expansion of volume from friday to friday we did 73,000 so silver's on its way up to this $28 level right now you're 2412 we go to the note and bond market we take a look at the note and bond market and we have out here with the tenure tenure had a decent sign of strength on friday you're pulling back today with light volume we did 2.6 or 2.7 million contracts on friday see we got and we were rolling also yeah 2.7 million contracts friday you roll them back with 1.1 that's a market that wants higher price and then we go to the dollar we take a look at this dollar this dollar is hanging tough man it's pretty wild it as I say it's hanging tough it is okay but it looks to me like this still wants to get in the lower range and we've been dealing with this now for almost two full weeks it got in the lower range once rejected it right away and then three other days it got to the top of the lower range and then rejected it now the cool thing about that folks is that if it does break into the lower range you are going to see volume explode because when that's the building cause of it the longer that you build cause the more destruction that it can down on the downside the longer you build cause on the upside the higher it can go down industries right now down 13 as X up 12 recipes are up 11 stay right there folks come back when I'm at Mr. Steve Rhodes


So, I feel like most of my audience is familiar with Jesse Lee Peterson. If not, then he is a far-right pro-Trump radio host, and he is known for saying a lot of outright extreme and outrageous things. And one of the main things that he focuses on is rebuilding manhood. He wants men to be men, and women to be women. And if you are a man, you're supposed to be heterosexual and act masculine, because if you don't, then you would be, as he put it, Beta male! Beta! Yes, a beta male. So, he's produced a number of videos explaining the evils of homosexuality, and how nobody is born gay, and that if you want to stop being homosexual, I mean, basically, it's as easy as flipping a switch, as that thumbnail implies. So, let's listen to his advice on how to stop being homosexual, specifically the video titled, How to Stop Being Homosexual, and hear what he has to say. The spirit of homosexuality is of their father the devil. It's not them the person, it's the spirit that made a home in them, and it came from them overreacting to some sort of a situation in life, whether it's from someone messing with them when they were kids, or overreacting to an angry mother, because you become like what you hate. So, it's an evil spirit, and they can overcome it, if they don't accept it as a right. They have to see that, yes, they're stuck into it, but it's something wrong with it, and in that, they're able to overcome it. God will remove that spirit away from them. Okay, so in short, you can overcome it. I think you know where I'm going with this. So, on his Twitter page, as you can see, you know, he has all of his likes here, and he liked a tweet from Donald Trump, where Trump says, make America great again. And then if you scroll down a little bit more, you see that he liked a tweet from your daddy-only fans, XXLboy, where there is a video of part two of a collaboration with Sean Boy and Antony D.  But as you can see, there is one man eating out another man's booty hole.  He liked this on his public account for his show. Oh my God, this is like, this is hilarious. This is hilarious. And he left that liked tweet up for hours. It was there for hours, and immediately after he realized that he liked that tweet, he set his account to private, probably because he's going to scroll through and make sure that he didn't like any other gay porn tweets because he maybe thought that he was using his throwaway account, not logged into his public account. And I'm sure he's got to make sure everything is okay. So I don't know if he's even going to speak to this. But if he does speak to this, then I would imagine that he is going to say he was either hacked or one of his interns or staffers did that, and they've been fired because they're homosexuals and that's evil. But we all know what this is about. And I think that lands from the serfs put it best, how it started versus how it's going. I mean, there's only one word to explain all of this. It is amazing. Amazing! Now, obviously, it is not a bad thing if Jesse Lee Peterson is gay, because, of course, it's perfectly natural contrary to what he would have you believe. So, you know, there's this question, is it is it ethical for us to point this out and and shame him and make fun of him? I would argue yes, because look, and there's disagreement about this within the LGBTQ plus community.   