Space-like categories / 1st part of Iekke Moerdijk: Classifying toposes, descent, groupoid representations

0:00 You told me that there was somebody in the next room who had the answer to non-morph. I'd give you two to one odds. But, uh, I get at it. So, but, uh, the point then becomes, well, so why is Fermat's Last Theorem hard? Well, it's not that it requires strong axioms. I mean, if this is true, and for all we know it might be true, it's not that you need big axioms, axioms about the infinitums. I mean, in principle, you may well be able to prove them. So then the question is, well, you know, in particular, by the way, it is a theorem of logic. I mean, it has been shown that the prime number theorem, the head of mathematics has the prime number theorem, trying to make sense of the conceptual resources given to you by analytic number logic is coming back to, so those are two, the two philosophical puzzles, so now, so again, in both cases you want to say that somehow you need, different proofs give you different understandings and somehow to prove things. Again, you need conceptual resources that give you an appropriate understanding. So now let me try to give you some suggestions as to how one can start talking about the topic. So now I'm going to go back to the first paper and pick up roughly... Yeah, so there, remember I had talked about what it is that you get from different... So we do have some fairly good intuitions as to some of the reasons that one may appreciate a particular... For example, we often value a proof when it exhibits methods that are powerful and informative. That is, we value methods that are generally and uniformly applicable, methods that make it easy to follow a complex chain, or provide useful information beyond the truth of the whole thesis, however, this claims.

2:30 I've just said that we value proof when it exhibits powerful methods. I have not said what it means for a proof to, quote, exhibit, quote, let alone what it means for a method to be general and uniformly applicable. Nor have I said anything about how methods can render a proof intelligible, or the types of information they can convey. The first projection is that it is safe. A second objection is that the claim is rather toothless. A few would deny that the attributes are not saying, look, I'm going to prove to the world that, you know, that mathematical methods, powerful methods are useful. I mean, there's not much of a claim. My goal here is sort of one of vagueness. To that end, I will discuss a model of proof that is currently used in the field of methods. So I focus on the problem of methods of understanding proofs. And the second objection noted above will instead become an asset. Which is to say, insofar as the terms can be made sense of, the result will be a philosophical claim that stands a good chance of being correct. So if you can, again, so I'm not going to tell you the story that's developed. As I mentioned, it was very much a...

5:00 That is now recording. And to switch it off, you press that. I suggest completely new class. In any case, the examples were very useful for refuting various naive conjectures. And it was surprising, actually, how often a very simple example serves to refute various naive conjectures. And the example which Chandra and I have found most useful is none of the reflexive graphs. So, the way we usually think of this is in terms of the three-element monoid, delta-1, and the three elements are the identity element, d0 and d1, and the multiplication rule is d-i-d-j-d-u-t.

7:30 The other, the constant element in the form of C, in section Cx, Cx, C, and so on and so forth, is factored through the terminal object, which we have on the various ways in which the constants occur. So, this monoid is very, very special in that it has the property that every element is either an identity or else a constant. The rule for identity is Ux is equal to x for all of them. And so, actually, it turns out that there's a whole class of examples of this property that, in some suitable generalized sense, everything is either an identity or a constant or a relatively constant or something that means that we have a three-element identity that says x, y, x, x, y, all x and y. The homologous and general categories that satisfy an identity of that sort have many properties in common with this one example. On the other hand, they express, in some sense, multidimensional information, whereas, in an intuitive sense, graphs are one-dimensional shapes.

10:00 For example, sets, Brodendieck talks about, Brodendieck never talks about topos, he talks about eutopos. All we did was to say, well, eut itself can be an eutopos, just like algebras can be z-algebra. So you take the contravariant representations of this monoid into u, which we write as right, and then you have the, of course, the unative embeddings, which says that This is a standard interpretation and an example, or if you like, two standard examples. By delta-1-bar, the result of splitting the input, Dryad-Karuthi-Ando, as it is sometimes known, is one of the most trivial-looking, yet non-trivial, operations. In any case, the United Amendment automatically extends to that, and we could consider this topos equally as the splitting of inputs. So that gives us two canonical objects, one of which is the terminal object, because in any finite monoid like this, if you take the product of all the elements and then split that, then you get the terminal object. So the two objects are, well, one of them is the terminal object, that's the one that came from the splitting, and the one that is already here, I might call it, for example, I for interval. Witten has two points. The C-max from 1 corresponds to E0 and E1 in the sense that by composing the unique map here, I get the input. These are the splittings of those two values.

