As you can probably tell, I’m hugely excited about Joachim Kock’s paper. I apologize to those of you who read it a year ago, and think I’m a bit late to the party.

Most late-stage PhD students, I imagine, have written an imaginary textbook entitled *Things I wish I’d known three years ago*. My own contribution to this genre has a snappier title – it’s called *Monoidal categories* – and since yesterday it’s been frantically rewriting itself. (The advantage of imaginary books over real ones is that they can do that.)

The traditional approach to monoidal units has relegated itself to a historical section, and ‘Saavedra units’ now take pride of place. But how radical to be? The radical approach is just to say that a unit is an object *I* with the property that the functors *I⊗–* and *–⊗I* are equivalences naturally isomorphic to the identity.^{*} (Of course we also want an associator that satisfies the pentagon condition, just as before.) What’s radical about this is that it doesn’t *uniquely* determine a monoidal structure in the traditional sense: each isomorphism *I⊗I → I* gives a different monoidal structure. But any two of these are isomorphic via a unique isomorphism, so really it’s unique enough for any self-respecting category theorist. Should you want a traditional monoidal structure for some reason, you can just pick an isomorphism *I⊗I → I* (we know there is one) and there you are! This is just like the attitude we have always taken to things like cartesian products.

(I think there’s a good case to be made that this is a *better* definition, as well as a simpler one. We never *really* cared about the unit isomorphisms *λ* and *ρ*, it’s just that with the traditional approach we needed to know what they were so we could impose a coherence condition on them!)

Continuing the radical theme, let’s define a strong monoidal functor *F: C → D* to be a functor

*F*such that

*FI ≅ I*, plus a natural isomorphism with components

*F*satisfying the usual associator condition. If we pick a particular isomorphism

_{A,B}: FA⊗FB → F(A⊗B)*F*, then for every traditional monoidal structure on

_{I}: I → FI*there is a unique traditional monoidal structure on*

**C***for which*

**D***(F, F*is a traditional monoidal functor, which is more than reasonable. (The situation for lax monoidal functors is not so obviously pretty, so I’ll sweep them under the rug until I’ve thought about them some more.)

_{A,B}, F_{I})Having tamed the unit so splendidly, it’s tempting to wonder whether the same can be done for the associativity. Is it possible for the same tensor functor to admit non-isomorphic associators (where by ‘associator’ I mean the thing to satisfy the pentagon condition)? Off the top of my head, I can only think of (essentially) one example where I know two different associators: say the category of abelian groups under biproduct, where in addition to the usual associator you can take the map *(a,(b,c)) ↦ ((-a,b),-c)*. However, this is uniquely isomorphic to the usual associator^{**}, so it doesn’t answer the question. But I have an inkling that the answer may (disappointingly) be yes, because of something I once heard, but still don’t understand, about isomorphism classes of associators corresponding to 3-cocycles in some cohomological sense…

***** For which it suffices to check that they’re full and faithful, and that *I⊗I* is isomorphic to *I*.

****** In the sense that there is a monoidal isomorphism whose underlying functor is the identity that takes one to the other.

I am also fascinated by this line of thought. As far as associators are concerned, have you read Tambara and Yamagami’s paper?

Tensor Categories with Fusion Rules of Self-Duality for Finite Abelian Groups, D. Tambara and S. Yamagami, Journal of Algebra, 209, 692-707 (1998)

This is a most revealing paper in as much as it does the one thing category theorists are loath to do : it performs explicit computations. The only sad corollary is that it is not always brutally explicit in the details – for instance, it seems to interchange the use of skeletal categories with run-of-the-mill categories, and so on.

Nevertheless, the mindblowing result (to a novice like me) is that the associators on the monoidal categories of representations of the dihedral group D_8 and the quaternion group Q_8 are not equivalent – even though they have the `same underlying tensor category’ (one way to interpret this last phrase is to say that their underlying fusion algebras are the same, as everyone knows).

Moreover, this paper seems to me the culmination of an idea which can be tracked back to Maclane’s original example in “Categories for the Working Mathematician”. At the end of Section VII. 1, he describes a counterexample pounted out to him by Isbell, showing that the associator on a skeleton of Set is nontrivial – *provided you define it to commute with the projections as usual*. For, if you forget about the projections, there is no problem in getting out a trivial associator (I think).