Now, I'm sure that he's struggling internally. It's probably a difficult battle with him. But the problem is that the things that he's saying publicly, we have to make sure that he is delegitimized. We have to make sure that people know he's a fraud and not to take him seriously because the things that he says can be deeply damaging.  You should suppress it. Like that can lead to really horrible things. People end up hating themselves. They, you know, end up doing self harm. We don't want that to happen.  We have to put that aside because we have to make this utilitarian decision. This is for the greater good. People need to know that what he's saying is bogus.   It's not like we are going out of his out of our way to out him. He outed himself and it may have been accidental, but nonetheless, he's still outed himself. This is his own fault, right? And look, speaking from a position of a gay dude who was outed, I came out to about like three to 5% of all of the people who I wanted to. And then I was outed to everyone. I mean, we're talking parents of childhood friends, old people that knew me from the church that my family used to go to when I was a kid. Like everyone knew, including people, siblings who were hostile and could potentially physically harm me. Like everyone knew I was outed to everyone. So speaking from a position of experience, I don't feel bad about people like this getting outed. I don't think it is ethically a bad thing if people like Jesse Lee Peterson out themselves.  Because again, the damage that he can cause by spreading anti-gay propaganda, talking about how bad it is and how you should suppress those feelings or not like yourself, that can lead to someone hating themselves. So the best thing that Jesse Lee Peterson should do or can do is to just own up to it. Just admit, look, yeah, I'm gay. I'm struggling with it. This has been something that I've always dealt with. And I've kind of taught myself to suppress it, but I'm going to accept it now. I'm going to embrace it now because this is who I am.  No, I can't change it because I probably would have being a Trump supporter who is vehemently anti-gay, at least publicly. So I mean the whole story like there really is some poetic justice in this because it's always the biggest homophobes that end up inadvertently outing themselves. I mean, who was it?  I mean, they always end up outing themselves. It's an inevitability at this point because you can only hold up that facade for so long until you end up making a mistake and you end up exposing yourself. And I feel like hear me now, quote me later. The next one is definitely going to be Steven Crowder because he focuses so much on LGBTQ issues.   Like you're inadvertently revealing to us that really you're projecting. Like you're the one who's struggling.  Like you're saying more about yourself than you are about the gay people. So he's going to be the next one, like, I just feel it. Maybe it'll be like decades down the road. But at some point, Steven Crowder is going to expose himself, not just him, like other homophobes, because again, like there's no reason for you to care this much about an issue, to focus so much on homosexuality or transgender issues, unless you yourself have some skeletons in your closet that you're trying to suppress. I mean, another example, Alex Jones, a huge transphobe. And when he's on a live stream, we see that he has transgender pornography on his phone in one of his browsers. So they always end up doing this. And sooner or later, they're going to get busted.  I think this is necessary so people know that we shouldn't take individuals like this who preach hate seriously. They are frauds and hypocrites almost always. And anything that they say is bullshit, because if you could suppress your sexuality to the point where you become straight, like they would be doing it, right? We wouldn't find Jesse Lee Peterson liking porn where two dudes are eating each other's asses out. Like we wouldn't see that. So I find it funny. And this is what you get. You be anti-gay, then sometimes you just end up exposing yourself in the most hilarious way possible publicly. And it's embarrassing, like I kind of feel bad for him because there's no doubt that he's struggling. But we've got to point this out to people because what he says about homosexuality is deeply troubling and it could really lead people to harm themselves.


