Here's some more stuff some of you might like, taken from http://math.ucr.edu/home/baez/week174.html Again, pardon the tone - it's written for nonexperts. If any of you know literature on the "walking biadjunction", I'd be interested! Best, jb ....................................................................... Now I'm going to dive in and pick up right where I left off in my discussion of the ideas behind this paper: 2) Michael Mueger, From subfactors to categories and topology I: Frobenius algebras in and Morita equivalence of tensor categories, available at math.CT/0111204. My ultimate goal is to take you to an elegant understanding of Frobenius algebras by means of a 2-category called the "walking biadjunction", but first I'll play around a bit with a simpler but more famous 2-category called the "walking adjunction". This may sound scary, but if you can stick with it, you'll see that I'm really just using these 2-categories to describe fun games that you can play with certain 2-dimensional pictures. Even if you don't read the words, please stare at the pictures - I spend my Thanksgiving weekend drawing them, and I don't want that work to go to waste! Category theorists love to talk about adjoint functors, but 2-category theorists know that these are just a special example of an "adjunction". An adjunction is something that makes sense in any 2-category; if we take the 2-category to be Cat we get adjoint functors. There are lots of other nice examples that make this generalization worthwhile. For example, in "week83" I explained how a pair of dual vector spaces is also an example of an adjunction. To study adjunctions, it suffices to study the "walking adjunction". This is a little 2-category containing exactly the stuff any adjunction in any 2-category must have: not a jot more, not a tiddle less! It was first studied by Schanuel and Street: 3) Stephen Schanuel and Ross Street, The free adjunction, Cah. Top. Geom. Diff. 27 (1986), 81-83. In a bit more detail, the walking adjunction is the 2-category freely generated by two objects: a and b, two morphisms: L: a -> b and R: b -> a, and two 2-morphisms, called the "unit" and "counit": i: 1_a => LR and e: RL => 1_b satisfying two relations, called the "triangle equations". I wrote down these equations already last week, but let me do it again using "string diagrams", as explained in "week79" and "week92". In a 2-categorical string diagram, objects are denoted by 2d regions in the plane, morphisms are denoted by 1d edges, and 2-morphisms are denoted by 0d points. If the dimensions look sort of upside-down, you're right - that's exactly the point! Instead of explaining the whole theory, I'll just plunge in with the example at hand. The unit i looks like this: i / \ L R / \ a / b \ a while the counit e looks like this: b \ a / b R L \ / \ / e Note that as you cross a line labelled "L" from left to right, you go from region a to region b, which is our way of saying that L: a -> b. Similarly, as you cross a line labelled "R" from left to right, you go from region b to region a, since R: b -> a. In terms of string diagrams, the triangle equations just say that we can straighten out a zig-zag: | | i | | / \ L | a / \ | | / \ | | | R / = a L b | \ / | L \ / b | | e | | | or a zag-zig: | | | i | R / \ | | / \ a | | / \ | \ L | = b R a \ / | | b \ / R | e | | | | We can build any 2-morphism in the walking adjunction by vertically and horizontally composing units and counits, which corresponds to sticking together string diagrams in a vertical or horizontal way. Thus, a typical 2-morphism looks like this: \ \ a / \ a / / | \ R L R L / i | \ \ / \ / / / \ L \ \ / \ / / a / R | b \ e e / / \ | a L R \ \ / \ b / i \ \ / \ / / \ L e \ / L R \ \ / / b \ \ By the triangle equations, we could straighten out the zig-zag without changing the 2-morphism. As you may know, the word "anaranjado" means "orange" in Spanish - there was no word in English for "orange" before people in England started importing oranges from Spain. And this is a nice mnemonic, because if we take the above picture and paint the regions labelled "a" orange, and paint the regions labelled "b" black, the above picture has a roughly tiger-striped appearance. In fact, these tiger stripes tell you everything you need to know about the 2-morphism! For example, starting from just this: \ \ a / \ a / / | \ \ / \ / / _ | \ \ / \ / / / \ | \ \_/ \_/ / a / \ | b \ / / \ | a \ / \ \ / \ b / _ \ \_/ \ / / \ \ \ / / \ \ \ / / b \ \ you can figure out where everything else should go. By the way, note that orange stripes can disappear can appear as we go down the page, and they can split, but they can't appear or merge. Black stripes can appear or merge, but they can't disappear or split. As a result, there can never be any orange or black *spots*. We'll change these rules