Subject: Preprints

From: categories <[email-masked]>
Date: Fri, 2 Feb 1996 14:55:46 -0400 (AST)

Date: Fri, 2 Feb 1996 10:29:02 +0100
From: Marco Grandis <[email-masked]>

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: [email-masked])
***

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.