Welcome to ToffeeTV, it is the player ratings for Neil Arsenal Neil, a game that I can barely remember and that wasn't just because of the Christmas party after the game and utterly forgettable Premier League game between two teams with new managers in the stands and it showed why really. I think forever and I think obviously after the four games that Duncan Veig's in the talk this was sort of the first game where I think there was just a weird mix wasn't it? Players coming into the team, some players tired, players unfit and it was just a weird mix where really the team on the pitch is probably not really what Duncan would have wanted and probably for the first time in well not not for the first time because obviously every game he's had we struggled but I think it was just a it was just a bad mix and that led to a game where really you could see we had no legs and you know once a woby had gone off having that those extra legs in my field we became such a slow team and the option he had on the bench really didn't help with that either so we've become an incredibly slow team very very quickly and I think that completely hampered us but the opposition we were playing probably the worst Arsenal team I've I can ever remember playing against and and the mad thing is they got two world-class strikers or at least one world-class striker I think and Abamian and Lacazette who's pretty good as well so yeah Jordan Pickford I've given him a six and he's kicking he made one really good save from Abamian I think it was in the second half but some of his decision making that's letting him down he's kicking was poor I thought he's coming he's staying on his line all the time now it's something I've noticed keeping to his game and I don't know whether that's confidence and the people playing in front of him or his own self-confidence but he I don't know what it is but he needs something and whether the change of the goalkeeper coach going forward or whatever I think there needs to be something just to get more out of him more trust that man as I said that might be the centre back partnership in front of him changing all the time but I honestly feel like he needs to start having a higher line because the amount of times we seem to get or he seems to be caught in that space and especially when it's from a place of longboard you've got Michael Keane chasing the ball and it does leave us very susceptible and honestly don't know why he started doing that so I've given him a six. Gilles Sadibi I've given him a seven I thought it was a good return not a perfect return but a good return and defend them wise brains on the one-on-ones going forward not as good as this week but you know after missing you know a couple of games through illness that's always going to be the case and yeah so seven I've given on the other side Luka Dean I've given him a six I mean he got through the 90 minutes so that's a bonus wasn't particularly great going forward the set pieces were absolutely dreadful absolutely dreadful football has come to evidence suddenly laying that they can't beat the first man on set pieces defensively was okay so I've given him a six Mason Holgate I've given him a seven and a half defensively thought he was he did pretty much okay and then but when he went into midfield I thought he was brilliant I thought he gave us the energy and it's I honestly think he could craft out a position there I genuinely do I think he's got a lot about him that could do well in that position I think he likes to stay in the game I think he likes to be part of the game I actually think he gets sometimes maybe gets bored playing centre back and it was it's just that I think that's probably one of the issues you before the game I mean the energy you provide in central field I really think that could have helped us win the game but then you know Michael Keane comes in at centre back and let's be honest recently he's been very very poor and confidence is this season has been absolutely on the floor and he's second and we'll get into his performance in a minute but might you know Mason Holgate thought he was excellent so I've given him a seven and a half and it'd be very very interesting to see if we can bring a centre back in to play alongside yeti Mina where the Mason Holgate could go midfield because I really think at the moment he could be the future in that position for us I really do I'd have him ahead of a lot of other well most other players that's let's be let's say but we need another centre back yeti Mina I've given him a seven pretty pretty straightforward game for him I thought yeah defended pretty well and a few little moments I mean a few maybe one or two maybe little moments but I honestly don't think he's fit and he went down and he went down and I think that that shows you where we are at the moment it's all hands to the pump so I don't think he's fit and I think he's playing through the pain body so fair play to him but pretty much you know as I said a couple of little moments but apart from that I thought he was pretty consistent so I've given him a seven Tom Davis I've given him a five he was stuck out there on the right hand side and strange one for me that him playing there really really was a strange one but with having Delph back and having Sigarton back maybe Duncan thought that would said to be playing behind him and no one else really to play there other than maybe Bernard maybe Duncan had looked at it tactically but I don't know they had they didn't they didn't have their first or second choice left back and I thought that's somewhere we could have really hurt them and I think having Tom there just and I can understand what he was thinking maybe coming in off the right but Tom just