# The Unapologetic Mathematician

## Monoidal Structures on Span 2-Categories

Now we want to take our 2-categories of spans and add some 2-categorical analogue of a monoidal structure on it.

Here’s what we need:

• An object $\mathbf{1}$ called the unit object.
• For objects $A_1$ and $A_2$, an object $A_1\otimes A_2$.
• For an object $A$ and a 1-morphism $f:B\rightarrow C$, 1-morphisms $A\otimes f:A\otimes B\rightarrow A\otimes C$ and $f\otimes A:B\otimes A\rightarrow C\otimes A$.
• For an object $A$ and a 2-morphism $\alpha:f\Rightarrow g$, 2-morphisms $A\otimes\alpha:A\otimes f\Rightarrow A\otimes g$ and $\alpha\otimes A:f\otimes A\Rightarrow g\otimes A$.
• For 1-morphisms $f:A\rightarrow A'$ and $g:B\rightarrow B'$, a 2-morphism $\bigotimes_{f,g}:(f\otimes B')\circ(A\otimes g)\Rightarrow(A'\otimes g)\circ(f\otimes B)$ 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 $\underline{\hphantom{X}}\otimes\underline{\hphantom{X}}$ 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 $\mathcal{C}$, and further that this monoidal structure preserves the pullbacks we’re assuming exist in $\mathcal{C}$. That is, if $D_1$ is a pullback of the diagram $A_1\rightarrow C_1\leftarrow B_1$ and $D_2$ is a pullback of the diagram $A_2\rightarrow C_2\leftarrow B_2$, then $D_1\otimes D_2$ will be a pullback of the diagram $A_1\otimes A_2\rightarrow C_1\otimes C_2\leftarrow B_1\otimes B_2$.

So what does this mean for $\mathbf{Span}(\mathcal{C})$? Well, the monoidal structure on $\mathcal{C}$ gives us a unit object $\mathbf{1}$ and monoidal product objects $A\otimes B$. If we have a span $B\stackrel{f_1}{\leftarrow}X\stackrel{f_2}{\rightarrow}C$ and an object $A$, we can form the spans $A\otimes B\stackrel{1_A\otimes f_1}{\leftarrow}A\otimes X\stackrel{1_A\otimes f_2}{\rightarrow}A\otimes C$ and $B\otimes A\stackrel{f_1\otimes 1_A}{\leftarrow}X\otimes A\stackrel{f_2\otimes 1_A}{\rightarrow}A\otimes C$. If we have spans $B\stackrel{f_1}{\leftarrow}X\stackrel{f_2}{\rightarrow}C$ and $B\stackrel{g_1}{\leftarrow}Y\stackrel{g_2}{\rightarrow}C$ and an arrow $\alpha:X\rightarrow Y$ with $f_1=\alpha\circ g_1$ and $f_2=\alpha\circ g_2$ then the arrow $1_A\otimes\alpha:A\otimes X\rightarrow A\otimes Y$ satisfies $1_A\otimes f_1=1_A\otimes\alpha\circ1_A\otimes g_1$ and $1_A\otimes f_2=1_A\otimes\alpha\circ1_A\otimes g_2$, and similarly the arrow $\alpha\otimes1_A:X\otimes A\rightarrow Y\otimes A$ satisfies $f_1\otimes1_A=\alpha\otimes1_A\circ g_1\otimes1_A$ and $f_2\otimes1_A=\alpha\otimes1_A\circ g_2\otimes1_A$. And so we have our monoidal products of objects with 1- and 2-morphisms.

When we take spans $f=A\stackrel{f_1}{\leftarrow}A''\stackrel{f_2}{\rightarrow}A'$ and $g=B\stackrel{g_1}{\leftarrow}B''\stackrel{g_2}{\rightarrow}B'$, 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 $\mathcal{C}$ 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 $C_1$ and $C_2$ are, they’re both isomorphic to $A''\otimes B''$, and that those isomorphisms play nicely with the other arrows we’ve drawn. And so there will be some isomorphism $\bigotimes_{f,g}:C_1\rightarrow C_2$ 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 $\mathcal{C}$ with pullbacks such that the monoidal structure preserves those pullbacks, we get the data for the structure of a (weak) monoidal 2-category on $\mathbf{Span}(\mathcal{C})$. Dually, we can show that given a monoidal category $\mathcal{C}$ with pushouts, such that the monoidal structure preserves them, we get the data for a monoidal 2-category $\mathbf{CoSpan}(\mathcal{C})$.

[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.

October 10, 2007 - Posted by | Category theory