later, when we talk about the "walking biadjunction". Okay, so we've got this 2-category, the walking adjunction: let's call it Ad for short. It's pretty simple. How can we understand it better? Well, for any two objects a and b in a 2-category we get a "hom-category" hom(a,b), whose objects are the morphisms from a to b, and whose morphisms are the 2-morphisms between those. If we work out these hom-categories in Ad, we get some cool stuff. First let's look at the hom-category hom(a,a). In this category, the objects are 1_a, LR, LRLR, LRLRLR, .... and all the morphisms are built by sticking these two basic generators together vertically or horizontally: \ \ a / / \ \ / / L R L R \ \ / / a \ \ / / a \ e / \ / | b | | | L R | | | | and i / \ a | | a | b | | | L R | | | | In tiger language, we're talking about pictures of black stripes on an orange background. The two basic generators are the merging of two black stripes and the appearance of a black stripe. If you read "week89", you'll know another way to describe this! Our ability to stick together pictures vertically and horizontally makes hom(a,a) into a "monoidal category". LR is a "monoid object", with merging of two black stripes being "multiplication", and the appearance of a black stripe being the "multiplicative identity". Being a "monoid object" simply means that these operations satisfy the left unit law: / / | | / / | | / / | | /\ / / | | \ \ / / | | \ \ / / | | \ \ / / a | | \ \/ / |b| | / = | | a | | | | a | | | | |b| | | | | a | | | | | | | | | | | | | | and its mirror image, called the right unit law, together with the associative law: \ \ a / / / / \ \ \ \ a / / \ \ / / a / / \ \ a \ \ / / \ \/ / / / \ \ \ \/ / \ / / / \ \ \ / \ \ / / \ \ / / \ \_/ / \ \_/ / \ / \ / | | | | a | | a a | | a | | = | | |b| |b| | | | | | | | | | | | | | | | | There aren't any other laws, so hom(a,a) is the "free monoidal category on a monoid object", or if you prefer, the "walking monoid"! I touched upon the immense consequences of this fact for algebraic topology in "week117" and "week118". They mainly rely on another way of thinking about hom(a,a): it's the category of order-preserving maps between finite ordinals! For example, these black tiger stripes on an orange background: 0 1 2 3 -------------------------------------------------------- | \ \ a | | a / / | | | | \ \ | | / / _ | | | | \ \ | | / / / \ | | | | \ \_/ \_/ / a / \ | | | | \ / \ \ | | | | a \ / \ \ / / | | \ b / _ \ \_/ / | | \ / / \ \ / | | \ / / b \ \ b / a | | \ / / \ \ | | -------------------------------------------------------- 0 1 2 correspond to the order-preserving map f: {0,1,2,3} -> {0,1,2} with f(0) = 0, f(1) = 0, f(2) = 0, f(3) = 2. Just read the stripes down! A more geometrical way to say the same thing is to call hom(a,a) the category of "simplices", usually denoted Delta. Here the object |---n+1 of them---| LRLR..........LRLR corresponds to the n-simplex, and these morphisms: -i.LRLR--> --i.LR-> -LR.i.LR-> 1_a --i--> LR --LR.i-> LRLR -LRLR.i--> LRLRLR ... <-L.e.R- <-L.e.RLR- <-LRL.e.R- are the basic "face" and "degeneracy" maps between simplices, which you'll find in any book on algebraic topology. The n-simplex is a face of the (n+1)-simplex in n+1 ways, and there are n basic degenerate ways to map the (n+1)-simplex down to the n-simplex. These aren't *all* the morphisms; just enough to generate all the rest by composition - i.e., sticking together pictures vertically, but *not* horizontally. Perhaps I should explain the notation here a bit more. Readers of "week80" will know that I use a dot to denote horizontal composition of 2-morphisms. For example, when we have a couple of 2-morphisms like this: f f' ---->---- ---->---- / || \ / || \ S: f => g x || S y || T z T: f' => g' \ \/ / \ \/ / ---->---- ---->---- g g' we get a 2-morphism like this: ff' -------->------- / || \ x || S.T z S.T: ff' => gg' \ \/ / -------->------- gg' But sometimes we can also horizontally compose a morphism and a 2-morphism! We can do it whenever our morphism f looks like a little "whisker" f sticking out of the 2-morphism T: f' ---->---- f / || \ x----->-----y || T z T: f' => g' \ \/ / ---->---- g' and what we get is a 2-morphism f.S like this: ff' -------->------- / || \ x || f.T z f.T: ff' => fg' \ \/ / -------->------- fg' This process, called "whiskering", is not really a new operation. f.S is really just the horizontal composite of these 2-morphisms: f f' ---->---- ---->---- / || \ / || \ x ||1_f y || S z \ \/ / \ \/ / ---->---- ---->---- f g' Similarly we can define T.f in this sort of situation: f' ---->---- / || \ f T: f' => g' x || T y----->-----z T.f: f'f => g'f \ \/ / ---->---- g' Anyway, once you're an expert on this 2-categorical yoga, you can easily see that these morphisms in hom(a,a), which are really 2-morphisms in Ad: -i.LRLR--> --i.LR-> -LR.i.LR-> 