On the homotopy structure of strongly homotopy associative algebras

Abstract

We study here the homotopy structure of Sha, the category of strongly homotopy associative algebras (sha-algebras for short) and strongly homotopy multiplicative maps, introduced by Stasheff (1963) for the study of the singular complex of a loop-space. Sha extends the category Da of associative differential (graded) algebras, by allowing for a homotopy relaxation of objects and morphisms, up to systems of homotopies of arbitrary degree. The better-known category Dash of associative d-algebras and strongly homotopy multiplicative maps is intermediate between them. To study sha-homotopies of any order and their operations, the usual cocylinder functor of d-algebras is extended to Sha, where we construct the vertical 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 acquires thus a laxified version of the homotopy structure studied by the author in previous works; the main results therein, developing homotopical algebra from the Puppe sequence to stabilisation and triangulated structures, can very likely be extended to the new axioms.