From Klein to Kan

0:00 Right, I think we're ready to start again. On this occasion, I can happily say, lady and gentleman, because Myla has now rejoined us. One of the greatest centres of activity in category theory, and indeed in reflections on the foundations and significance of category theory for mathematics, is Montreal. all. And our next speaker, Jean-Pierre Marquis, is very much at the forefront of philosophical reflection on that category of theory. His, I think, was just about the first ever article that I read on foundations of mathematics by a philosopher, which really impressed me as having something very wise and penetrating to say. He carefully distinguished between several dimensions in the notion of a foundation for mathematics, ontological, epistemological, think, methodological, as my memory says, and traced in a very interesting and I thought in a beautifully controlled way what the distinctions and interrelationships between these different dimensions and senses of the foundation of mathematics are, relating those distinctions very much to the activity of real mathematicians. He has a book forthcoming, I should say, he He did his PhD in the University of McGill University. The teacher's now in the University of Montreal and has a book forthcoming shortly called From a Geometrical Point of View, which is a general study on the philosophical significance of category theory, which I think is to be strongly recommended. There's some reading for everybody here. I look forward certainly to reading it very much. He's going to speak to us on a thing which has already come up a couple of times in the course of our discussions, between the vision of the original founders of category theory, Eilenberg and MacLean, and earlier ideas, such as particularly those connected with the Erlanger program, both as regards the continuity and the new elements which category theory brought. His title is From Klein to Kahn, The Algebra of Space and the Space of Algebra. Thank you, John. Thank you. First of all, I want to thank the organizers Michael and John for organizing this evening. This is a wonderful opportunity to discuss these issues with the listeners' speakers.

2:30 I'm not going to talk about Columbeck, as you can see from the title, although, as Professor Cartier said this morning, the focal paper in 1957 is a landmark in the history category. I agree entirely with that. I'm going to talk about a contemporary event, which is Khan's Discovery of the Patriot Founders, which was in 1956, published in 1958. It's a different story, related, so this talk will be mostly historical, but with a philosophical twist, I hope, with mathematics in the background. Let me start with, this is from the paper, 1945 paper by Thalbert Reclaim, where we have an explicit reference to Kline. This category may be regarded as a continuation of the Kline and Laguer program in the sense that a geometrical space, with its group of transformations, is generalized to a category with its algebra of mappings. So what I'll do, so there's a reference here. The first thing I'll try to do is to present what they had in mind, because they actually developed this idea a little bit in the paper, and explicitly say, here is the connection with Klein and how we think it works. What happens afterwards is that it disappears. No one refers to it in the year afterwards. And then it comes up again in the 60s. There's an explicit, I'll show you later, there's an explicit reference to the 60s. Something's applied. That comes again. So what I want to do is to sketch related why I think it just stayed there for a while, Then moved very quickly to Kahn, where he was coming from, what he did before the adjoint paper, the adjoint function paper, and the connection, and then some general comments on the significance to the notion of the space that appears to be. So, here's another quote from that paper in the same paragraph. This emphasis on the specification of the type of mappings employed gives more insight

5:00 onto the degree of invariance of the various concepts involved. For instance, we show that the concept of commutator subgroup of a group is in a sense a more invariant one than that of the center, which in its turn is more invariant than the concept of the automorphism group of a group, even though in the classical sense all three concepts are invariant. So that's the suggestion. Here's the connection with the Plano-Lingo program, explicitly made there in the paper. It has to do with the invariance of concepts. So in the same way that Klein was saying, we'll use transformation that will reveal the invariance properties of the geometric figures. Here, Oliver McLean is saying, we'll used category theory in the algebra of mappings, and that will be revealed in various of certain concepts. And we give the examples. We'll see that that's the only examples I've actually given. I don't go very far with that. The parallel is explicit, and I'll present it in such a way. Here's a quote from Klein himself, in the translation, of course, given a manifold and a transformation group acting on it to investigate those properties of figures on that manifold which are invariant under all transformations of that group. So that's a group of automorphisms and we investigate the properties that are invariant under that group that gives you information about geometry and you use that to prove certain results. Just a remark here. As far as I know, there are very, very little philosophical analysis appliance program, even the historical analysis, there are some debates going on as to whether it was intellectual or not a wet way, and that's interesting, but the philosophical aspect has not been investigated, I think, deeply. Although I do think that it is a turning point in the way of thinking about mathematics at the end of the 90s, in various ways. so given the type of objects so in parallel with category theory here is given the type of objects so groups, rings, topological spaces, bonach spaces in fact in their paper most of the examples are groups, topological spaces, bonach spaces together with their mappings