didn't have the legs for it he really didn't he looked like he was running in treacle again so not not not totally down to him that he's got that score but you know he's running looks to me like he's running on a little bit looks on on running on empty but so I've given him a five and give he said this and I've given three of those he was terrible absolute not even terrible just like wait where are you where are you playing or are you not playing he was allowed to this another game where he's allowed to disappear and again listen for he might be fit either he might be fit you know he might have been rushed back for the sake of just having bodies but to me just he looked lost and he looked he looked somewhere to know where in this game and I think that's what that's the worst thing that can happen to Guilfysig it's in his games where he can just go missing and you can't quite pinpoint what he's supposed to be doing and I don't know whether Duncan had maybe looked at you play like a diamond or something I don't know but it just never quite worked for Guilfysig and he just went missing again so I've given him a three Femian Delph I've given a seven I thought I thought for his comeback game I thought he played pretty well loves that arching ball doesn't he back get himself an angle and arch as your back loves that he absolutely loves that I think there's a player in there if we can keep on and obviously there is a player in there we can keep him fit but they're keeping them fit part of it that's the worry and I think he is someone that could just play in front of the back four and actually with him in midfield with someone like Mason Hallgate I think you've got a little bit of everything in there if we depending on how we set up going forward the fact that he hasn't played for what I don't know what's he not well I can't remember where his last game was but I think he did okay and obviously went off but I think he just went out of legs so I've given give him a seven Alex a woby obviously went off injured hadn't really done anything really but disappointing for him to go off with that injury and it's disappointing to pick up another injury with so many games in a short space of time and I've given him a four because he really didn't do anything so yeah hopefully it's not too serious but those type of injuries do take a couple of weeks and at this time year a couple of weeks is well four games five games nearly so hopefully he'll be he'll be back sooner rather than later with Charleston I've given him a six lots of work lots of endeavour very little end product and I could say the same for Dominic Avalune I've also given a six lots of hard work but no real end result no shots on target very very little attack on wise out of them just hard-working graft and you need a little bit more in games like this didn't really play to either of their strengths really we never really went long that much but we never went over the top for them to run on to and we never went sure for them either so very very set staying set as air circumstances and would you know with Charleston going out wide as well and it just it just it was just the whole game was just a strange strange setup really for me so neither of them benefited from anything for me. Towson came on and went off I've given him a three I thought he was garbage absolutely garbage I thought he was genuinely and honestly if he's kicking off about being subbed then honestly he was terrible I've no idea where he was actually playing he was terrible he was honestly so so bad and I think it was I don't know if it was a strange decision to bring him on I think maybe what Duncan had looked at was who in those forward areas could last the game or maybe not last the game and he probably thought you know what if I bring Mines Keenan and I have to take him off later on how's that going to look no doing it once to one player to a player is bad but doing it twice in a week that's that's where people tongues do start wagging and he asked there you know listen it didn't bother me doesn't bother me that he took him off at man United doesn't bother me that he took Towson off it really doesn't but if you do twice then you at my start effect and not only not only for the player and even though there's a new manager but how people how people look at you and he just might have thought that Towson was the right man to bring on I just shows you that the bench was just very very limited in terms of what you know that that kind of injury and I mean he could have brought Bernard on I appreciate that but obviously he obviously must think there's a issue with Bernard in terms of what he wants from Bernard whether it be fitness and injuries or just general stature I'm not sure so I've give Towson a three Michael Keen I've give a four and I actually thought this was the right sub I thought it was the right sub and I actually called it while I was sitting there told that was the right to make bringing Keenan and pushing Hallgate into my field I did think he gave us those extra legs we needed in there and it got us a bit more of the ball and so I thought it was the right sub but he was poor wasn't he I've given my four because I just thought he was poor again balls bouncing bouncing off and kicking it into touch never really seems to be in control never feels looks he never looks comfortable he doesn't look like you look at yet he may not and yet he may not always you always look at him thinking he knows what he what he wants to do what he's trying to do Keen you don't get that same impression you really don't so I've given my four and Moise Keen I've give a six again I've given six because it's probably like with Charleston and Dominic Carvin, Calvin Carvin living, work rate and endeavour no end product and that's it ultimately that's that's ultimately it so there you go these have been my play of ratings let me know if you agree or you disagree yeah it's it wasn't it was a it was a terrible game it was a terrible game of football and you get a lot of that at this time of year because squads are stretched and there's a lot of games there's a lot of injuries there's a lot of tiredness there's a lot of illness um hopefully boxing day it picks up against barely because we really do need three points on that one so let me know your thoughts in the comments you agree do you disagree with what I've said if you haven't subscribed please do click the bell for a later you know exactly when a video is coming out and if you want more great videos join us over on page and thanks for watching see you later


