Session: Lextensivity Rig Geometry / Internal Categories / Quantales (& others) — Part 2

0:00 We give some internal categories, descent, and data theory in monolithic categories. Thank you very much. Thank you very much for your attention, Professor Nadis. Since I'm going to contradict maybe a little bit what Ross said, let me begin by expressing my admiration to Ross Koch. He is a genius and maybe we need years to understand him fully. So, what I'm going to tell you is in some sense closely related, and I wanted to say that category theory is not a religion, maybe something much better will be discovered soon, but it didn't happen yet, and it will be a very very big change in mathematics and maybe in science in general. And it is very dangerous to think that you can renovate, so to speak, some important parts of mathematics avoiding category theory. And I think this is happening nowadays, and the example is mathematics which has adjectives like noncommutative and quantum. I think these are wrong words, sorry Ross, yeah, yeah, I know, otherwise I would say it, and the true words is non-Cartesian, and these things are old ideas which existed in category theory, so... Monoidal categories were invented by Sonders-McLean, as we know, and is a monoidal category.

2:30 A very important example is, if you take C1 for terminology, and it's good to call this Cartesian case, which I think many people do, certainly Max Kelley. Many things which are now pretend to be non-commutative of quantum is just straightforward generalizations of Cartesian to non-Cartesian monoidal and I'm going to give you one such example and I will not arrive to Galois theory but I will arrive to descent theory and there is an I also. One of the most mentioned relations in Ross's talk is that it is about, one aspect is about non-Cartesian internal categories. Now, Ross, why didn't you mention internal categories here? Never in your talk. You generalize them. How many percent? Yes, yes. But they exist. They, as we all know, their importance was first understood in topos theory. Internal categories are important in categories which are not topos, like groups and they are used in homotopy theory and so on, but the original things like internal families, internal actions, internal categories, the word internal itself comes from topos theory. So we should, if we are talking about internal categories in the monoidal category, we should look here first and see how can we generalize this. Now, let me recall two things which Ross mentioned. Category of common objects.

5:00 When C is Cartesian, it is just C. Actions, co-action of common object A, to A, but this is co, so it goes from Cartesian is tensor, and from the axioms, You immediately see that this is just of the form of, say, alpha 1, where 1 is not this 1, it's the identity morphism, and alpha is any morphism from A to M. So the monoidal, non-Cocasian concept of the internal family is this co-action. Now, there is a lot of things changed now. In the completion case, we simply consider the former category, this last category, as we said in the last years, and this thing is normally called, as I said, our originating topos theory, this thing is normally called the category of internal aiming experiments. Now, it would be reasonable to write it like this. You can, and you can motivate one. This notation is even more obvious. It's the category of M co-actions. Only M is a monoid, A is not. And the morphism is not just from A to M, but like this, it satisfies the action axioms. So this is the monoid replacement of this, and they coincide in the Cartesian mathematics,

7:30 say, in a category with finite limits. And you want to develop non-Cartesian mathematics. Well, a thousand times you will see a morphism, and every time you have a choice. Again, to take a morphism in your category, or take a homomorphism of monoids is the same, in the Cartesian case, or require only one of them to be a monoid and consider an action instead of such a morphism, or a homomorphism. Sorry, if I said monoid, yes, yes, it's always commonoid. And maybe I want to have a thousand different commonoid structures on B. So, it is not immediately obvious how to generalize things in a clever way. Marek here, Marek told me something. It seems that there are good motivations. Now, I said not obvious, but there's a word yet because The internal categories in this context were not introduced by MacLean in 1964, so it's not, should not be considered as autism. Sorry? Erasmus. Yes, yes, yes, but not in monoidal context, I mean. But in a much more primitive way, it will be a different concept, much, much simpler. The concept is this. Because we have a lot of choices here, decide where to begin. And I think the right way to decide where to begin is internal categories as monoids, which Ross also did. They are monoids with a set of morphisms. You would see it as a graph, but now you see it as a span.

