Mac Lane’s Coherence Theorem
Okay, as I promised yesterday, I’m going to prove the Coherence Theorem for monoidal categories today. That is, any two natural transformations built from , , and between any two functors built from and are equal once we know the pentagon and triangle identities are satisfied.
Here’s an example. Brace yourself for the diagram.
Okay, the outer pentagon is the pentagon identity for the quadruple . The triangles in the upper-right and on the left are the triangle identity. The quadrilaterals are all naturality squares for and . And the last triangle near the center is the identity we’re proving. It says that . There’s a similar diagram to show that , which you should find.
Now we’re ready to really get down to work. We can get any functor built from and by writing down a parenthesized sequence of multiplications like and filling some of the open slots with .
First, let’s get rid of those identity objects. Each of them shows up either to the right or to the left of some product symbol, so we can use either a or a to remove it, but there’s some choice involved here. For example, near the we’re trying to get rid of the functor might look like . We can either use a right away, or we can hit it with an associator and then use a . The triangle identity says that these two ways of removing the are the same. Similarly, if nearby the functor looks like or we have some leeway, but the two new triangles we proved above say we’ll get the same natural transformation to remove the either way. The upshot is that any two natural transformations we use to reach a functor without any slots filled with identity objects are the same.
So now we just have to deal with functors built from monoidal products. Each of them is just a different way to parenthesize a multiplication of some number of variables, and the associator (or its inverse) lets us move between different ways of parenthesizing the same number of variables. One convenient way of parenthesizing is to have all pairs of parentheses at the end, like , and what we’re going to do is try to make every functor look like this.
Now imagine a big diagram with all the different ways of parenthesizing a product of terms as its vertices, and draw an arrow from one vertex to another if you can go from the first to the second by applying one associator. When , this is just the pentagon diagram. Try actually drawing the one for (there should be 14 vertices). We’re allowed to walk forwards or backwards along the arrows, since the associator is an isomorphism. Every path is a natural isomorphism between the two functors at its ends, since it’s a composition of natural isomorphisms. What the Coherence Theorem says is that any two paths between the same two points give the same natural isomorphism.
Let’s say that we have a path from a vertex in this diagram to a vertex , which might consist of some forward arrows and some reverse arrows. Each time we change directions from forward to reverse we can pick a path using only forward arrows from the turning point to the vertex , which has all its parentheses to the right. Here’s a sketch of what I’m talking about
Each arrow here is a path consisting of only forward arrows. We start with the path along the top. At the two points where we change from walking forwards along the arrows to walking backwards, we pick a path made up of only forward arrows to the point . Now the path we started with is the same as walking forwards a step from , down to , back the way we just came (undoing the isomorphism), forwards to the next juncture, down to and back, and on to .
It seems we’ve made our lives more complicated by adding in these side-trips to , just to undo them immediately, but something really nice happens. There are two paths here from the middle point of our original path down to , and each of them consists only of forward arrows. If these two paths are the same, I can cut out that whole middle loop. Then our path walks straight down from to along forward arrows, then back up to along backward arrows. If there are more changing points in the original path, then there are just more loops to cut out. So if any two paths from a vertex to consisting of only forward arrows give the same isomorphism, then every path from to is the same as one passing through . And even better, the paths from and to themselves consist entirely of forward arrows, so there’s only one of each of them. That is, if we can show that any two paths made of forward arrows from an arbitrary vertex to the fixed vertex give the same natural isomorphism, then we’re done.
Now here’s where it gets a bit messy. We’ll need to use inductions on both the number of terms in our functors and on how close they are to . We know the Coherence Theorem works up to terms, because that’s the pentagon identity. We’ll prove it for terms assuming it’s known for fewer than terms and work our way up. Similarly, for a fixed we’ll start by establishing the result for the vertices closest to and work our way out.
Let’s start with a functor , which is the product of functors and . Let and be two arrows out of to functors and , respectively, which them continue on to paths to . We need to find a and paths from and to to make the square commute. Since and are both closer than to , we can assume that both paths from to in the diagram give the same isomorphism, as do both paths from to . Then since is even closer to we can assume there’s only one transformation from to , so the result follows for . Here’s another sketch:
If and are already the same there’s nothing to be done. Pick and go home. Otherwise there are three possibilities for
- applies an associator inside
- applies an associator inside
- , and is an associator
and the same three for .
If and both act within or within then we’re done because of the induction on the number of terms. If acts in one factor and in the other then they commute because is a functor of two variables. All that’s left to see is if is an associator and acts in either or .
If acts inside , sending it to , then we get by the naturality of . A similar statement holds if acts inside or . Finally, if is itself the associator , then and are actually the first steps around the pentagon identity, so we can use the far vertex of the pentagon as our .
And after all these cases, we’re done. If two paths start out differently from we can guide them back together so that the square where they differed commutes, and we can keep doing this until we get to . Thus there’s only one natural isomorphism built from associators taking a functor the the fully right-parenthesized functor on the same number of terms.
Thus there’s only one natural isomorphism built from associators between any two functors built from monoidal products on the same number of terms.
Thus any two natural transformations built from , , and between any two functors built from and are equal, given that the pentagon and the triangle identities are satisfied.