12:30 The category that I obtained by splitting the inputs. In other words, there are always two ways of thinking of it, purely formally. It's the category whose objects are the impotent endo-maps of the previous category, natural motion and morphism, or maybe not so natural and representable as the results too average in the matter. Or you can take out of this inside depreciate category, you just take all retracts of representable. Retracts of representable functions are morally almost like representable functions. Of course, again, if you look at conjugate examples, this almost turns into... Batch theories of mathematics, because if you go from open sets when you're playing in space to manifolds, that's an example of that operation. Every single manifold is a retract of an open shell or whatever. There's a lot of content. We're going from free objects to projective objects. Okay, so one of the nice things about reflecting graphs is that you can picture them as graphs. In other words, I can think of this object, I, as an arrow. What this arrow really is, is the identity map on I. These two points are these two points. So, a point, perhaps, I can think of as a dot. And so the two morphisms are simply placing the dot on one or the other end of the arrow. The thing about this very simple picture is that it's a generic picture. At least in the case that the view doesn't have too much structure, in general it's more to the full picture than the abstract set.

15:00 So if you take any object X, put it with the right action, then, of course, there are the equivariant morphisms, the natural maps. And these one can think of as figures. Figures of this shape in this object. By figure, I just mean math, except that usually figures have a special... Special domains, in this case, one or two special domains, arbitrary objects. Figures of a given shape and a given object. And then by composing such a figure with these two points, we get two points of x. So actually, since every free sheet is a co-limit of representables, All these figures and their interrelations merely mean composing. So, a particular arrow inside X might happen to be, might happen to have a few endpoints the same, in which case you'd write it as a loop, but any, essentially any picture you can think of that involves dots and arrows that connect to dots is really any such object as Pictures of that sort can be represented in a certain picture. It's not exactly unique because you can squish it around in all sorts of ways without changing its isomorphism type. But essentially, this is what's called a reflexive graph or network. They call them networks, I've been told. But why are they called reflexive?

17:30 The fact that you have the terminal object in there means that at every dot there is a canonical loop. There may be lots of dots, lots of loops. It's like a category, but without composition. There's the identity part of the trivial loop, and you should be able to draw the trivial loop. And we gather, in the sense that there are three elements, but two of them are degenerates. Degeneracy is the word that's used in common material topology, again, to denote these retractions. Incidents are relations which are retractions of other reasons. Okay, so, now, there are many, many things to be said about this. Points that I'd like to emphasize, and which are useful, is that Of course, there is a relation between the category of reflection graphs and view itself. For example, there is the inclusion, which corresponds to the category of one bar with the terminal category. And this means that any object could be thought of as a trivial graph of the more general situation. If you have two sites, or let's just say two small categories, and you look at the pre-shades on each, then you want to say, well, this function introduces a relationship, and if this were a full inclusion, if it were a full, that statement is overlooking a very profound dialectic, namely, see, in some sense, that you might think of,

20:00 The last conclusion that you might think of is not the one that's part of the geometric morphism. Left adjoint, F lower string, which preserves, for example, the genative embedding, they're compatible with the left adjoint, not with the... ...involved into the geometric morphism. First, F lower string is left adjoint to inverse integer, the substitution F upper star, where we have alpha for star, and phi for star. And then there is the athlor star. Normally the definition of geometric morphism is truly this pair of athlons. The athlor star is the deep left exact that requires a very strong region that preserves all inverse limits because it has an angioma. But there is no, even in case this was full inclusion, the inclusion, the community and identity allowed it. There are two. If this is a common retraction for the two, these are actually two completely different subcategories, which are isomorphic as categories, but in a very definite sense, as subcategories.

22:30 If we didn't have this distinction between objects and sub-objects, we'd really be lost. I might say that, in many cases, another name which is suggested in some contexts should refer to the word sheet. If you start with the B sheet on D, the word sheet for X could be expressed as the last X. As for the word D with C, this makes very good sense. Instance relations from C that are joined formally together and then identified in exactly the same way. It's a composition of modules. In the same sphere, we have lower stars on the same deck. And a palm, which is a palm, meaning a natural mass of respect to D, comes from C, soon to be interpreted into that. In this spirit, and I don't think this is standard terminology, but I propose to say that at this flat, the lower string is left exact.

25:00 This conflict between the edges of this flat here means that the tensor is left exact. Of course, if that lower string is left exact, then that would imply that there's another... If there's another geometric morphism, G, one in the opposite direction, maybe, okay, so G, G upper star, upper star, but then there's still an F lower star hanging around there, so what is that in relation to G when it comes to the upper sheet? One says that G itself, the morphism, always carries with it, which is essential, but it's more than essential. Essential means just that that lower sheet exists. Now, of course, this is one of these rotational yogurt with the shrieks, and you know for sure that the upper symbols means it goes backwards to the map of things, and lower means it goes forwards, stars means the things that you have all the time, and the shriek means the things that you don't.

27:30 Surprising. Or amazing. So that's why I talked about amazing. Basically, amazing has to do with local. Don't say more about that in the third lecture. Now, first, even though this indicates where f is exact, there is another more than g, but it may not be induced. It typically will not be induced. You might have a funder from D to C who was centered right at F. For example, if D had the terminal of it, then the terminal of it is right at the terminal of the funder there. And if that existed, then this material is really just coming back to that. In that case, the G wouldn't be induced. Well, if that is full and faithful, then both that floor street and that floor star, in fact, if one exists, then it's full and faithful, and the other one exists, then it's also full and faithful. So this is more or less just to, all right, and if that is full and faithful, then of course in particular, a composite, so that one gets a pair of two different interphones just by composing. So you get one interval, which is the left-axis is the cohomad, and the right-axis, and of course, they together, that's what I see in my head, in some contexts, these are called the skeletons, the death skeleton, and the death co-skeleton.

