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
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
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grandis＠dima.unige.it -
James Stasheff