Session: Lextensivity Rig Geometry / Internal Categories / Quantales (& others) — Part 3

0:00 My categories are Hewitt's, Hilbert. Actually, the talk is in two parts. The first part, for being as objects in Cartesian bicategories, is work done with Bob Walters last year after the CTO7. And the second part, with Ignacio de Lourdes Franco and Ross Streep, was done more recently when I was visiting Ross in Australia. It's maybe a bit ambitious to try and record on two pieces of work in one short talk, but these things are quite related, so here goes. For the third CTNO, I'm going to begin by talking about Cartesian bicategories. Cartesian-ness, thank you, George, is a property of a bicategory. A bicategory is pre-Cartesian in the first instance. As you can see, each of the categories has five products. As soon as you have properties 1 and 2, you can construct the hints of product according to this formula. And this formula will enable me to explain my notation a bit. I consistently use p for first projection and r for second projection at the level of maps, and I consistently use pi as first projection and rho as second projection at the level of the local products. So you see you can construct what looks like a tensor product and what looks like a unit, just having one and two as I said, but what you get are lax bunkers. I mean, the eye is the easiest one. A lax vector from one into a bicamera is determined by picking out an object and a monad on it. Okay, so the object we pick out is the global terminal object, one, and the monad on it is the local terminal object, top, which has a unique monad structure.

2:30 Okay, so the third axiom, then, is that the constraints that one has for these things as lax functions are, in fact, convertible, so that they're pseudo-functions. Okay. That's what Cartesian bicategory is. Now, for any object A in a Cartesian bicategory, we have the equality shown there, because in a Cartesian bicategory, just as in a Cartesian category, George told us every object is canonically a commutative co-model. This is the co-associativity for the co-model A. Well, the identity two-cell there has a pair of lengths, delta zero and delta one. So delta one, you see, goes up like that and across. I'm using upper star for the right adjoint. Because the map means the left-back. I mean, to say these things are maps is to say the left-back does it. And so that's what it means. d star d is, in the case of delta one, related to one cross d, one tensor d, a star, d star, tensor one. Okay, so either one of those deltas being invertible makes the other invertible. And we say that in that case that the object A is Frobenius. So you can say a lot of things about Frobenius objects in Cartesian bicategories. One of them is this first figure I'm showing here, that an arrow between Frobenius objects is a map, if and only if it's a comonoid homomorphism, if and only if it has the right adjoint actually given by the mate with respect to the adjunctions, x left adjoint to x. A, left adjoint to A, the one has. Now, okay, what do I mean here? Well, we have the tensor product on a bicategory, so we can regard it as a one-object tricategory, and by saying that X is left adjoint to X, what I mean is that in the one-object tricategory, one has a unit, a co-unit, and so on, satisfying the coherence axioms.

5:00 And, uh, for, for joiners. Okay, so the thing is that if you have an arrow from X to A, then you have something, a main going from A, A, A ring to, to X ring, that's R ring. And, uh, well, that's just what the right adjoin is, as it turns out. But it's, it's, it's interesting that it's at a completely different level. Now, okay, that's one theorem about, uh, Cartesian bicategories. But the one I want to talk about... If you have a Frobenius object in a Cartesian bicategory, then for every x, the hom category, map b, xa, is a groupoid. A special case of this would say that if you have a Frobenius object in the Cartesian bicategory of proactors, that the Frobenius object is a groupoid, is itself a groupoid. The proof of that, which was given a long time ago, both by Aurelio Carboni and by me independently, the proof of that for pro-functions is what we have shown to be generalized to this general situation. So, with reference to the two-cell, the delta-1 that I introduced, I want to first of all note that the domain, dd star, can be more conveniently written as a composite of products of projections and their right adjoints. And the codomain is that threefold product that you see there, b star b, wedge, d star r, wedge, r star r. By the way, I should say that the wedge is just a convenient notation for product, so I don't mix it up with other things. I'm not suggesting that this is locally ordered. Okay, so delta 1, the provenience mate, which is hypothesized to be invertible, is in fact, gets identified with the three component arrows shown there, pi pi, pi rho, rho rho. So, let me explain what I mean by that. The typical one, pi rho, is shown in the two lozenges below there.