30:00 We have the common tensor, we have streets and stars, we have skeletons and co-skeletons, these are all names that are used in various contexts, more or less the same. And we have flat and local, which aren't the same, but they're agile. And again, caution that the, in some sense, the naively natural way to include this into this is to use the depth of the tree, and that has nothing to do with what's called the so-called direct image and sheath theory. I often thought that someone told me I was wrong about this, but I always believed that in some sense there was a mistake made The meaning of that lower star is that that lower star is like a universal form of science. A left-handed wave doesn't even exist in all cases. It's more like the existential equation, or it's just a direct image. That lower star is a direct image for students. That's a long time ago. And then someone tried to explain to me that actually that's not right because the original thing that one wanted to take the direct image of were... Modules of, for example, ideals of definition of the sub-states, and what is its direct image in this sense of its corresponding to the direct image in the geometric sense. That was never very convincing. It might justify that Serre or Hoover were... Coming back to the example, when we say that the original topography of the set was included in the structural graphs,

32:30 Well, we see when we're talking about something that's induced by a function connecting delta 1 to 1, or delta 1 bar to 1, if you like, because that does have a terminal object, but in any case, so we're actually going to get all four functions. There are two different, there is the trivial graph in the center, which seems to be, as I said, the left-hand one, the most obvious direction, where you just take the original set and you think of it as dots. And there are no non-trivial arrows. You just see a bunch of dots. Of course, they're exclusively the identity arrows on each, loops on each dot, otherwise known as an arrow. And that we call the discrete. Now, the opposite one is the, of course, the co-discrete. Co-discrete is the upper street. And, of course, in the middle, the... Unity and identity that unites those two alphabets is, of course, just the taking of the points, where the points, nodes, vertices, are just called points. They're matching lines. So that's a very simple example of unity and identity of adjoint alphabets, being in a two-category, three-dimensional adjoint mathematical system. And moreover, it's composites by identity, so we have a two-dimensional world. Now, but there's also the lower street, which is taking the components. Now, this is sometimes called pi zero, or just the...

35:00 Now, one of the important features of this particular example is that pi zero, by rights, must reserve direct limits after all that's given up to adjuncts. But, if we actually take the product of two complex graphs and then take the components of that, that always has to be synonymous with the product in view of the sets of components. But that's the equivalence in this case. On the other hand, pi is not preserved equally. That thing is not flat, right? That thing is not flat, but pi is not preserved in all five of them. Wax is an Australian design for industrial products and again it's helpful to talk about this example, as I said, is that we can picture the inside of the objects and we can picture products, for example, just by drawing little rectangles, at least for a particular occasion. So if we, for example, there's an obvious potion. The standard representable, which is just one non-trivial loop, can identify the two ends of the loop, L, and follow this I. So, L cross I, you can count how many dots does it have. How many dots does it have? Well, it has two, because L has one and I has two, and maps from one to the product, so it's still two. On the other hand, when we talk about the loops of the arrows, we have to count them all, so this has three arrows in general, and this has two, so this must have six arrows, but then two of which are...

37:30 What does that mean? Well, you can picture, here's a picture of L, here's a picture of I, and you draw a rectangle, so you only have to have two dots, which are... And then the pair consisting of the interesting arrows here is a kind of loop. I'm not really sure, because the picture is bulging out. And on the other hand, there are these loops in each. And then, of course, the pair consisting of this non-trivial arrow and this trivial one, like that. So by just drawing the picture, we get, let's see, one, two, three, four. Non-trivial arrows, the two trivial ones, that's six as we got them. So, at least on the simpler examples I'm going to... Oh, and by the way, the left adjoint, the fact that it's called components, suggests correctly that it's exactly the components that you see when you look at the graph. I mean, the graph is sufficiently... You can spread out so that you can actually see it, and you can also see the components in the same way as the first thing you would see if you approached it from a distance and back to that perception. So anyway, so what is I times I? Look at the product space. We can first make an abstract count, and then we can also figure out by composing the incidence relations what the picture should be of the incidence relations that are contained in it. But just by counting, we see, well, it has to have four dots, and each of these has actually three arrows, so three times three is nine, so it has to be nine arrows, but four are the amendments, so five non-trivial arrows.

