```II.  Lessons from Homotopy Theory

8.  The Associahedron and Little k-Cubes Operads

A.  The associahedron operad.   If K_n is the associahedron governing ways to
multiply n things, K_n is (n-2) dimensional.

1.  Defined combinatorially using trees with n leaves.  To do this right,
it's best to apply the bar construction to the operad for semigroups,
getting a barycentric subdivision of the associahedra which is a
simplicial operad.  But, we need to explain the bar construction
before doing this; there may be a more lowbrow explanation.

2.  Defined in terms of paths in the (n-1)-cube.

a,b,c

ab,c                                    a,bc

abc

a,b,c,d

ab,c,d                    a,bc,d                   a,b,cd

abc,d                      ab,cd                   a,bcd

abcd

There's some work involved in making the "directed path space" of an
polytope with partially ordered vertices into another polytope (with
partially ordered vertices, so we can iterate?), but perhaps we can get
the idea across without too much formalism and treat the details in some
guided exercises.

3.  Defined in terms of paths of paths in the n-simplex.   Each point in
the above cube is a directed path in the n-simplex.  Here we are using the
same "directed path space" construction!  So, we need to do this construction
twice to get from the n-simplex to the (n-1)-cube to the (n-2)-dimensional
associahedron.

A topic for later: how are A_infinity categories related to Segal categories,
and does the relation between the associahedron operad and the simplex shed
light on this?  It should!

Also: what happens if we do this "directed path space" construction again?
(It also shows up for permutahedra.)

4.  Combinatorial digression on Catalan numbers and their ilk.   Counting faces
of various dimensions using structure types.

It takes n-1 distinct points in the interval to chop it into n pieces, so we
get an open (n-1)-simplex of ways to compose an n-tuple of paths.  How is this
related to the (n-2)-dimensional associahedron? - perhaps this is treated in
Sternberg's book.  Do we get an operad homomorphism?

It takes 2n points in the interval to specify n little intervals with disjoint
interiors inside it:

0 <= a_1 < b_1 <= a_2 < b_2 <= .... <=1

so we get a not-quite-open, not-quite-closed 2n-simplex of ways to compose
an n-tuple of paths.   This gives a topological operad containing the "big

Could this be more directly related to the associahedron than the "big
intervals" operad?   It's definitely better to use this when we get to
studying E_n spaces.

D.  The general concept of an A_infinity operad; what makes some better than
others.  Theorems saying that any algebra of an A_infinity operad becomes a
loop space.  Perhaps the heavier aspects of this should be put off until we do
E_k spaces and the little k-cubes operad.

i. Defined geometrically.  General theorems saying that any algebra of the
little k-cubes operad is a k-fold loop space.

ii. Relation to the k-fold tensor product of the little intervals operad.  We
need to explain the tensor product of operads here, and how an
(A tensor B)-algebra is an A-algebra in the category of B-algebras, so we
get a "loop space in the category of loop spaces" as a double loop space
(roughly).  Jerry Dunn's theorem that the k-fold tensor product of the
little intervals operad is isomorphic to the "decomposable little k-cubes"
operad, where a configuration of little k-cubes is "decomposable" if we can
build it as an operation like

(a o_2 b) o_3 (c o_1 d)

where o_i is any operation in the ith copy of the little intervals operad.
E.g.,

------------------
|   |............|
|   |xxxxx|      |
|___|xxxxx|      |
|.........|      |
|.........|      |
------------------

is a nondecomposable 4-ary operation in the little 2-cubes operation.
(See Zig Fiederowicz's talk at the 2004 IMA summer session on n-categories;
I have some notes of his on this.)

Sketch how this operad is homotopy equivalent to the little k-cubes operad
and thus "just as good" in some sense - a sense we must explain!

iii. Relation to Fiederowicz et al's work on the categorical operad for
k-fold monoidal categories.  For this we need to define the category of
"strict monoidal categories and half-lax monoidal functors", where half-lax
means that the unit is strictly preserved but the tensor product is preserved
only up to a not necessarily invertible morphism.  We can iterate this
construction and say an (n+1)-fold monoidal category is a strict monoid
in the category of n-fold monoidal categories and half-lax monoidal functors.
The category of these is equivalent to the category of algebras of a certain
operad M_k in Cat, where the objects in the category of k-ary operations look
like

(a o_2 b) o_3 (c o_1 d)                        (k = 4, n = 3 or more)

Taking the nerve of this categorical operad we get a topological operad which
is homotopy equivalent to the k-fold tensor product of the little k-cubes

Drawing the nerve of the category of 2-ary operations in M_3 is very
convincing, since it's an octahedron and thus homotopy equivalent to the
space of 2-ary operations in the little 3-cubes operation.  In general,
the space of 2-ary operations in the little k-cubes operad is homotopy
equivalent to a (k-1)-sphere, while M_k gives a cross-polytope.

Note that the category of n-ary operations in M_k is a poset.

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