1_a --i--> LR --LR.i-> LRLR -LRLR.i--> LRLRLR ... <-L.e.R- <-L.e.RLR- <-LRL.e.R- are obtained by taking our basic tiger stripe operations - the "merging of two black stripes", or L.e.R, and the "appearance of a black stripe", or i - and drawing some extra black stripes on both sides. That's what those LR's are for. After all, no tiger is complete without whiskers! Okay. Now, having understood hom(a,a) in all these ways, let's turn to hom(b,b). Luckily, this is very similar! Here the objects are 1_b, RL, RLRL, RLRLRL, .... and morphisms are pictures of *orange* stripes on a *black* background: \ a / \ a / / | \ / \ / / _ | \ / \ / / / \ | \_/ \_/ / a / \ | b / / \ | / \ \ / b / _ \ \_/ / / \ \ / / \ \ / / b \ \ These orange stripes can only split: | | | | R L | | | a | / \ / i \ b / / \ \ b / / \ \ R L R L / / \ \ / / b \ \ or disappear: | | b | a | b | | R L | | | | \ / e as we march down the page. This means is that hom(b,b) is Delta^{op}: the *opposite* of the category of simplices, the *opposite* of the category of finite ordinals, or the walking *comonoid* - which is just like a monoid, only upside down! Here is another picture of hom(b,b): --R.i.LRL-> --R.i.L-> --RLR.i.L-> 1_b <--e-- RL <--e.RL-- RLRL <--e.RLRL-- RLRLRL ... <--RL.e-- <--RL.e.RL- <--RLRL.e-- If you're a devoted reader of This Week's Finds, you'll know I secretly drew this category already in section N of "week118". There I was talking about specific adjoint functors instead of the walking adjunction, so as not to prematurely blow your mind. I was also writing horizontal composites backwards, for certain old-fashioned reasons. But the idea is exactly the same! The morphisms above give the usual "face and degeneracy maps" we always have in a simplicial set, since a simplicial set is a functor F: Delta^{op} -> Set. By the way, you may have noticed that to get from hom(a,a) to hom(b,b), we had to switch the colors orange and black AND read the pictures upside-down. The reason is that if we turn around all the 1-morphisms AND 2-morphisms in the walking adjunction, we get the walking adjunction again. Ponder that! We can summarize what we've learned so far using the "Platonic idea" jargon I introduced last week: The Platonic idea of a monoid and the Platonic idea of a comonoid are the hom-categories hom(a,a) and hom(b,b) sitting inside the Platonic idea of an adjunction! (By the way, to round this off we should really describe hom(a,b) and hom(b,a), too. I think hom(a,b) is the Platonic idea of "an object with a left action of a monoid and a right coaction of a comonoid, in a compatible way". If so, hom(b,a) would be the Platonic idea of "an object with a right action of a monoid and a left coaction of a comonoid, in a compatible way". By "compatible" I'm saying that we can act on one side and coact on the other side in either order, and get the same thing. Filling in the details requires concepts I'm not eager to discuss right now, so I leave this as an exercise for the highly energetic reader. The less energetic reader can just study the tiger-stripe descriptions of these categories.) Finally, here's Mueger's new twist on all these ideas! Better than an adjunction is a "biadjunction". This has some extra structure, which turns out to explain all sorts of fancy-sounding stuff people look at in the study of subfactors and TQFTs and the like.... But what's a "biadjunction"? A biadjunction is where you have a morphism L: a -> b in a 2-category that is both left and right adjoint to R: b -> a. More precisely, a "biadjunction" is a setup (a,b,L,R,i,e,j,f) where (a,b,L,R,i,e) and (b,a,R,L,j,f) are both adjunctions. In terms of string diagrams, our generating 2-morphisms look like this: i j / \ / \ L R R L / \ / \ a / b \ a b / a \ b b \ a / b a \ b / a R L L R \ / \ / \ / \ / e f and the triangle equations say all possible zig-zags can be straightened out. Now let's study the "walking biadjunction", BiAd. As before, 2-morphisms in BiAd can be described using pictures with orange and black stripes - but now *both* kinds of stripes can appear, disappear, merge or split as we march down the page: ------------------------------------------------------- | \ \ a | | a / / | | | \ \ | | / / | | | \ \__/ \__/ / a | | | \ _____ / _____ | | | \ / a \ / / \ | | | a / / ___ \ / / \ / | | / / / \ \ / / __ \_/ | | / / \ b / / / / / \ | | / b \ \_/ / / / / a \ b | | / \ / / / / \ | ------------------------------------------------------- This allows for quite arbitrary ways of cutting up a rectangle into regions of orange and black, with piecewise linear boundaries, subject to the condition that each vertical border has the same color all along it. The triangle equations and the rules for 2-categories say that we can warp such a picture around without changing the 2-morphism that it defines... I don't want to be too