7:30 So pi is the first projection from P star wedge R star into P star, and similarly Rho is the second projection, and I'm taking a horizontal composite of those. So we'll write delta just like that, to abbreviate the thing. Right, and one more time with feeling. Delta is across the top here, and we have something new coming down the left because, you see, we didn't yet look at the combination R star P. Well, delta being invertible, that gives us another arrow that I call mu, because it's reminiscent of a multicell operation. Now, before doing some work with this, I want to introduce sort of homonymization. So if S is a general arrow in the Cartesian bicategory, and little a is a map, then to give a two-cell from A into Sb is the same as to give a two-cell from The identity into a star sb, and I write sigma is an element of s of ab in that case, as I say, to make it look like profiles, but of course if s is the identity, and alpha goes from f to g in map b xa, so f and g are maps, then I can write alpha belongs to afg, meaning what is shown there on the bottom line, so it's nothing more than a notation, but it does help to tell us what to do. See, I want to show that MATBXA is a group point, so let me begin by explaining that if I have four maps, F, G, H, and K, like that, that if I whisker the delta, the defining Frobenius two-cell, with GK and FH star, that what I get is as shown in the bottom two-cell. With the left adjoint and post-composed with the right adjoint, you preserve all limits, in particular finite products. So the product diagram that I have in the middle there, on the lower part, comes out to this finite product, and in the home notation it looks like that.

10:00 So, a typical element, if you will, of the codomain of the Frobenius two-cell is alpha-beta-gamma, and it looks as you see there in the S configuration. Alpha goes from f to g, beta goes from f to k, and so on. And what mu is going to do is associate to such an s-configuration an arrow from h to g. So that's why I suggested this. It's like a Mozart operation. Okay, now, what about elements of the domain of the Frobenius? Capital anxiety. We don't know very much about them at first. It turns out we don't need to know very much. We only need to know about certain of these things. You see, if I had another map, little x going from t to a, then I could, I might have a two cell from, well, I might have an element of p meet r of x, g, k. I might have an element of p star meet r star, f, h, x. And you see, we know how proton decomposition works, and so. If I paste two of these things together, I will certainly get an element of b star, wedge, r star, after the wedge r. But it's easy to unravel what the components are here. In the first place, we have the first two-cell that gives us an arrow from x to g and a two-cell from x to k. So, you might say that some of the elements are given by equivalence classes and configurations of that with an x like that. Now, here's a limit. For an element x i arising from an x configuration like that, it's particularly easy to explain the effect of the Frobenius arrow.

12:30 And of course, mu of the configuration is the obvious composite there, A to Z. So when you have a single two-cell alpha going from F to G, you have an S-configuration. You can take the identity, alpha, the identity. So it's natural enough to try to see what happens with mu of one alpha one. And just as a prelude, just remember this particular x-configuration, which you can see composes to give alpha, alpha, 1. So to see that alpha dagger really is the inverse of alpha, just read across the bottom here. Alpha, alpha dagger is alpha of mu of 1, alpha 1 by definition. Yes, and you see that's because in the almost, well, that's the definition, but why is that mu alpha alpha one? Well, that's because, that's because the square commutes here because by naturality, if you will. I mean, it would look like naturality if you were working in proctors, but it works the same way in this general context because of the, just the composition of the bicategory itself. So, So going that way around the pentagon figure gives alpha-alpha dagger, but I'm saying coming down this way is just one alpha-1 followed by alpha in that slot there, so that's, we have mu of alpha-alpha-1, but alpha-alpha-1, as I said, is delta of xi, and mu-delta is nu, nu of xi was obviously the identity, so... To get the other equation, it's equally easy. So, you know, I was very happy with this, with this proof. In January, I was speaking at the Center of Australian Category Theory, and as soon as I gave this talk, Russ said,

15:00 it's got to be something you can prove by Servidro Bravado's 1972 result. I didn't believe it, but, of course, Russ was right. So, that's the next part of the talk. The Romano theorem says that if tau is a monoidal natural transformation between strong monoidal functors and V is part of a dual situation in the monoidal category V, then the component tau V is invariable. Okay, to say that V is part of a dual situation means that V has an adjoint with respect to the tensor product. And strong monoidal functors preserve such adjunctions. So, you see, not only do I have a component tau v, but I also have tau v plus, which goes from fv plus to gv plus, and if we take right adjoints, we get to gv and fv, and if there's an 8, you see, which goes from gv to fv, and that provides the inverse of the new component of tau, which is really easy. Right, so, at roughly the same time... Craig Castro, who Ross has mentioned earlier today, and Brian Day discovered the notion of Frobenius monoidal punctures. It turns out that you don't need F and G to be strong in Rabatis theorem, you only need that F and G be Frobenius. So Frobenius monoidal punctures are monoidal punctures that also have an op-monoidal component, and the op-monoidal and monoidal parts satisfy the obvious Frobenius condition. And you do need then to tell the both monoidal and co-monoidal as a natural transformation. So that was their improvement, and Ross and I then, before we started working with Ignacio, decided what we needed to do was study this in the context of a monoidal bicategory. So in a monoidal bicategory, of course, you can talk about monoidal objects, such as x, p, j, and so on, up and down the row. I'm working on a monolithic light category to suppress the big tensor, just write it as juxtaposition, and if little x is thought of as an a-indexed family of objects of x, and similarly for little y, then to take their tensor product in x, I like to write that as just x.y, as I show there on the first line.