So I'm Rose Mulan for those of you who haven't met me before. There's quite a few young people in Melbourne, so I've really enjoyed the day. Thanks again to Leon and Tom. I'm sure Tom contributed lots to organising these, so I didn't really. And it's been lovely to see so many old friends and colleagues, many of whom I've spent time with before. And people have come to you far and wide, obviously, and that's a great tribute to Mike and a great delight for the day. I think we've had a very good combination of old talk, sorry, not old. Sorry if Mike's word and as well as some really cutting edge new things, including a soup and a lot of equations and some wonderful, wonderful, dung beetles and some stuff that I absolutely didn't understand. I was once made a physicist who enjoyed optics, but this is all completely beyond me. And then I became a psychologist, which is quite interesting. I think had Mike been carried with a brilliant job. I think he would have appreciated the new stuff and the old stuff. And I think he would have engaged with a lot of questions with everyone. So I'm probably going to get upset now, but I'm trying not to. One thing that's been commented on several times is that he was very minded in the science and basically curiosity driven and just looked at things in fresh eyes and had a lot of fun, fun thing where it's been used a lot. So I just going to read something that he wrote, which is in his autobiography, which I don't know what it was written for. I think it's probably was society, but it's at the end of his autobiography when he's talking about working with eyewitnesses and has become, you know, had quite a lot of interaction with theoretical psychologists who don't come out very well in this. So excuse that. I'm sorry for the bad first one. So he's talking about the eyewitnesses with the device. He says, so far we've used the system to study driving, playing ball games and sight reading music. Our ethological colleagues would like to take it to a disco to start studying many choices, but so far we have resisted this. The questions, interests revolve around the way we sample the world, given that our regional occupation is only about one degree across. Where do we need to look to get the information? We need to do things. Answers are emerging, but we get into trouble with some of our psychologist's points. But this is a curiosity driven essay. Where are the hypotheses? The trouble with hypothesis driven science is you never get to make a discovery. And that's the real buzz.


Hello, I'm Leslie. I'm originally from New York and I have been living in Shanghai for close to 20 years. I've been an EFL teacher here in Shanghai for about 10 years now, and due to increased competition and an increased focus on trainer qualifications, I decided to earn my TEFL certificate. Another trainer friend of mine got this certificate with ITTT and highly recommended this 120-hour online TEFL program, so I thought I'd give it a try. Even though, as I just mentioned, I already have several years of experience as an EFL trainer here in Shanghai, I still learn several useful and important concepts and ideas in this training program, such as the ESA elicitation study and activate lesson structure and detailed descriptions of the different verb tenses. I was really impressed with the quality of the material in this course and I highly recommend the program to anyone who is either a new to training or even experienced trainers like myself. Thanks, IEEE. Thanks so much for watching. We are ITTT, the leading provider for TEFL and TESOL training courses. If you like this video, please subscribe by clicking the button down here and click on any of the videos here on the left for more interesting teaching tips for getting certified to teach English abroad and online.


What is conflict? Definitions and assumptions. Here's where we're going. We're going to discuss disagreement, parties involved, perceived threat, needs, interests, or concerns. Conflict and conflict participants, dimensions of conflict, conflict is normal, creative problem-solving strategies, conflict styles competing, accommodating, avoiding, compromising, and collaborating, how we respond to conflict, the role of perceptions and conflicts, and don't avoid conflict. Disagreement. There is some level of difference in the positions of the two or more parties involved in the conflict. The true disagreement versus the perceived disagreement may be quite different. If we can understand the true areas of disagreement, this will help us to solve the right problems and manage the true needs of the parties. Parties involved. There are often disparities in our sense of who is involved in the conflict. Sometimes people are surprised to learn they are a party to the conflict, while other times we are shocked to learn we are not included in the disagreement. Perceived threat. People respond