In my previous entry, I deferred the problem of defining lax monoidal functors between radical monoidal categories. But yesterday evening on the train I realised that there is a cute way to think about lax monoidal functors, which makes it possible to simply calculate the answer.
Suppose we have monoidal categories C and D, and a functor F: C → D. To give a lax monoidal structure on F is to give a monoidal structure on the comma category C↓F, such that the projections C↓F → C and C↓F → D are strict monoidal.
I wasn’t going to explain that, because it’s fun to figure it out and rather dull to explain, but in the end I decided to put an explanation at the bottom of this post (appendix B). Anyway, looking at it this way makes it obvious how to radicalise it. We want a radical monoidal structure on C↓F, such that the projections are strict monoidal in the obvious radical sense (i.e. a functor is strict monoidal when it strictly preserves the tensor, associator and unit object).
Such a monoidal structure amounts to the usual natural transformation φX,Y: FX⊗FY → F(X⊗Y), subject to the usual associativity requirement, together with an arrow φI: I → FI that is isomorphic in the comma category to
The other requirement is that the functors “tensoring with φI in the comma category” should be full and faithful. That works out equivalent to asking that the composites
(I ought to justify that last claim, because it’s not completely obvious, so I’ve done that in appendix A below.)
Ultimately all we have done here is to recover the expected definition, though this analysis makes it clear that the arrows displayed above need only be monic. In fact they will always turn out to be invertible of course, so in practice this difference is not terribly important. More importantly though, we have a compelling reason to believe that this is the right definition.
I’ll just do one of the two cases. Suppose we have a morphism, in the comma category, from (I,I,φI)⊗(A,X,f) to (I,I,φI)⊗(B,Y,g). Because tensoring with a unit in C and D is full and faithful, this morphism is (I⊗h,I⊗k) for some unique h: A → B and k: X → Y. So we have a commutative diagram like this:
The full-faithfulness condition thus boils down to the requirement that, whenever we have maps h and k that make the diagram above commute, the diagram
also commutes. Now consider the diagram
The centre and right squares always commute, and our condition requires that the left square commute whenever the outside does. It’s easy to see that this corresponds to the map
being monic, as claimed.
Suppose that C and D are traditional monoidal categories. We want to show that putting a traditional monoidal structure on C↓F, that is strictly preserved by the projections, amounts to putting a traditional lax monoidal structure on F.
An object of C↓F consists of an object A∈D, an object X∈C, and an arrow f: A → FX. A morphism m: (A,X,f) → (B,Y,g) is a pair of arrows m0: A → B and m1: X → Y such that the square
commutes. So, if we want to define a tensor product on this comma category that makes the projections strict monoidal, that means we need some operation that takes two objects (A,X,f) and (B,Y,g), and gives us a object (A⊗B,X⊗Y,h). The objects have to be A⊗B and X⊗Y if the projections are to be strict monoidal, so we only have to worry about defining h. Moreover, the projection requirement completely determines the behaviour of our tensor on morphisms. Now, to define the tensor on objects, in particular we need to define (FA,A,1FA)⊗(FB,B,1FB), which is an arrow φA,B: FA⊗FB → F(A⊗B). And that totally determines the general case, for the following reason. Suppose we have objects (A,X,f) and (B,Y,g): certainly the diagrams
commute, which means that (f,1) is a morphism (A,X,f) → (FX,X,1) in the comma category, and so is (g,1): (B,Y,g) → (FY,Y,1). Therefore the square
must also commute, where (A⊗B,X⊗Y,h) is the tensor product, in the comma category, of (A,X,f) and (B,Y,g). So h is equal to φX,Y(f⊗g).
For the unit, we need an object of the comma category that projects onto I from both ends, which is to say an arrow φI: I → FI. The three axioms in the definition of lax monoidal functor correspond to our requirement that the projections should preserve the structural isomorphisms.