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 FA,B: FA⊗FB → F(A⊗B) satisfying the usual associator condition. If we pick a particular isomorphism FI: I → FI, then for every traditional monoidal structure on C there is a unique traditional monoidal structure on D for which (F, FA,B, FI) 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.)
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.