to the perceived threat rather than the true threat facing them. People's behaviors, feelings, and ongoing responses become modified by the evolving sense of the threat they confront. If we can work to understand the true threat, in other words, issues, and develop strategy or solutions that manage it, in other words, agreement, we are acting constructively to manage the conflict. Needs, interests, or concerns. There are always procedural needs and psychological needs to be addressed within the conflict, in addition to the substantive needs that are generally presented. The durability of the interest and concerns of the parties transcends the immediate presenting situation. Conflict. A conflict is more than a mere disagreement. It is a situation in which people perceive a threat. This could be physical, emotional, power, or status related, or something else. Or scarce resource that impacts their well-being. Conflict participants. Participants in conflict tend to respond on the basis of their perceptions of the situation rather than an objective view of it. People filter their perceptions and reactions through their values, cultures, beliefs, information, experience, gender, and other variables. Conflict responses are both filled with ideas and feelings that can be very strong and powerful guides to our sense of possible solutions. Dimensions of conflict. As in any problem, conflicts contain substantive, procedural, and psychological dimensions to be negotiated. In order to best understand the threat perceived by those engaged in a conflict, we need to consider all of these dimensions. Conflicts are normal experiences. They are predictable and expectable situations that naturally arise as we go about managing complex and stressful projects in which we are significantly invested. If we develop procedures for identifying conflicts likely to arise, as well as systems through which we can constructively manage conflicts, we may be able to discover new opportunities and to transform conflict into a productive learning experience. Albert Einstein once wisely said, imagination is more important than knowledge. Consider the creative problem-solving techniques you used last time you were faced with a difficult situation. Creative problem-solving strategies. We need to transform the situation from one in which it is my way or the highway into one in which we entertain new possibilities that have been otherwise elusive. Conflict is normal, anticipating conflicts likely to arise in a workplace. Consider your work environment. What are some key sources of conflict in our workplace? When do they tend to occur? How do people respond to these conflicts as they arise? When we solve problems, do we do so for the moment or do we put in place systems for addressing these types of concerns in the future? Are there seasonal peaks in our workload that tend to occur annually? Do we have channels for expressing normal problems and concerns in a predictable, reliable manner? Are there certain factors in the environment that make problems worse, especially at times of conflict? Conflict styles and their consequences. Competing. This is a style in which one's own needs are advocated over the needs of others. It relies on aggressive style of communication, low regard for future relationships, and the exercise of coercive power. Those using a competitive style tend to seek control over discussion. Competing tends to result in responses that increase the level of threat. Accommodating. Also known as smoothing is the opposite of competing. Persons using this style yield their needs to those of others, trying to be diplomatic. They tend to allow the needs of the group to overwhelm their own, which may not ever be stated as preserving the relationship is seen as most important. Avoiding. A common response, feelings get pent up, views go unexpressed, and the conflict festers until it becomes too big to ignore. Like a cancer that may well have been cured if treated early, the conflict grows and spreads until it kills the relationship. Because needs and concerns go unexpressed, people are often confused, wondering what went wrong in a relationship. Compromising. This is an approach to conflict in which people gain and give in a series of trade-offs. While satisfactory, compromise is generally not satisfying. We each remain shaped by our individual perceptions of our needs and don't necessarily understand the other side very well. We often retain lack of trust and avoid risk-taking involved in more collaborative behaviors. Collaborating, also known as win-win. The pulling of individual needs and goals toward a common goal. It requires a sort of communication and cooperation in order to achieve a better solution. It offers the chance for consensus, the integration of needs, and the potential to exceed the budget of possibilities that previously limited our views of the conflict. It brings new time, energy, and ideas to resolve the dispute meaningfully. How we respond to conflict. We have emotional responses, cognitive responses, and physical responses. The role of perceptions in conflict. Culture, race, and ethnicity. Gender and sexuality. Knowledge, general, and situational. Impressions of the messenger. And previous experiences. All shape our perceptions in conflict. Don't avoid conflict. Our society tends to reward alternative responses to conflict. Negotiation requires profound courage on the part of all parties. It takes courage to honestly articulate your needs. And it takes courage to sit down and listen to your adversaries. So to review. Considering these diverse perspectives and dimensions will make you better prepared to deal with personal and professional conflict scenarios. Thank you for watching.


