Monoidal Structures on Span 2-Categories
Here’s what we need:
- An object called the unit object.
- For objects and , an object .
- For an object and a 1-morphism , 1-morphisms and .
- For an object and a 2-morphism , 2-morphisms and .
- For 1-morphisms and , a 2-morphism called the “tensorator”.
Notice that instead of defining the tensor product as a functor, we define its action on a single object and a single 1-morphism (in either order). Then if we have two 1-morphisms we have two ways of doing first one on one side of the tensor product, then the other on the other side. To say that is a functor would say that these two are equal, but we want to weaken this to say that there is some 2-morphism from one to the other.
Now let’s assume that we’ve got a regular monoidal structure on our category , and further that this monoidal structure preserves the pullbacks we’re assuming exist in . That is, if is a pullback of the diagram and is a pullback of the diagram , then will be a pullback of the diagram .
So what does this mean for ? Well, the monoidal structure on gives us a unit object and monoidal product objects . If we have a span and an object , we can form the spans and . If we have spans and and an arrow with and then the arrow satisfies and , and similarly the arrow satisfies and . And so we have our monoidal products of objects with 1- and 2-morphisms.
When we take spans and , we can form the following two composite spans:
where we use the assumption that the monoidal product preserves pullbacks to show that the squares in these diagrams are indeed pullback squares.
As we’ve drawn them, these two spans are the same. However, remember that the pullback in is only defined up to isomorphism. That is, when we define the pullback as a functor, we choose some isomorphism class of cones, and these diagrams say that the pullbacks we’ve drawn are isomorphic to those defined by the pullback functor. But that means that whatever the “real” pullbacks and are, they’re both isomorphic to , and that those isomorphisms play nicely with the other arrows we’ve drawn. And so there will be some isomorphism between the “real” pullbacks that make the required triangles commute, giving us our tensorator.
Therefore what we have shown is this: Given a monoidal category with pullbacks such that the monoidal structure preserves those pullbacks, we get the data for the structure of a (weak) monoidal 2-category on . Dually, we can show that given a monoidal category with pushouts, such that the monoidal structure preserves them, we get the data for a monoidal 2-category .
[UPDATE]: In my hurry to get to my second class, I overstated myself. I should have said that we have the data of the monoidal structure. The next post contains the conditions the data must satisfy.