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In reply to Stasheff's question on terminology for homotopy coherent algebras:

>but now what about e.g. 1-homotopy associaitve satisfying a STRICT
>pentagon??
>perhaps strict 1-homotopy

I would say:

"2-strict sha-algebra", as motivated below.
(sha = strongly homotopy associative)

However: after the strict pentagon, this structure has a second coherence
condition for the associativity homotopy
(which disappears for monoidal categories, just because their 2-morphisms
are trivial)

____________

In a paper [*] on strongly homotopy associative (differential) algebras, I
proposed this definition (4.2; pages 38-39).

Notation: a sha-algebra is a graded module  A  with morphisms
(sort of components of a global differential  d of bar coalgebras)

   d_1: A --> A  (degree - 1; the differential)

   d_2: AoA --> A  (degree 0; the product)

   d_3: AoAoA --> A  (degree 1; the associativity 1-homotopy)
   ........
   d_n: A^n  -->  A   (degree  n - 2; the coherence n-homotopy)
   ........

( o = tensor product;  ^n  = tensor power)

under axioms

(1)  d_1.d_1 = 0
(2)  ....

(expressing  dd = 0  for the global differential).

DEF. This is called an *n-strict sha-algebra* if  d_p = 0  for  p > n.

Equivalently,  the morphisms  d_1,..., d_n  have to satisfy the original
axioms (1)  ... (n)
plus n - 1 conditions obtained from the axioms  (n+1) ... (2n - 1),
cancelling the null  d_p's
(the remaining axioms become trivial).

This gives:

1-strict = differential module

2-strict = associative differential algebra

3-strict = 1-homotopy associative differential algebra
   with strict pentagon (from axiom (3)) and axiom (4) reduced to:

   (4)   d3 (1o1od3 + 1od3o1 + d3o1o1) = 0.

_______

So far in that paper.
The name is chosen to make  d_n  the last relevant component, in the
n-strict case.

I might now (more geometrically) prefer a - 1 shift in these names, so that
the last example would be named 2-strict, in accord with the fact that the
last relevant homotopy is a an ordinary ("one-dimensional") homotopy and
everything becomes strict starting with "dimension 2".

_______

Reference:

[*] M. Grandis, On the homotopy structure of strongly homotopy associative
algebras, J. Pure Appl. Algebra 134 (1999), 15-81.

_______

Regards    MG