40:00 But the crucial point about this picture is that it's an instance of this equation here. One time one is one, this is still connected. And the fact that the product is preserved by pi zero is a geometric fact, but in a way it has a little ingredient of bad infinity in it too, because you can represent pi zero as a sort of co-equalizer. In fact, it is a co-equalizer, if you like. The discrete inclusion is the diagonal. Limits and co-limits are just adjuvants to a diagonal functor, right? So this is actually the co-equalizer of the diagram that you see here in the graph, but the co-equalizer is only the generator for the equivalence relation, so there's a sequential process of building up the equivalence relation, and it's here that the, I mean, it's a very complicated graph, and it wasn't laid out clearly. You'd have to figure out when things are the same. And this might be a non-trivial process. It might take a computer thousands of years. But the point is that at least, so in some sense, given a pair of dots, for example, there are proofs, sequential proofs that these belong to the same component.

42:30 Steps. Well, this is connected to that, obviously connected to that, obviously connected to that, blah, blah, blah. So you have proofs, in some sense, so the proof-theoretic ideas you can kind of measure once with a possible explanation. So, but now if you have a product space, you could say, well, one, a proof that you have two pairs, two points in the product space, you want to see if they're equivalent to two other, to another one. You may have a proof that the first components are equivalent, but that proof takes n steps, that the second components are equivalent, but that takes m steps, so n is not equal to m. You need a proof, a single proof, but there's no way to get it, except by using the reflexivity, because you can insert trivial steps, the dots themselves, not only the arrows, but the dots themselves. So, given these two proofs of anything in them, you just fill in the right number of dots and they'll turn out to be the same length. So, a pair of proofs of the same length is really one proof without a pair. Now, the point is that this is definitely not true for an emphasis called Geo-reflection graphs without degeneracy. This has to do with the difference. In fact, these are often confused, in a way. You just take two pairs of arrows, but nothing coming back, and look at three sheets on that, then again you have a kind of generic arrow, but it's better to, it's better to, you know, it's a different style of picture, because you should draw a loop if you have it, because you don't always have it.

45:00 But more, more, more succinctly, the natural morphisms in this object having preserved the degeneracy, preserved the preferred loop. Whereas, even if you have an irrepressible graph, which happens to have moves, there's no reason why more business should reserve any future curriculum. If you see that the product preservation property is not reserved, all four functions exist. I'm sorry, pi zero, discrete, and twice exist. And so on and on and on and on and on and on and on and on and on and on and on and on and In this category, the dot is a sub-object. Let's call it U. U is a typical sub-object or terminal object, because the terminal object does have a loop. In this category, a map from one is not perhaps what you thought it was. It's not a dot, it's a loop. Or a chain basis or whatever. This equation is the reason why we thought there might be such objective numbers here.

47:30 Equations satisfied by objects and distributors. We've been working for six or seven years on a book called . It does go somewhat further than that. Now the other point that needs to be made about these examples is how does the truth value object look? The truth values can be identified with right ideals in the monoid, because right ideals is just the way they're named with sub-objects of the regular representation, and if you count those up, so how many sub-graphs are there of this arrow? Only, excuse me, there are five. This dot, that dot, two dots together, a whole arrow, or of course nothing, five. But again, how are these files put together? Again, you can calculate that by composing various ways, pullbacks along the incidence relations, because there's a general way to determine the structure of a sub-object, but it's helpful to have, again, a picture, because, you see, in any topological theory, we have a distinguished family of figures that generate. We always have a kind of picture. There is an internal picture necessarily of those figures. There is a difference. In the case of graphs, we have this internal picture. In some sense, one level more internal.

50:00 So this internal picture is a generic graph. This generic subgraph, A, is a truth value object. This map is A into true, so there's got to be a dot called true. Sometimes the miraculous sounding feature of omega, knowing what such a characteristic map takes to be true, determines what it takes to be everything else. And you can figure that out in the following way. Because if you take a typical element up there and then... The truth value, this should be the truth value in some sense. X belongs to A. So if X really does belong to A, then the value is true. But the typical figure is not a point, it's an arrow. And of course some arrows do belong to A, that's true. But some arrows, and some arrows are completely outside of A, have nothing to do with A, even though they are X. So of course, C of A must take them to another dot called false. Now, some arrows are actually neater, they enter A. The phi has to preserve the graph's structure, so phi applied to such an arrow must give a value which is an arrow from false to true, and also the opposite. Some arrows exit from A, and so the characteristic function of such an arrow must give a value which is an arrow from true to false.

52:30 Namely, an arrow which, while itself not innate, nonetheless both scarves and diminishes it. Something I call, sometimes, a for-ing. It's not very good, so I don't know if that makes sense to anybody. But in any case, we need another loop here to serve as the truth value for any arrow which is not in the sub-object. That completes the whole truth value object, so there are actually five elements. In the reflexive case, in the reflexive case we can draw it this way, not in any other way. In the irreflexive case, we have to draw it rather this way, to actually show these. And this is actually true, like true, but a little bit less than true. Again, in terms of, this is one picture, the only different one is the border structure of right ideals, which is the same as the border structure of, as a hiding column. So that has the empty at the bottom, that's false, and it has the true at the top. These are the two not seen separately. This is actually the union of the two tops in just below. Oh, by the way, the word irreflexive doesn't mean they're not reflexive. This is the strong deviation. Not, should be not? I don't know.

