The following two preprints will soon be available. A "hard" copy will be sent on request. With best regards, Marco Grandis Dipartimento di Matematica Universita' di Genova Via Dodecaneso 35 16146 Genova, Italy (E-mail: grandis@dima.unige.it) *** 1. M. Grandis, Categorically algebraic foundations for homotopical algebra, Dip. Mat. Univ. Genova, Preprint 293 (1996). Abstract. We investigate a structure for an abstract cylinder endofunctor I which produces a good basis for homotopical algebra. It essentially consists of the usual operations (faces, degeneracies, connections, symmetries, composition) together with a transformation I^2 -> I^2, which we call lens collapse after its realisation in the standard topological case. This structure, if somewhat heavy, has the interest of being "categorically algebraic", i.e. based on operations on functors. Consequently, it can be naturally lifted from a category A to its categories of diagrams A^S and its slice categories A\X, A/X. Further, the dual structure, based on a cocylinder (or path) endofunctor P can be lifted to the category of A-valued sheaves on a site, whenever P preserves limits, and to the category of internal monoids in A, with respect to any monoidal structure of A consistent with P. 2. M. Grandis, On the homotopy structure of strongly homotopy associative differential algebras, Dip. Mat. Univ. Genova, Preprint 294 (1996). Abstract. We study here the homotopy structure of Shad, the category of strongly homotopy associative d-algebras (shad-algebras for short), also called A_infinity-algebras and introduced by Stasheff ([St], 1963) for the study of the singular complex of the loop-space of a pointed topological space. Shad 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 differential algebras and strongly homotopy multiplicative maps (Stasheff-Halperin [StH], Munkholm [Mu1-4]), having strict objects (the ones of Da) and lax morphisms (the ones of Shad) is intermediate between them. A crucial advantage of Shad over its subcategories Dash and Da is the homotopy invariance property proved by Gugenheim - Stasheff [GuS]. In order to study shad-homotopies of any order and their operations, the usual cocylinder functor of d-algebras is here extended to Shad, 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). Shad 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, so to be available for Shad.