**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).