55:00 So now the other feature I want to underline here is that the truth value object is connected. The truth value object is connected. Now that's something completely unknown to classical logic. In any Boolean topos, the truth value object is one plus one. It's got two connected. And indeed in most topologies. Yet in both these examples, it is. Now, these two special properties, both having to do with the notion of components, both having to do really with the contrast between the definition of products, but not equalizers, and the connectedness of the truth-telling model, were judged to be plausible in the general definition of the purpose of split-out spaces as opposed to a generalized space. Then, as a topos of spaces, Leberkin, on the contrary, themselves, very, very special in all of his observations, he used on the parallel arrows, I slice it by the standard object, of course, I always get a topos that way, and since J is big to one, we call it covering, morphism, so it's a covering.

57:30 How about the original? How about the non-reflexive? But actually this one is equivalent to the sheaves on a space. So the non-reflexive graph is locally homomorphic to, is locally, locally the sheaves, is locally localic. What is that space? Again, you can figure it out by computing. But again, it's, I like to think of it that way. We can collapse that space to a three-pointed space, which is a very important space, not discrete, this is a continuous map, which is strictly positive, it becomes open, two or both open sets. So we have a space with three points and five open sets. Now, how do we understand sheaves on this? This is a very important space, by the way, people are always worried about this. Or the open space you consider on the whole set of bases. So instead of, in this case, five open sets is equivalent to, we have the whole space here included.

1:00:00 And the claim is that any, any constant variant is essentially achieved on five point space. Sorry, three point space, with the five open sets. So what does it mean? We have to give a set, which is the global section. Two other sets, which are the sections on these points, and then two restriction maps, restriction maps basically. So how do we get from that spatial picture to this somewhat slightly less spatial picture? We just coalesce these two sets, but we do not coalesce the two. The second graph can be spaced together with the information that these two sets are given. Having given, I should work with it. Bill, can you describe the space again? It's five points and three are open, is that it? Three points are open. Pardon me? No, no, three points. And two are open. And two are open. Two points are open. And there are more than five open sets. I mean, this is very light. It's sort of next to the... If nobody's going to ask, are you going to do that? Yeah. I think you're calling it a graphic. Right. I made a mistake for an entirely different reason than I was confronted with the same thing as I had to do with... I made a mistake of typing the opposite of, no, oh, typing into, um, x, y, x, z, y, type that in with the word semi-reverberate.

1:02:30 And I'm afraid the name could leave the damn thing overlaid, just left, right, you were banned. Yeah, in that case, no, they were originally discovered, uh, were again by Schutzenfresche. Some of these papers are reviewed by Garrett Kirchhoff. The original, the original, the original motivations were non-communicative logic. I slowly, of course, yes, that's regular advanced. Oh, for a long time they were called Schutzenberger. Actually, Schutzenberger was not the first of these German guys. Schutzenberger proved to us that free ones are finite. And I remember you told me in less than a minute that not only are they finite in the exact number, but in factorial times e. The truth is that Bernoulli's contribution was not really that great. He never talked about representing them. There are lots of literature on these things which doesn't even get considered. But they've come up recently. For me, it's quite a different subject, mainly that the famous hyperplane arrangement actually forms such a model. You can take this system of hyperplanes, because essentially the positions of hyperplanes in the generated hyperplanes can be determined. You ask for each one, is it on the left or on the right, or is it on the same side or opposite side, or on the...

1:05:00 And so it's a family of those things with three possibilities, and actually they multiply. Actually, Chips had noticed this in a very, very complicated way. He was barely visible in that regular band. Since we're talking history, if you want to go that further, besides Google the way to find the newest Sloan's encyclopedia, I was actually looking at the semigroup case rather than the monolith case. And so that number, you had to subtract one from it. That dates to 1713 in the paper by Bernoulli, for the briefing. What was new in this talk for me was the graphs as presheaves on a monoid. I hadn't seen that before, that you can do this as presheaves on that particular monoid. Is there a way that I could have known that by looking at the topos? Is it easy to say, or is it possible to say when topos of presheaves is presheaves on a monoid? Well, there are two different ways. Take in terms of the figures and relations. What is a graph? It's always reflexive, right? What is it? I mean, it's made up of all these variables that are linked together. So therefore, there's one generator in the joke box. You need a very narrow one. A small project of generals. Anyway, see me at lunch.

1:07:30 Thank you for your attention. Thank you for your attention. If you have a selection of topos, and down to the entry you have a mathematical diagram, what is the likelihood that you're going to get a topos?