10:00 C0 graphs, he has all graphs in his book, form a monoidal category, and an internal category is a monoid there. Let us use this idea to define internal category and monoidal. The first step is what do we want to take instead of spends? And French experience, Bernabeu, Australian experience give very definite answer. So, if our original category was called C, we can take commonoids in C, we can consider them as a bicategory, but bimodal, not taking a monoid in that category. We have a monoidal category, so we have to fix C0. We would call C0 cat in C.

12:30 So, C0 stands for the fact that we consider all categories whose object of object is fixed C0. In this concept of an internal category, you have again object of objects, object of morphisms, but unlike the very sophisticated concept Ross gave us, that's why I asked this question, only the object of objects is a monolith, and not the object of morphisms. Given the fact that we have a bi-category there, we can go ahead and copy. Internal category theory to this context. Now, the question is, what shall we do in real life? Shall we use this primitive concept which I explained to you in a few minutes, or it should be that sophisticated concept which Ross invented and which is related to many things in various branches of mathematics? I think I have to give arguments for the one I introduced. Excursion to descent theory in the Cartesian case. We call P an effective descent morphism. The pullback factor along P. Of course, there are two conditions, but that's one of them. And this means that if you take here the category of algebras,

15:00 Which is also called the category of descent data, and I have canonical functor and canonical functor, and this comparison functor and this comparison functor should be an equivalence of categories. Now, I must build more about this yesterday remark on John Beck. There are various ways to define the category of descent data, which apparently, Walter, we didn't know Beck was thinking about, of various ways. Now, what we did, we called them not various ways, but facets of this entire world, and indeed, there are many ways which come from different independent branches of category theory which give you a good description of this thing, and one of them, apart from being category of algebras, is this, determined by p, and this occurrence relation determined by p. It's, of course, distinct. Thanks to the case where C is the opposite category of commutative links, we will get those cases of descent which Grotendieck wanted to do. Some years ago, Stefan came from Brussels, telling me that somebody invented non-commutative descent. And I said, isn't it just a monodicity? And he looked at the paper again and said, yes, yes, it's just monodicity. So at that time I didn't think that it can be done this way. And all you do is do this. Non-Cartesian means internal category in this sense, in a monoidal category.

17:30 The object of objects is a commonoid, but this thing is not, because it's a category of algebras. This morphism, which will be morphism back, will... And you can take tens of products of algebras, even in non-commutative cases, and get an algebra. In the definitive case, not every homomorphism gives you an algebra structure, it should be central. And if it is just a homomorphism, even though E and P are rings, E is an algebra over P, this is not an algebra, it's only a model. So, this straightforward generalization of an internal equivalence relation determined by P is such a non-Cartesian internal category. And then you can repeat this facet of descent. And again, you can show that the category of Algebras is equivalent to the category of Actions, which is defined in statistical order. Is it not? There is one thing I did not check carefully. I never thought of non-symmetric monoidal case, but I think it equally works in non-symmetric case. At least it doesn't use anything. And doesn't seem to use anything.

20:00 This result is known for algebras. You will not see this known Cartesian internal category as a generalization of internal category, because they don't know internal categories. You can see this structure, for example, in Canockwell's paper in Fields Proceedings. You can see it in many other papers. And what do you think it is called? It is called a coring. It's a very typical and very clear case of ignorance of category theory producing seemingly new mathematics, but because the people who did it were not so bad after all, they produced a good thing because it agrees with the... I had a conversation with Greg Castro about his joint research with Ross, and I sort of also came to this conclusion that this was a good notion of the internal category, and then Greg responded that this idea actually appeared in a PhD thesis in the mid-90s of an algebraic topologist, whose name I believe was Marcelo Aguiar, so... Thank you, and it would be nice if this idea appeared five more times, and it is certainly in these people who talk about co-links, because they say absolutely every word I say, except that they don't mention internal categories, and they tell that product is just a product of ordinary models. But thank you.

22:30 The interpretation is to say everything, but how can I say everything? The fact that you have internal actions, the fact that internal actions are monadic, all these things come very easily from the fact that bimodules form a bicategory. All right, what I can say is, if you open topos theory by Peter, and if you say what he says before taking your internal category inside of a topos, everything can be covered. Thank you for your attention.