## Abstracts

On the homotopy structure of strongly homotopy associative differential algebras We present the homotopy structure of Shad, the category of strongly homotopy associative differential algebras (shad-algebras for short), also called A_infinity-algebras, introduced by Stasheff (1963) for the study of the singular complex of a loop-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, with strict objects (those of Da) and lax morphisms (those of Shad), is intermediate between them. In order to study shad-homotopies of any order and their operations, the usual cocylinder functor of d-algebras is 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 thus acquires a laxified version of the homotopy structure studied in previous works. The main results therein may very likely be extended to the new axioms, developing homotopical algebra from the Puppe sequence to stabilisation and triangulated structures, so to be available for Shad.

Liberi, coliberi, alberi Le coalgebre sono diventate uno strumento fondamentale in molte branche della matematica: combinatoria, logica, quantum group, ecc. Ma la maggior parte dei matematici non ha ancora abbastanza familiarità con i concetti basilari che riguardano questi oggetti molto naturali. Tratteremo coalgebre da un punto di vista elementare e mostreremo la connessione tra coalgebre coliberi e famiglie di alberi.

Coalgebras have become fundamental tools in many branches of mathematics - combinatorics, logic, quantum groups, etc - but most mathematicians are still unfamiliar with the basic concepts of these very natural objects. We will treat coalgebras from an elementary point of view and show the link between cofree coalgebras and families of trees.

A new model construction for higher type systems The second order lambda calculus has been introduced by Girard for proof theoretical studies and reinvented by Reynolds as a tool for the investigation of polymorphism in programming languages. As has been shown by Reynolds there are no models of this calculus in the category of sets with arbirtrary maps as morphisms. But there are models in other categories, e.g., in the category of indexed sets with effective maps as morphisms and in the category of Scott domains with continuous maps as morphisms. In this talk a new model is presented in which types are interpreted as inverse limits of chains of finite sets. The appropriate morphisms are then those mappings which can be represented as sequences of finite maps that map the i-th approximation of the domain to the i-th approximation of the codomain. The construction can be done in such a way that not all types are inhabited.

A non-replete directed complete poset The notion of S-replete object in a category introduced by Martin Hyland and Eugenio Moggi determines a "nice" reflective full subcategory: one may think it consists of those objects which have the same "algebraic" properties as S. It is used in Synthetic Domain Theory as an intrinsic notion of domain for denotational semantics. In the case of the cartesian closed category of directed complete posets, we show that S-repleteness is non-trivial, where S is the non-trivial poset on two elements. The construction of a non-replete, directed complete poset is quite general, and we discuss possible further developments. This is joint work with Michael Makkai.

Variables in categories and homotopy categories Variables in a category X are introduced, extending subobjects. Variables are well related to weak limits, as subobjects to limits; and they may be viewed as a replacement of subobjects in categories just possessing weak limits, typically homotopy categories. The Freyd embedding X --> FrX (introduced to embed the stable homotopy category of spaces into an abelian category, in his La Jolla paper on "Stable homotopy") allows one to reduce variables in X to distinguished subobjects in FrX (with respect to a canonical factorisation structure) and, loosely speaking, weak limits to limits. In particular, "homotopy variables" for a space X, with respect to the homotopy category HoTop, form a lattice Fib(X) of types of fibrations over X, which can be identified to the lattice of distinguished subobjects of X in Fr(HoTop).