1:10:00 And you can approach this from either end. You can try and describe what the initial structure of a topos should preserve, or you can look at the examples and say what maths actually is. The fascinating thing is that these both mean that we have the potential for topos theory. And it is a bit of a surprise. So, if you start from the examples, well the first examples you think of are Sheetron topological spaces, which is actually where the name of the topos came from. If we have two spaces x and y, do we know what it means to have a continuous function from x to y? The topos at least became the model for that notion of continuous function. So you ask, if you're given a function from x to y, what sort of function do you get? The answer is you get two of them, and you can think of these ordered sets as categories, this is the functor, it's not a preservative, but it's better than that, those are unions, but in the context we're in, those co-limits have become in the coverages, they're closed on these two, so you can then ask what happens to cheese when you have those two, and that is actually going to be part of Morin's lecture, and the answer is you get two functors out of this. One of them, in the same direction, is the one that's called the direct image, with a bill that is not the best name that could have been chosen for this,

1:12:30 because it directly does suggest the left-hand one. This is simply composition with this one to here. So, I mean, this is... So, given that we've dualized twice, it becomes a covariant thing. This is the operation that humans achieve on x, doesn't simply have a puncture on an x-off, and then composes that puncture with x-1. And observed result really is a sheet on Y because of these preservation properties. Where the preservation comes in is the absolutely crucial fact that this left-hand point itself is observed. Because you can think of this as an extension of X minus 1 itself. Remember, O of X sits inside sheets on Y. These are code complete categories and this is actually just an extension. I'm thinking of these as the negative evidence of it. So that's what we get. The justification from one end, of course, is topos, but of course that example alone wouldn't be enough. It is the fact that there are lots of other examples, and every way you look at examples, that's all very common as well, because induced geometric morphisms. Precise topos is, there, of course, you get more morphisms than just these.

1:15:00 You get, in this case, the third lower streak. Continuous acts of the topos, continuous non-morphisms, carry through points as well. So, in every case where you have a class of examples of topologies arising in nature, it seems to be that you get that the natural motion of morphisms between those examples are at least geometric morphisms, sometimes a bit more structured. So that's one point, just to capture all the examples you want to study. But there is a more serious... ...are important. Well, from one point of view, they really are. This is... Rather more subtle, because of course they don't preserve the structure that we first think of when you define topos. They don't preserve the structure that's involved with the definition of an elementary topos. They don't preserve exponentials. The other point of view involves the categories in a very strong sense. But in there, there's genetic embeddings. Say that the genetic embeddings say that the category is co-complete in a very strong sense.

1:17:30 The logarithmic toposses are categories of all the co-limits and all the co-limits commute with the finite limit. So, and of course that is exactly how you preserve finite limits, you preserve the co-limit because it's a manufacturing point. Toposses, those two are equivalent because for a logarithmic toposs you always have... So from that point of view, this is the really important factor and this is the... If you're viewing toposses in algebraic structure, this is the algebraic structure preserved. And this one is just a sort of decimal, which exists because of this. But there is still a sense in which that remains true, even when we go away from proof matrices to elementary ones, where you don't have all of them in it, and so you certainly don't have the answer to some of them in it. I imagine you'll also take a couple of hours of lecturing to get to the point of explaining that, but the basic idea is that given any category you're on here, you can look at the world of ethics not just as categories, but as categories equipped with the notion of substitution of families. You always have S itself providing finite limits, because you take the arrow category S for the two, S by the codename vibration, which corresponds to the notion of an I index family of objects of S, which is to be an object of S over I.

1:20:00 This is S itself in the world of S index categories. To say that S is a topos is precisely to say that that vibration is complete and well powered, or co-complete and well powered. So, these are also the constructions in the sense that they preserve the index category notions. The problem then is there aren't enough of them. You know, since a has preserved the logic, you can encode it, of that kind, connecting them all. They come up as the inverse images of local homomorphisms. Oh, sure. I mean, if you have... They may go a long way out if you use just one of them all for something. So, obviously, I'm sure there are other contexts, but... Questions? So, in a way, in a way, there's underlying all this. They're already there. Most people invented topology.

1:22:30 You know, like, why infinite equivalence and finite... I mean, another example that seems to have nothing much to do with it, if you take a monoid and look at the right ideals in the monoid, that's the truth value of something. It's corresponding to Topo's... It's also worth pointing out that the F to the minus one on the black color has a right eye joints to the minus one for various unions which are the supremum in those things, so that you've got a miniature example of a geometric morphism right there. You don't believe, because it's guaranteed to be there whenever you want it, a change of space. You were saying that the cosmic toposys have own coordinates and finite limits, and moreover these coordinates commute towards finite limits. And so that this definition of geometric morphism arrives quite naturally, because these families preserve what they have to do, what they have to preserve. Is there encoded in the definition of geometric morphism the fact that... These collimates commute with the finite limits. Is there a code in the adjunction or in any sort of definition of the matrix? Well, not in the definition of the metric morphism, no. I mean, that's encoded in the definition of topos. In fact, the fact about topos is that the original state of the Turov's theorem makes that very clear in the statements. Apart from the size conditions, the conditions that appear in the state of the Turov's theorem are all about the existence of finite limits, the existence of arbitrary coordinates.