The virus does not understand your emotions. This commitment you must take for yourself, for the sake of the nation and for the sake of humanity. There is no need to panic, what is needed is sense. All of us absolutely lockdown, lockdown. We would have in many ways contained it. This question is from Chetanya. Sadhguru, the cases in India are increasing and with the lockdown opening, it might pick up the pace. I would like to know if I get corona, how do I deal with it emotionally? Oh, the virus does not understand your emotions. So, don't deal with it emotionally. First thing, don't get it, but it may happen. Once you're out on the street, it may happen. So please don't deal with it emotionally. No, no, no, that's not what I'm saying. How should I handle my emotion? This is what I was saying all this time. Your emotion, whatever is the sweetest point you hit in your life, the high point, you must keep it there, don't bring it down. So, if virus gets you, you must deal with it sensibly. The most sensible thing is, if you get it, nobody else should get it from you. This commitment you must take. For yourself, for the sake of the nation and for the sake of humanity, you must take this. In case you get it, it should not hop to one more person. You must stop here.  This responsibility we must take. So there is no emotional stuff to it. You deal with it, it'll be little hard to breathe. People say it's quite terrible because it feels like you're drowning, but then now it is showing other symptoms also, various other kinds. So, we don't know what kind of sweet emotions virus has for you or not. We do not know. We do not know if he has any emotions, he's just doing his business. Simply, he's just going about his business of survival. So you go about your business of survival, that's it. Just like the virus, all right? The virus is trying to survive upon you. You survive upon this planet, don't go away. No emotion needed for this. Yes, what is needed, you hand yourself over to a doctor, whatever they have to do, they will do, you take instructions and just do that. Or before the lockdown is over, fully inform yourself what you must do, what you must not do, absolutely, 100% clear. But if you keep yourself, we have taught you Sema Kriya, but if you're already doing Shakti Chalana Kriya, your immune system is good in terms of food. Maybe we can put it out with little more, you know, little rejig that a little bit. How to get the immune system up, use these fifteen days to really get the immune system into a strong place. In spite of that, we can get it. But now they're saying 80% of the people who get it don't show any symptoms. They just go through it, nothing happens, it's just that they're giving it to other people, that's all. So if you're in the vulnerable age group, you must be super careful, for you lockdown is not over. If you're over sixty, sixty-five years of age, if you have any kind of immune deficiency disorders, for you lockdown is not going to get over on 17th of May. It's not going to get over. For next six months, eight months, nine months, we don't know, maybe twelve months. Those who are young and healthy, yes, you have to get back to activity and do things. Others have to maintain. Those who are in the vulnerable groups, they have to maintain lockdown. And everybody, you need to understand, even if the lockdown gets over, you must understand this is not because we got on top of the virus. Simply because we can't afford to stay home, both in terms of economy, in terms of various other social things, we cannot continue to stay back. So we're going out. Virus is not saying, yes, I'm done, I'm defeated. There's no such thing. Nothing has been done about the virus. Forget about vaccination. We still don't have a treatment protocol which is clear and everybody says, this is the way to do it. We've not arrived at that. Simply we're trying many things that we normally do when we get something else and managing those who are strong are coming out of it. Those who are not are going down. So these numbers may definitely increase. There is no need to panic. What is needed is sense. Keep sufficient distance. It doesn't matter your friend or foe. If you're friend, you don't have to hug them. Enemy don't have to slap them. A slap could be very dangerous. So don't do anything, all right? Do minimal contact with people with distancing. This must be maintained for next few months. Doesn't matter. A government says you can go out. They are saying that out of the compulsions of national economy and various other things that need to happen. You can't keep people locked down forever. But it's phenomenal that more than six weeks we have successfully locked down largely very successful, fantastic. Today the Indian forces, all the three tri-forces of army, navy and air force have been around the country honoring all the COVID warriors as they are called today. The doctors, the medical professionals, the police and others who have played a significant role in managing it this way. With the kind of diversity of people that we have, diversity of languages, because let's say in United States if the president or somebody speaks in English language, everybody understands. This is not like that. You have to speak in twenty-eight languages, all right? And there are many people who don't understand any language. In spite of the complexity of this social structure, which no other nation has this kind of complex structure in terms of language, race, religion, everything. In spite of that, it has been, you know, managed in a fantastic way with minimal amount of force. And people have cooperated in a significant way. We really have to congratulate the Indian people. But virus can pick up even if two people have in the country slowly, it will again grow. But now there is a significant way of identifying and isolating people who are infected.  Nobody can test 1.4 billion people. You can't test all of them for this virus. So the best thing is we just maintain the distance even after the lockdown is over, till we feel the wave is completely gone, till then we have to maintain this discipline. This is the way forward for all of us.


AB ducting nystagmus are a variety of dissociated nystagmus, usually seen as part of an inter-nuclear ophthalmoplegia. This patient has a subtle inter-nuclear ophthalmoplegia. What's most evident is a dissociated nystagmus.  Smaller oscillations can be seen in the left eye. An inter-nuclear ophthalmoplegia localizes to an intrinsic lesion of the brainstem.