On the homotopy structure of strongly homotopy associative algebras

We study here the homotopy structure of Sha, the category of strongly homot opy associative algebras (sha-algebras for short) and strongly homotopy mul tiplicative maps, introduced by Stasheff (1963) for the study of the singul ar complex of a loop-space. Sha extends the category Da of associative diff erential (graded) algebras, by allowing for a homotopy relaxation of object s and morphisms, up to systems of homotopies of arbitrary degree. The bette r-known category Dash of associative d-algebras and strongly homotopy multi plicative maps is intermediate between them. To study sha-homotopies of any order and their operations, the usual cocyli nder functor of d-algebras is extended to Sha, where we construct the verti cal composition and reversion of homotopies (also existing in Dash, but not in Da) and homotopy pullbacks (which exist in Da, but not in Dash). Sha ac quires thus a laxified version of the homotopy structure studied by the aut hor in previous works; the main results therein, developing homotopical alg ebra from the Puppe sequence to stabilisation and triangulated structures, can very likely be extended to the new axioms. (C) 1999 Elsevier Science B. V. All rights reserved.