1:25:00 You can get it down to a minimal set of such conditions. You could say a homework as a non-associated range has the same definition of homework as a non-associated range. Distributivity, which is a case of not being in the field. Can we talk a little bit about the open masses between topics? Maybe they can grow as quickly as the market can grow. I don't know if anybody wants to say anything. You want me to do it? Well, third of all... In addition to the notion of a nearest map, you have the notion of an open map. It's telling you something interesting. Again, let's go back to this picture. It's telling you that this s to the minus 1 has, in addition to its right-hand point, always there while you're going to think about it, it also has a left-hand point. It's not saying you've got a left-hand point. It's saying that you have a left-hand point. The smallest load containing each image in life just happened to exist, even though the actual set that had the image wasn't there.

1:27:30 The theoretical image here can be stated in terms of the interaction between sub-objects of one, but also between sub-objects of any object, whatever. So essentially it's saying that if I take, slicing this map over the object, pulling back along the level of homomorphism, that you get the same sort of left-gauge, so you can take a sub-object f half a star over a. So that is one thing that for every object f, the Codename Topology, has this left-gauge line.

1:30:00 It turns out there are all sorts of nice... There are other ways of it. It turns out that it preserves, guaranteed to be preserved, but that's why. Just the existence of these within left-hand joints ensures that they're not the star of reserves. The logical operation is not to preserve, that's the implication of quantification. It's sort of obvious why this tells you that it preserves universal quantification. It's just the same at left-hand joints. The Provenience Respiratory Commission says that this left-hand joint commutes with an intersection with an exponent. So this is another way of saying it, which indicates why it is important from a logical point of view. And further, is that this is equivalent to, not the resolution of Euler, which is the same as the F-upper star. That is stronger, and certainly that is insufficient, but what it says is that the F-upper star preserves,

1:32:30 And to take the inverse of interest back under F, it's precisely because this map is always a monomorphism, so that the higher order logic is basically embedded, and I don't really understand it myself, why for preservation of the first order logic, preservation up to the monomorphism of the higher order logic. Does anyone want to? Well, if you want to say more in depth why open maps are important, open coverings are important, because that's the end of it. Invisibly comes with topology. Open subjective maps, in the worst case, is something I don't know how it's basically reflected in the alphabet, but then it's true that the topology, geometry, and physics essentially are reconstructed from those. Precisely, essentially, you've got this novelist theory that's been ancient.

1:35:00 It remains a novelist theory that's been ancient. Semantic sort of stuff. So take the binary general theory, then the fact that... Well, being toughest theorists, we don't want to stop here. There's a Cogar and Tucker category going to the right. Very confusing. I'm talking about Cogar and Tucker. This was Kripke's category, and it is a spatial... Take all the right ideals on here and take those as your open sets. Take that lattice. Don't think of this as the space. Take that locale. That is in fact the locale. It is a spatial locale. It ain't sober.

1:37:30 When you're sober-fied, and that doesn't change the category of chiefs, what you get way down here, infinitely far away, is a copy of Anderson's. If I can reinterpret the Kripke theorem, a sign that you're getting into a category of sheaves and you put a little top with spatial sense on a very particular space which I'll call K for Kripke, it has, it has this counter space here, open subset... And so the idea of what we can do with this, I mean, how, I mean, that's, it was the day-to-day, so.

1:40:00 Yeah, yeah. We will have some stuff. I can, yeah, I can interweave with that. Right. By the way, there's an excellent publication on this theory and application in the category section on the internet. It has a section called reprints. I'm reprinting it there, 40 years back or so, together with commentary. So these commentaries are intended to answer part of the question, but particularly the theory of the category of sets, which is a paper published in Signora's, a much longer version of it, always available in the library of the University of Chicago, which was since 1960. That has been included with the commentary. There is an interest in the foundations of continuum content in the attempt to find a framework in which the simple physical reasoning could be carried out correctly in a way independent of particular choices about the new foundations of analysis project and say something about it.

1:42:30 So, yes, it mentions in that commentary that, yes, that's right. You see, Rotenbeek had axiomatized already in the 50s what is sometimes called an abtopos or Rotenbeek category. It's a co-complete category with exactness features, a set of generators, except it's additive, which is an abelian version, having just a figure of the number. For the category of sets, I thought, well, there are also variable sets, just like there are variable abelian groups, and this is also a definite class of categories that should have, in some sense, analogous. There is also the strong influence of Voltaire and Horowitz and so on about extreme importance. By the way, most mathematicians still have not properly recognized. The extreme importance of the internal notion of the mapping state, the fact that the natural notion, that is, say, appreciative of the intrinsic notion, of course, but it turns out concretely to mean naturality and so forth and so on, that that's actually the correct notion in various applications and analysis. This analysis has not realized this. On the opposite, they have still many of them. The mythology is impossible to realize that simple. And this is because they start with the wrong idea of the default category in which to talk about cohesion and analysis. So this also plays an important role. So when I launched the program of categorical dynamics in 1967, this was certainly one of the very important ingredients.

