The n-Category Café

Associahedra

The first pattern is the sequence of Stasheff polytopes, or associahedra. These are geometric forms that tell how to move parentheses around. There’s only one way to parenthesize one or two objects. There are two ways to parenthesize three, so we need an associator. There are five ways to parenthesize four objects, so we need a pentagonator. There are fourteen ways to parenthesize five objects, so we need a cube with three of its edges beveled.

Unitors

The next sequence tells how to cancel the unit. When we write an equation like $ab1d = abd$, we’re going from the product of four objects to the product of three. This sequence gives us a series of prisms that connect the associahedron for the product of $n$ objects to the associahedron for the product of $(n-1).$ Here are the right unitor; the left 2-unitor that connects the associator “line” to the single way of parenthesizing two objects; and the prism connecting the pentagonator for four objects to the associator for three objects.

Shuffle polytopes

Next we get ways of shuffling objects into each other. To shuffle a deck of cards, we cut it into two parts, splay the cards, and allow them to slip past our thumbs. The cards on the left hand in the same position relative to the other cards in that hand, but an arbitrary number of cards from the right hand may appear between them; by symmetry, the same is true for the cards in the right hand. The shuffle polytopes show how the ways of shuffling two lists of objects into each other are related by uses of the braiding. The smallest shuffle polytope is the braiding itself, the two ways of shuffling a single card in each hand. Next comes a triangle, the three ways of shuffling one card into two cards. After that come two polyhedra: a tetrahedron for shuffling one card into three, and a six-vertex polyhedron for shuffling two cards into two.

The polytopes above assume that the tensor product is strictly associative; when we account for the associator, it truncates corners and bevels edges:

Permutahedra

A permutohedron tiles the space it inhabits: in one dimension, it’s the braiding, which is topologically a line segment; in two dimensions, it’s the Yang-Baxter equation, a hexagon; in three dimensions, it’s a truncated octahedron. The shuffle polytopes imply that the permutohedra commute until we get to the bicategory level. In a compact closed category, the tensor product automatically satisfies the Yang-Baxter equation—but it does so in two different ways!

In a compact closed bicategory, these two hexagons form the front and back face of a coherence polytope. Kapranov and Voevodsky didn’t notice this was necessary when they defined braided semistrict monoidal 2-categories; Breen noticed the omission, so I call this the “Breen polytope”. Again, when we account for associators, various vertices get truncated.

Syllepsis

The syllepsis comes with a coherence law that relates the two triangle shuffle polytopes to each other; to make a symmetric monoidal bicategory rather than a merely sylleptic one, we also need a coherence law relating it to the identity.

Duals

In an abelian group, $aa^{-1} = 1$; in a compact closed category, this equation gets weakened to an adjunction, a pair of morphisms $e_A:A\otimes A^{\ast} \to 1$ and $i_A:1 \to A^{\ast}\otimes A$ that satisfy some “yanking” or “zigzag” equations. In a compact closed bicategory, these equations get weakened to isomorphisms and we need a “swallowtail” coherence law.

Other examples

Groupoids, spans, and maps of spans generalize the example of sets, relations, and implications. They give us a wonderful generalization of linear algebra; the groupoidifcation program is exploring the possibilities it offers. Alex Hoffnung has a paper where he shows that given a 2-category $C$ with finite products and weak pullbacks, one can form a monoidal tricategory Span($C$) of objects, spans, weak maps of spans, and maps of weak maps. If we take isomorphism classes at the highest level, we get a compact closed bicategory Span₂($C$).

There’s a deep relation between spans and profunctors. Prof, the 2-category of small categories, profunctors, and natural transformations is compact closed.

When $R$ is a rig category, we can form Mat($R$), the compact closed bicategory of natural numbers, matrices with elements in Ob($R$), and matrices with elements in Mor($R$).

These examples are all rather algebraic in flavor, but there are topological examples too, like the compact closed bicategory $n$Cob₂ of manifolds, manifolds with boundary, and manifolds with corners. (Well, it’s actually a double bicategory, but there’s a simple way of turning it into a bicategory, and that thing is compact closed.)

Monads

One can define a monad in any compact closed bicategory; the result is a category-like object.

A monad in Rel is a preorder.

If $C$ has finite products and pullbacks, a monad in Span($C$) is a category internal to $C$.

A monad in SymMonProf is an Arrow in the sense of Hughes.

A monad in Mat($R$) is one way to get an $R$-enriched category.

Traces

Simon Willerton has a great slide deck on 2-traces in compact closed bicategories. Since every object has a dual, we can take any 1-morphism $f:A\otimes X \to B\otimes X$ in a compact closed bicategory and bend the output $X$ around backwards and feed it into the input. This brings up all kinds of fun stuff about fixed points and recursion.

My paper

The reason for this post is really to advertise my new paper. After several years of working on this paper a few hours a week, it’s finally at the point where I feel like it’s ready for comments. I intend to submit it eventually to Mathematical Structures in Computer Science. It still needs some work on tuning it to that particular audience.

In the paper I give all the examples above and lay out the complete definition of a compact closed bicategory. I prove that Span₂($C$) is compact closed and work out the canonical round 2-trace with all its associators and unitors and such. I’m sure I’ve missed some recent work; I haven’t yet read Ponto and Shulman’s paper on fixed points that I mentioned above.