7:30 there's a category of objects But if there's a category in this, the problem with what's going on here could be obvious at the level of the category. So here's how McLean and Allen Verdict work in the paper. So there's a section in the paper where they actually spell out how you can show that one concept is more invariant than another. And I said, how am I going to have to change invariants? What they did is not wrong, it's just that it was not very fruitful, in some sense. So, given a manifold S, one considers the group of automorphisms of the manifold, and then you get the various various properties of the objects. And here, the idea is to have two groups, for instance, And you consider all the morphisms between them. This is the algebra of morphisms. So when Annenberg and McLean talk about a category and its algebra of morphisms, that's what they have in mind. So it is replacing the optomorphisms with something which is more general, all the homomorphisms. And this is an algebra. It is in a very loose sense. And, of course, it is a fun term. So, here's how they actually work out the invariance. They start with two parallel fungers, F and T. And they give the definition of a sub-funger. That's ends of it. And you'll see this is very sub-theoretic in spirit. But G is a sub-factor of F, if for all objects of C, we have an inclusion from G in the objects of F, and for all morphisms from X to Y, then we also have an inclusion in the set-the-graph sense. So that's the notion of a sub-factor, which is another notion that was not preserved afterward. it was replaced by for instance with in 1957

10:00 the notion of a sub-natural transformation is defined similarly so it's also set theoretical I don't have to do it here but it's exactly the same spirit so they used this to get finally the notion of a quotient and they specify now you have to work arbitrary category, but you work in the category of proofs, in which you can actually work quotients with normal subunctures. So, suppose again that you have two functures, F and G, and now from the category of proofs to the category of proofs and the functures, and you suppose also that F is a sub-functor of G, the definition I just gave, but also that F is a normal subject, and then you find a quotient factor, Q, in this fashion, so it's just a normal quotient that you do in the category, and now they are groups. And finally, this is just a set, you fix the identity factor, okay? And now you start. The classification now can actually begin, and here it is. so for any group G this is also well known the commutator subgroup is a normal sub-factor of the identity and now they consider the quotient factor of I-quotient by the commutator subgroup and this is called by then the factor commutator subgroup now the main point is that this is a functor over the whole table. And that's the main point, that it is functorial. Because then, we move on to the center. And the concept is well-defined, here's the definition, the center of G, instead of G, is all the elements that commute with the others. But now we make the remark, this is not a functor on the whole category. But it is a functor if you restrict the morphisms to the subjective form. In this sense, according to them, it is less invariant than the previous concept.

12:30 Finally, the construction automorphisms of G, this is also a construction, is functorial only if you restrict the morphisms to the isomorphisms. Conclusion, this is even less invariant than the construction of the center. So this is interesting because not only is their idea that, oh, look, we're looking at morphism. This is very much in the spirit of what Klein was trying or was saying that we ought to be doing, for instance, in geometry. There is more, because Klein was just not saying, if you have a geometry, look at a group of automorphism, and you'll get interesting information. He was saying you can use the proof of automorphisms to actually prove things, and, moreover, you can actually see how various geometries are related to one another via the proof of automorphisms. And that was his main goal. So, here the notion of invariance was playing a key role. And here, Heidelberg and MacLean are doing something similar in the sense that now we can actually say that one concept is more invariant than another within group 2. So that's the idea. And it seems to be from a philosophical point of view, a very interesting article, as some concepts are more invariant than another. How do you know? Well, you look at the category of objects. If the construction is functorial, then it is invariant in a strong sense, quote, unquote. If the construction, for it to be functurial, you have to restrict the morphism to a subclass then it is not as imperative as the previous one and so forth. So that seems to be the idea that they're proposing in the 1945 as to how they actually are generalizing in a very specific manner in a way that will yield a classification of concepts or mathematical concepts. Okay, so, as I said, we have three group theoretical concepts

15:00 whose invariance is revealed by the algebraic values. This is similar to what happens in geometry, where the invariance of certain constructions is relative to the transformations it has, for instance, projective versus plebeian or apine transformations. So you have all these relationships that are very interesting in revealing. At that level, they are suggesting that if you do the assignment, you'll see that there are different experiments that are related to that. Okay. They say a little bit more. They say, here's a quote. The invariant character of a mathematical discipline can be formulated in these terms. Thus, in group theory, all the basic constructions can be regarded as the definitions of co- or contravariant functors. Someone may formulate the dictum. Subject of group theory is essentially the study of those constructions of groups which behave in a covariant or contravariant manner under induced homomorphisms. But this is a very strong claim. The subject of group theory is those instructions that behave in a coherent or contrary matter for more. So, that's my guess is that it is my claim. I would think that this is good. You discuss that afterwards, but I agree. one thing that will be remembered and is really important and will become more and more important in the years coming they emphasize the functorial nature of loop-boretical constructions and that's clearly an important aspect work. This is not what the paper is about. Remember, the paper is about natural equivalences. But in these sections, they do emphasize the fact that certain constructions are important. And that's a revealing fact about constructions. However, they do not give a general analysis of the various mathematical concepts. In the previous group, remember, they said in a mathematical discipline we could blah, blah, blah. The only example they give is group