It is this last sentence which seems the link between Maclane’s example of Set and the representation categories of groups. In both cases there are canonical projections floating around, which the associator is supposed to commute with, and it seems to me that the whole issue of non-trivial associators is wrapped up with them.

One final thing which has always bothered me : isn’t the very notion of a `tensor product’ of two objects against the philosophy of higher categories? For it seems that all the coherence diagrams, etc., are consequences of some cruel dictator who has forced us to *choose* a tensor product for each pair of objects (much like the unit issue you talk about above). I know it comes from a monoidal category being a bicategory, where one must actually choose what the composition of two arrows is… so what I’m asking about is whether this notion is a good idea – should we really explicitly say what the composite of two morphisms is? I know it sounds like heresy, but it does seem the homotopy theorists have a more sensible way of working with these things.

B. Bartlett: Thanks a lot for this. I don’t know the paper you cite, but from your description it sounds as though it may answer my last question. I have downloaded a copy, but I’ll have to read it slowly because it’s coming from such a (to me) unfamiliar direction. I confess that I didn’t even know there

wassuch a thing as a fusion algebra, let alone what one is!On your final point: if you don’t like the notion of having to choose a tensor product for every pair of objects (and I think you’re right not to like it!) you will probably be interested in Hermida’s

representable multicategoriesapproach, which does precisely what you’re asking for – and does indeed dispense with explicit coherence conditions. (Also Street’s Lax monoids, pseudo-operads and convolution maps out the landscape between multicategories and monoidal categories in an interesting way.) Many (most?all?) of the general approaches to weak n-categories use some version of the same idea, so that‘s not a heretical suggestion at all. In fact Hermida’s paper was directly motivated by Baez and Dolan’s definition of n-category.Bruce Bartlett writes:

One final thing which has always bothered me : isn’t the very notion of a `tensor product’ of two objects against the philosophy of higher categories?

More precisely, it’s not the notion of

atensor product of objects that’s the problem – it’s the notion ofthetensor product that’s the problem.For it seems that all the coherence diagrams, etc., are consequences of some cruel dictator who has forced us to *choose* a tensor product for each pair of objects (much like the unit issue you talk about above).

To some extent this is true, and this why James Dolan and I came up with our opetopic definition of n-category, in which all composites are defined only up to canonical equivalence. As bosker notes, one can chop this down to get Hermida’s version of “representable multicategory”, where objects only have

atensor product. And, our work was heavily influence by Makkai’s concept of an “anafunctor” F: C -> D, which assigns to each object in C not a specific object of D but only theuniversal propertyof an object in D. A good example is the “tensor product”, normally treated as a functor from Vect x Vect to Vect: it’s really just an anafunctor.Hermida and Makkai have subsequently worked on opetopic n-categories, calling them “multicategories”. So have Tom Leinster and Eugenia Cheng. It’s a philosophically attractive approach, but leads to technical problems of its own, which have not fully been resolved yet.

bosker writes:

Is it possible for the same tensor functor to admit non-isomorphic associators (where by ‘associator’ I mean the thing to satisfy the pentagon condition)?

Sure! Otherwise we wouldn’t care much about associators.

But I have an inkling that the answer may (disappointingly) be yes, because of something I once heard, but still don’t understand, about isomorphism classes of associators corresponding to 3-cocycles in some cohomological sense…

That’s exactly right, at least for

2-groups: monoidal categories where every morphism has an inverse and every object has a weak inverse. 2-groups are classified up to equivalence by: 1) the group of isomorphism classes of objects, G, 2) the group of automorphisms of the unit object, an abelian group A, 3) the obvious action of G on A, and 4) a functiona: G^3 -> A

coming from the associator. This function a satisfies an equation – the pentagon identity in disguise – which is also known as the “3-cocycle condition” in group cohomology. And, two associators give equivalent 2-groups iff these cocycles differ by a coboundary.

This is great, because generalizing it to higher categories, we ultimately see that cohomology theory is part of n-category theory, and everything makes wonderful sense.

The idea that 2-groups are classified by 3-cocycles goes back to the thesis of Madame Sinh, a student of Grothendieck. (I follow the sexist but charming tradition of calling her “Madame” – everyone seems to do that.) Joyal and Street gave a very precise statement and proof in their paper

Braided tensor categories, but then they didn’t publish this version of their paper – they left it out of the final version. So, Aaron Lauda and I decided to include a complete proof in our paper on 2-groups. It should be easy to follow even if you’re not familiar with group cohomology.John: This is very cool! Thanks a lot.