precise here, since it would be boring. Hopefully you get the idea: BiAd has a purely topological description! Now for the punchline: in BiAd, what is the category hom(a,a) like? As in Ad, the objects are 1_a, LR, LRLR, LRLRLR, ... but now the object LR is equipped not only with multiplication: \ \ a / / \ \ / / L R L R \ \ / / a \ \ / / a \ e / multiplication: \ / L.e.R: LRLR => LR | b | | | L R | | | | and multiplicative identity: i / \ a | | a multiplicative | b | identity: | | i: 1_a => LR L R | | | | but also a "comultiplication": | | | | L R | | | b | / \ / j \ comultiplication: a / / \ \ a L.j.R: LR => LRLR / / \ \ L R L R / / \ \ / / b \ \ and "comultiplicative coidentity": | | a | b | a | | comultiplicative L R coidentity: | | f: LR => 1_a | | \ / f which make it into a monoid object *and* a comonoid object. Even better, there are some extra relations between the multiplication and comultiplication, which make LR into a so-called "Frobenius object"! In short, hom(a,a) is the walking Frobenius object! So is hom(b,b), since there is no real asymmetry between the objects a and b in a biadjunction, as there was with an adjunction. I haven't thought much about hom(a,b) and hom(b,a) yet, but one obvious thing is that they're isomorphic. Next time I'll talk about examples of Frobenius objects and why they are so important in subfactors, TQFTs and the like. This is what Mueger is really interested in. Right now, I want to wrap up by saying exactly what it means to say LR is a "Frobenius object". What are the extra relations between multiplication and comultiplication? There are various ways of describing these relations. Mueger uses a pair of equations that are popular in the TQFT literature: \ \ / / | | | | \ \ / / | | | | \ \_/ / | | | | \ / | \ a | | | | | \ | | a | | a a | |\ \ | | a | | | | \ \ | | |b| | | \ \ | | | | = | | \ \ | | | | | | \ \ | | | | | | a \ \| | | | | | \ | / _ \ | | \ b| / / \ \ | | | | / / \ \ | | | | / / \ \ | | | | and its mirror image. People sometimes call these the "I = N" equations, for the obvious reason. So: one definition of a "Frobenius object" in a monoidal category is that it's a monoid object / comonoid object satisfying the I = N equations. Where can you read about this? Well, besides Mueger's paper, there are these: 4) Frank Quinn, Lectures on axiomatic quantum field theory, in Geometry and Quantum Field Theory, Amer. Math. Soc., Providence, RI, 1995. 5) Lowell Abrams, Two-dimensional topological quantum field theories and Frobenius algebras, J. Knot Theory and its Ramifications 5 (1996), 569-587. The I = N equations are cute, but personally I prefer a more conceptual description of a Frobenius object. This may be a bit mindblowing to the uninitiated, so if you're just barely hanging on, please stop now. Hmm! If you're still reading this, you must be brave! Okay - don't say I didn't warn you. Let's start by pondering LR a bit more. This guy is its own adjoint, with the unit and counit as follows: _ a / \ | | | | unit for LR = | b | multiplicative identity composed with / _ \ comultiplication / / \ \ / / \ \ / / a \ \ \ \ a / / \ \ / / \ \_/ / counit for LR = \ / multiplication composed with a | b | comultiplicative coidentity | | | | \_/ It's easy to check the triangle equations by straightening out the relevant zig-zags. Now, whenever a monoid object has a right or left adjoint, that right or left adjoint automatically becomes a comonoid object, by the magic of duality. But if a monoid object is its *own* adjoint, it becomes a comonoid object in *two* ways, because it is both its own left *and* right adjoint! So, our guy LR is a comonoid object in *three* ways! Huh? Well, we already knew LR was a comonoid object before this devilish paragraph began, but since LR is its own adjoint, it becomes a comonoid object in two other ways. Amazingly, the I = N equations are equivalent to the fact that all three comonoid structures agree! I leave this as an exercise for the insanely energetic reader... I've worked it out before, and I rechecked it this morning in bed. I don't know if a proof exists in the literature, but from what Mueger writes, I suspect maybe you can catch glimpses of it in Appendix A3 of this book: 6) L. Kadison, New Examples of Frobenius Extensions, University Lecture Series #14, Amer. Math. Soc., Providence RI, 1999. Anyway, the upshot is that we can equivalently define a Frobenius object in a monoidal category as follows: it's a monoid object / comonoid object which becomes its own adjoint by letting unit = multiplicative identity composed with comultiplication counit = multiplication composed with comultiplicative coidentity and has the property that the resulting 3 comonoid structures agree. Or, equivalently, that the resulting 3 monoid structures agree! There is much more to say about this, but let's stop here.