nLab
associahedron

Idea

The associahedra or Stasheff polytopes $\{K_n\}$ are CW complexes that naturally arrange themselves into an topological operad that resolves the standard associative operad: an A-infinity-operad.

The vertices of $K_n$ correspond to ways in which one can bracket a product of $n$ variables. The edges correspond to rebracketings, the faces relate different sequences of rebracketings that lead to the same result, and so on.

The associahedra were introduced by Jim Stasheff in order to describe topological spaces equipped with a multiplication operation that is associative up to every higher coherent homotopy.

Definition

Here is the rough idea, copied, for the moment, verbatim from Markl94 p. 26 (for more details see references below):

For $n\ge 1$ the associahedron $K_n$ is an $(n-2)$-dimensional polyhedron whose $i$-dimensional cells are, for $0\le i\le n-2$, indexed by all (meaningful) insertions of $(n-i-2)$ pairs of brackets between $n$ independent indeterminants, with suitably defined incidence maps.

Illustrations

• $K_1$ is the empty set, a degenerate case not usually considered.

• $K_2$ is simply the shape of a binary operation:

  $$x\otimes y,$$x \otimes y ,

  which we interpret here as a single point.

• $K_3$ is the shape of the usual associator or associative law

  $$(x\otimes y)\otimes z\to x\otimes (y\otimes z),$$(x \otimes y) \otimes z \to x \otimes (y \otimes z) ,

  consisting of a single interval.

• $K_4$ The fourth associahedron $K_4$ is the pentagon which expresses the different ways a product of four elements may be bracketed

One can also think of this as the top-level structure of the 4th oriental. This controls in particular the pentagon identity in the definition of monoidal category, as discussed there.

• $K_5$ is the dual polyhedron to the triaugmented triangular prism

(image from the Wikimedia Commons)

A template which can be cut out and assembled into a $K_5$ can be found here.

• Rotatable illustrations of some Stasheff polyhedra can be found at

  • Les polytopes de Stasheff

  • This is part of a larger website Petites pages sur divers sujets which contains illusrations of other polyedra, too.

Relation to other structures

Relation to orientals

The above list shows that the first few Stasheff polytopes are nothing but the first few orientals. This doesn’t remain true as $n$ increases. The orientals are free strict omega-categories on simplexes as parity complexes. This means that certain interchange cells (e.g., Gray tensorators) show up as thin in the oriental description.

The first place this happens is the sixth oriental: where there are three tensorator squares and six pentagons in Stasheff’s $K_6$, the corresponding tensorator squares coming from $O(6)$ are collapsed.

It was when Todd Trimble made this point to Ross Street that Street began to think about using associahedra to define weak n-categories.

Categorified associahedra

There is a categorification of associahedra discussed in

• Stefan Forcey, A catgorification of associahedra (web)

References

The original article that defines associahedra and in which the operad $K$ that gives $A(\mathrm{\infty })$-topological spaces is implicit is

• Jim Stasheff, Homotopy associative H-spaces I, II, Trans. Amerk. Math. Soc. 108 (1963), 275–312

A textbook discussion (slightly modified) is in section 1.6 of the book

• Martin Markl, Steve Shnider, Jim Stasheff, Operads in Algebra, Topology and Physics (web)

Further explanations and references are collected at

• AMS entry on associahedra