Sorry your second comment didn’t come through immediately, btw. I had to approve it, because it contains more than one link! I’ve changed that setting now, so it ought to work in future. (I’m still getting used to this software – I’ve only just discovered how to make it call me by my name!)

Last night I figured out a simple example where there are non-isomorphic associators (which is in fact a 2-group). Take two objects

IandX, with one non-identity arrowx: X → Xwithxx=1. Define a tensor ⊗ for whichIis a strict unit, andX⊗X = I. There is a trivial associator, but you can also define an associatorαwhich is trivial except forα_X,X,X = x. These monoidal categories are not equivalent, for suppose we had a monoidal equivalenceFfrom one to the other. The underlying functor has to be the identity (that’s the only equivalence there is!) and, by considering the associator condition for the casesI⊗I⊗XandX⊗I⊗I, the monoidality part must be trivial too. But then the associator condition fails forX⊗X⊗X.Using the correspondence explained in your 2-groups paper, I guess I’ve just described a non-trivial normalized 3-cocycle on

with coefficients inZ_2. Put like that, it sounds very fancy!Z_2Thanks for posting my second comment – I tried to post it repeatedly, because it wasn’t going through. Sorry.

You write:

Last night I figured out a simple example where there are non-isomorphic associators (which is in fact a 2-group).

Cool!

Using the correspondence explained in your 2-groups paper, I guess I’ve just described a non-trivial normalized 3-cocycle on Z/2 with coefficients in Z/2. Put like that, it sounds very fancy!

Yes, but it’s all relative: I believe more mathematicians would think

thissounds very fancy: “a monoidal category with Z/2 as objects and Z/2 as the endomorphisms of any object, whose associator is not the identity”.Luckily, they’re the exact same thing! And Mac Lane played a big role in developing both group cohomology and monoidal categories:

In fact,

H^3(Z/2,Z/2) = Z/2

so there are just two inequivalent monoidal categories with Z/2 as objects and Z/2 as endomorphisms of any object.

Here’s how Mac Lane would figure this out: he’d show that the infinite-dimensional projective space

RP^infinity

was a space with fundamental group Z/2, and all higher homotopy groups

trivial. He showed this implies

H^n(RP^infinity,A) = H^n(Z/2,A)

where on the left we have cohomology of spaces and on the right we have cohomology of groups! Using this, and using standard methods of calculating cohomology of spaces, one gets

H^n(Z/2,Z/2) = Z/2

for all n.

This implies some fun stuff! For example

H^4(Z/2,Z/2) = Z/2

classifies monoidal

2-categorieswith Z/2 as objects and Z/2 as endo-2-morphisms of any object! So, there are just two of these, up to equivalence. They differ in their “pentagonator”, which defines a 4-cocycle.And so on, ad infinitum….

H

Hmm,

somethingis wrong here. Presumably with my example, though I can’t see what.In the example I gave, the only automorphism of

Iis the identity, which makes the abelian groupHtrivial. So actually I’ve described a non-trivial 3-cocycle on Z/2 with coefficients in the trivial group, which is plainly impossible. But I can’t see what I’ve done wrong…Oh yes, I get it. The problem is that the

trivialassociator fails to be natural! We havex⊗(1_X⊗1_X) = x⊗1_I = xbut(x⊗1_X)⊗1_X = 1_I⊗1_X = 1_X. The way to fix this is to add another automorphism onI, and then we do get the pair of examples that you describe.If you want more explicit calculations in the vein of Tambara & Yamagami, I did an extension of their work which you can find here:

http://www.msp.warwick.ac.uk/agt/2003/03/p025.xhtml

It gives rather explicit associators for tensor categories where there is only simple, noninvertible object. D8 and Q8 Representations, as in T-Y are an example. A

familyof examples is given by representations of the affine group over a finite field. The paper shows it’s possible to have “nonstandard” associators in these categories that look like representations of the affine group.Almost everybody that messes with these – ie, trying to solve pentagons by giving explicit matrices – inevitably goes through a stage where they say “Hey, wait, why can’t I always do this just by using the identity for all my associators!” And the answer is “the identity matrix is not the same as the identity map.” Or, “there IS no identity map between DIFFERENT OBJECTS.”