17:30 to medical. Don't sketch how you could apply that in an arbitrary category because then you would have to specify that you are able to do or not certain constructions, for instance making quotients. They don't say anything about that. And as I said, it's not that important because that's not the point of the paper. That's not what the point of the paper was about. I'll come back to that later because there's actually something to be said about that. Alright, so that's Aliberg and MacLean, 1945, we have natural equivalences, we have defunctures, categories are not very important in the paper itself, if you read it, actually there's a section in which when you consider foundational issues as well, in practice we can dispense with a concept altogether of a category, but we have to represent this more for conceptual And then afterwards, in algebraic topology, very quickly, Eilenberg and Steenrod will apply categorical ideas and actually write a textbook that is based on categorical ideas. And that will be very influential. So what becomes more and more clear is that certain mathematical constructions are seen as being functors. of course in Eilenberg and Steenrod the whole machinery is in terms of functures so a homology theory is a functur and a co-homology theory is a functur so suddenly things are becoming functures however in the early 50s and I think this is correct though we might discuss this I do believe that all the functures that are considered in the early 50s they all go in one direction in Eilenberg and Steenrod's book on algebraic topology. They do present the whole thing as being a translation of topology into algebra. So you go from topology, you add an algebra, you work in algebra, you have results about topology. You don't really go back by a function. And lastly, the algebra of mappings was not used as such to characterize constructions, although McClane did it in 1948, 1949, in a paper published in 1950, where he actually introduced universal constructions,

20:00 so products are defined in the way we now know, and so on and so forth, but that was not read very much. It's not a well-known paper, but he was aware of that already in 1950. Okay, now we're going to switch, because we're not going to switch towards Dalia Kahn. And Dalia Kahn is coming from homodonic theory, an algebraic topology. His main source of knowledge in algebraic topology is is Allenberg in Steamrock. That's how he learned algebraic biology, right from the start, from a categorical perspective. Now I want to point out a very simple thing that was known, but also emphasized by Kuhirich right in the beginning of his paper, 1935-1936. That's something that took a little while to be clarified, Bill mentioned a few things there, is the fact that, so the first line you have the product of x and the unit interval, so this is a homotopy, it's a continuous map, so you can define a homotopy like that with qualifications, and there's an alternative way of defining a homotopy, is from the space acts to the space of functions from the unit interval to the paths. Kurevich does mention that. He has to apply some mild conditions on why. Why is it metric space in Kurevich's paper? It takes a little while to clarify what are these conditions in general. Even McLean, Oliver and McLean in 1945, in their paper, they have to specify how you construct a topology in general the example at the bottom is mentioned here but this is a standard the reason I mention this is that these two ways of defining a homotopy was known Kahn knew it and in his paper on adroit functors this becomes one example but he knew it in this way. This is just a special case of the more general

22:30 relationship that Bill mentioned at the bottom between spaces. And in the 30s and 40s people working in algebraic topology really knew that that was an important relationship. And they knew that they had to restrict the topology, but they knew how to define it so that it could work. So this is in the background. So Connick publishes in 55 and 56, four short notes. They are called the original abstract homotopy theory, one, two, three, four. He comes through the hole, and they are, and he's coming from, he's coming from there, okay, Arlenberg and Zilber. That's one of the references that comes in each of the notes, okay, and every later. This is one of his main sources of inspiration. One of other sources of inspiration is Jean-Pierre Serres' physics, which he knew, read, and found very stimulating, and other papers by James, also in the early 50s, on loop space. So what he does in 1955-1956, is really trying to build combinatorial homotopy theory. Using algebraic information and only algebraic information or combinatorial information to define homotopy groups and do homotopy theory. So here's a long quote from an early paper. It's the very beginning of the paper. Most theorems of homotopy theory, in particular those singular homology groups may be divided into, sorry, let me see. Let's think two parts. A. A theorem on abstract complexes and maps. B. A translation of this abstract theorem into topological language by means of a singular functor, simplicial, or cubical. As we mentioned, Bill, here it is in 1954. Such an abstract theorem, however, concerns which are the singular complexes of a topological space. In this note, it will be indicated how

25:00 a homotopy theory may be developed for all abstract cubical complexes, which satisfy only a certain extension axion. Homotopy groups will be introduced for all such complexes. Now, these abstract cubical complexes satisfying these extension axions will become what are now called Yes, well, he will consider it after in the next paper.