In `higher homotopy theory', terminology has not setled down nor is it transparent homotopy ___________ algebra can mean a variety of things letting ______________ = associative it can mean JUST that there is a homtopy for associaitivity or some authors use it to mean A_\infty which I initially tried to indicate by strongly homtopy associative _\infty seems to have caught on to mean the presence of higher homtopies of all orders in most but not all cases, such algebras have a homtopy invariant defintion so I would suggest the following revisionist terminology 1-homotopy associative means JUST that there is a homotopy for associaitivity similarly n-homotopy associative would mean homotopies of homotopies of... homotopy invariant ___ algebra would mean just what it says so far so good but now what about e.g. 1-homotopy associaitve satisfying a STRICT pentagon?? perhaps strict 1-homotopy open to suggestions Jim Stasheff jds@math.upenn.edu Home page: www.math.unc.edu/Faculty/jds As of July 1, 2002, I am Professor Emeritus at UNC and I will be visiting U Penn but for hard copy the relevant address is: 146 Woodland Dr Lansdale PA 19446 (215)822-6707