1:45:00 It's much more important than whatever particular foundation you give it. It is the way of computation. So this was one. But then it was coarsed by Gabriel at the United States. And so then I was 65. Of course I had spent this year in Berkeley. I learned something about set theory and logic. Lane spent a while. The idea that maybe logic would be the foundation. It turned out it isn't for all three of us. But it does play a certain important role. So on January 1st, 1966, I well remember there's a handwritten manuscript that was that day, but so dated, by Dana Scott, where he had discovered Boolean-valued models of set theories. So I immediately said, aha! This is really a great thing to post. Now it took years to actually verify this, but the Boolean-valued model was the same sort of thing. This was really the, and then of course, as said, the discovery that we, in some sense, as Rodendieck told me later, the only thing he didn't know about theory was about theory. He never noticed that there was those truth-telling objects. He immediately, immediately recognized it was important, but he didn't see it for some reason. He noticed that, then we were really off the ground. Of course, it was Miles who really showed it. How simple that is, because I had a much more complicated idea of how things should go, but the observation that was just classified by just these simple endo-maps, this was also somehow the idea that there needed to be a telemetry action, to complete in themselves without relying on something external, this seemed to be very important, even when we were dealing with Grotendieck toposes in the narrow sense.

1:47:30 The fact that a single map that expresses this complicated notion of, as you called it, a whole family of families and families of covers and all that, which of course is one important guide, but there's one useful guide, which is just this single little modal operator operating on truth. This had the elementary aspect of being finite, finite reactions. I just wondered, I don't know if this problem has ever been really solved. I mean, as an elementary topos, it's sort of an elementary version of a rotary topos. Something more, more detailed than just abelian geometry. It's not just abelian. Injective unknowns. A few more. Injective unknowns. I don't know exactly, but there should be an analog there. It's an interesting problem, although at least the way we all think of it intrinsically, implicitly, is that it's related to some topos, at least something Cartesian. Yeah, see, the fact, of course you can have interesting English categories like tensors and Hobbes, but unless you somewhere have a diagonal map, you cannot even state the definition of naturality. The whole definition of naturality depends on having a diagonal map somewhere. So you cannot be completely additive. You have something like floating around underneath this, but I do agree with you, as intrinsic as possible. Examples, of course, are... Yes, of course, so that's, yeah, that's the obvious example. You have the two things related, but somehow these moving groups don't quite fly on their own. You needn't have injectors. So, that's why I mentioned injectors. I mean, you told us, you needn't have injectors.

1:50:00 Yeah, that's a comparison there between, just because F preserves finite limits, you can't have that two together. Well, yeah, I mean, there is, as you say, that other comparison. You get in the junction between tens of thirds and like that. I think the one with the lower star has a left edge on it because of the process. The practical motivation for always trying to enlarge the theorems from being small to large in this sort of general process of, I guess it applies to small categories and if you try really hard you can make it in some kind of way apply also to large categories. I can see the intrinsic aspect of it, of just trying to make the theories more general. Those sorts of tricks. The only answer is the fact that once you allow them to have a canonical site,

1:52:30 that every shift of canonical politics is an economical choice. But on the other hand, it's not really a generalization at all. It's just extending the type of presentation that you like. It's a predicature to use the thing itself. It's not really something more general. Excuse me. Somewhat apropos is that Peter Freud wasn't here to defend himself when he said that Lex Total is a doubly dead pencil. And since he's the author of the theorem about Lex Totals being always... I'm not getting the deal. What? I'm not getting the deal. That's not my name. I think I went along with the defense of my use of the word cart. See, my position is that Cart gave us not just products, but he certainly gave us equalizers. Well, I mean, that's not what I'm going to give you. There's a whole terminology for cartography. Yeah, I'd call them, I would, see, the word limitless, as we're going into such things, the word limitless is not a big thing. I like limitless. Landbeck would use inf. I like calling it that. The reason the products was because of the equalizer.

1:55:00 ...the cohesion of the magnetic process, you know? But now, people always say that you are, when things are preserved, you feel like you're negative, you know? I just wonder about the definition of science, how you're including this in your commentary. Do you think it could be selected by some Germanic people to follow them? No, it's not. And the predictive version of the regenerated ones? I just wanted to have a comment about that. I'll be saying a bit about that in my third lecture. And what does it present? It makes a difference. I mean, all of them with respect to physics. Yeah. Which is not at all related to that. About your talk, actually, you were talking about some of the classifiers for diagrams. You discussed arrows that started in A and finished in... You had this little sub-A. Yeah. The arrows that started in A and finished in A but went outside of A. Yeah. What happens to the arrows that start outside and finish outside but go in? Yeah. They don't remember. They can't remember. They can't do that. Some object is closed under the operation of taking sources. It's not a general subset. There is a classifier for general subsets, but it's a...