17:30 So, in order to generalize Rivaldo's theorem, the first thing that we had to do was to So we come up with a suitable notion of dual situation in an object in a monoidal bi-category. So what turned out to be a good thing to do was to introduce the notion of exact pairing for arrows A to X and from A ring to X. So we need A to have a right bi-dual with respect to the monoidal bi-category structure. And then, you see, we ask for a unit. All of these things are related to each other, and they are related to each other, and they are related to each other, and they are related to each other, and they are related to each other, Thus, that data is required to satisfy two equations, which are just the triangle equations for a junction somewhat disguised. By kappa sub x there, I simply, in the first one, I simply mean the canonical map that you have for the identity tensor x into x tensor the identity. That's really the contents of that kappa. Steve Black made an interesting observation when we spoke about this in Sydney. He said, you're just looking at the right bi-dual of the object little x in the monoidal bi-category m over x, and that's helpful in some respects to make sure that you've got the right idea, a useful notion, but we didn't find it simplified the calculations. But anyway, we're happy with Steve's comment about that. If in the monominal bi-category we have an object A that has a right bi-dual, and x and y are monoidal objects, and if we have an exact pairing where little x is an arrow from A to x, and if tau is a monoidal and co-monoidal two-cell between Frobenius arrows, then tau x is invertible, and the inverse is given by the formulas you see there.

20:00 Of course, Tau X plus is also invertible with a second pointer. So, equalization as a monoidal object of M is to say that the identity is part of an exact parent. To say that the identity has a left adjoint at this extra left. And in that situation, if X is a monoidal object and you have Tau going from F to G, Key terms may include mathematical physics. Speakers include mathematical physics. Key terms may include mathematical physics. In that situation, alpha and alpha inverse, the associativity arrows, two cells, have mates, phi and psi, and we say that A is naturally Frobenius if both of those, phi and psi, are invertible. Okay, so, when you have those, you see you can define an n, as I have in the upper left-hand corner, and define two cells, eta and epsilon, and... You see, what they're doing is as follows. The n is giving you the unit to say that A is left adjoint to A. And E is giving you the co-unit. The eta is an invertible two-cell which gives you one of the triangle equations.

22:30 And so is the epsilon. And of course, there's a coherence condition that has to be satisfied for that. And then you can also construct beta and alpha, as I show there, because j and p are the same. So the epsilon sub j is the co-unit for the j left diagonal to j star at the junction, and similarly epsilon p. Okay, so, theorem. For a, p, j, and naturally for b, it's a mathematical object in a monoidal light category. The arrows N and E, together with the two cells eta and epsilon, exhibit A as a right-by-dual for itself, and the two cells beta and alpha exhibit the identity in itself as an exact pairing, so that A has the left symbolization given by the identity as a mathematical object. There was one complicated thing to show, and that was that the A is right by rule to itself. And I really like this diagram because it's got all the coherence conditions of the normal category theory in it. The pentagon is on the ladder, the triangle for identity is in the center, and on the right side, the three regions are the so-called redundant McLean axioms for coherence. The little one in the center, for example, says that rho i is equal to lambda i. Okay, let's recover the Walters and Wood theorem in the following way. For B, a Cartesian bicategory, every object x has essentially, uniquely, the structure of a strict, co-monoidal object. So in the monoidal bicategory B, co-op, every object is a strict, map-monoidal object. Maps in B are still maps in B co-op, but the unit of the code unit didn't change. And you see, this map monoidal object is naturally Frobenius, in the sense I just gave, if and only if the object was Frobenius in the Cartesian sense. And then, you know, every arrow in B is co-monoidal, so in B co-op every arrow is monoidal. And it's strong monoidal if and only if R is a map, and so on and so forth.

25:00 Then Math.B.X.A. opposite, which is Math.B.CoA.A.X. That's a group point by Rabanos here. And then one more trivial dualization says that Math.B.X.A. is a group point. Ross was right, but there was a little bit of work to sort it out. I still like the first proof as well because, you see, it teaches us something about how to think in a Cartesian bicategory, and it supplies us with new techniques. But on the other hand, what we've done in the second part is hypothesized ourselves into some sort of Cartesian co-reflection. Almost, because Cartesian co-reflection undoubtedly would involve commutative, and it would be the... And so, we see several things here. We need to understand better the co-op dual of a Cartesian bi-category, and we also, I think, need to understand the relationship between these proofs. The first one is sort of Maltsev, and the second one is all about mates. So is there a deeper connection between the Maltsev operation and mating? I don't know. Thank you. Thank the speaker again.