**Nearly Representable Monads and Artin Glueing**
The free monoid monad F has the following three properties:
1) Preserves connected limits
2) The extension of F to the product completion of Sets is representable
3) The Artin Glueing of F is a presheaf topos.
In a recent paper with P.T. Johnstone (`Connected limits, familial
representability and Artin glueing', to appear in Proceedings of CTCS-5
(Amsterdam, 1993), Math. Structures in Comp. Sci.) we show that in fact all
the three properties are equivalent for a functor F:P --> Set,
when P is a presheaf category, and more generally, that 1) and 3) are
equivalent conditions when also the codomain is a presheaf category.
When F is the functor part of a monad for which unit and multiplication
are cartesian, then we also provide a syntactic characterization of the
previous equivalent conditions ("strong regularity"), and we show that
the conditions on the unit and multiplication can not be omitted.
The same class of monads also appeared in a recent paper of G.M. Kelly,
where they were characterized as the clubs over the free monoid monad
in the category of Sets.
In proving that 3) implies 1), we develop a general technique for
reflecting categorical properties of the glueing to properties of the functor;
for example, we show that cartesian closedness of the glueing is already
sufficient to imply that the functor takes products to pullbacks over F(1),
and that local cartesian closedness implies that the functor preserves
pullbacks.

**Some remarks on hybrid systems**
An important part of concurrency, the study of systems with
many activities occurring together, is the study of hybrid
dynamical systems. Hybrid systems are dynamical systems with
two levels of description. On the one hand they are continuous
systems whose behaviour is governed by differential equations.
But they also have an internal structure allowing them to be
viewed as discrete machines -- they undergo discrete activities
passing through discrete states. The change from one discrete
state to another occurs over an interval of time which may be
state-dependent, and hence successive discrete actions cannot
be identified with regular interval of time. A simple example
is a heating system with a temperature controller. The
continuous activity is the heating and cooling; the discrete
activity is the switching on and off of the thermostat.
Asynchronous circuits provide another class of examples
of particular interest in this talk.
Several automata models have been proposed for analysing hybrid
systems. The aim of this lecture, reporting joint work with
Piergiulio Katis (Sydney) and Nicoletta Sabadini (Milano), is
to give a precise definition of such systems, to provide a
calculus for constructing and analysing them using a structured
kind of automata called distributive automata -- automata
constructed using the operations of a distributive category
-- and to prove a sheaf representation theorem for distributive
automata which makes the spatial distribution of state
explicit. We relate our work with the event logic of
asynchronous circuits.
Notice that this work is not the analytic study of the
observation of instantaneous events in continuous systems as
some models of concurrency appear to be. It is instead the
study of continuous systems tempered to behave in a discrete
way, and their use in synthesizing hybrid machines.

**Aspects of cosheaves**
The notion of a cosheaf on a topological space is given by duality
from that of a sheaf. It appears in the Borel-Moore Homology theory obtained
by duality from sheaf cohomology (1960). The covering space equivalent to
the notion of a cosheaf is discussed implicitly by R.H.Fox (1957) in his
theory of covering spaces with singularities. A comparison between the two
versions of cosheaves is given by G. Bergman (1991) for complete metric
spaces and extended by J. Funk (1993) for arbitrary locales.
In my paper "Cosheaves and Distributions on Toposes" (to appear in the A.Day
issue of Algebra Universalis) it is shown that Lawvere distributions on a topos
are cosheaves on a site of definition. Thus, results about distributions are
useful in order to understand cosheaves and vuiceversa. The following are
proved
Theorem 1. The forgetful 2- functor from the 2-category of bounded S-toposes
to the 2-category of cocomplete (small presented) S-indexed categories has a
left 2-adjoint.
Corollaries are: (1) the existence of the symmetric topos of a topos; (2)
calculation of limits of toposes and geometric morphisms as colimits of
cocomplete underlying S-indexed categories and inverse image parts of
geometric morphisms; (3) the theory of S-valued cosheaves on a site is
coherent.
Theorem 2. (Structure of cosheaves). For any given site, the category of
cosheaves on the site is equivalent to a category of G-cosheaves for a localic
groupoid G, where a G-cosheaf is a localic cosheaf with an action by G (unitary
and associative).
Further aspects of categories of cosheaves will be discussed. In particular,
a new construction of the symmetric topos (done in collaboration with
Aurelio Carboni) will be discussed, whereby the symmetric topos of a topos
X is given as the topos of sheaves on the lex completion of a (suitable)
site of definition for X. Unpublished work of A.M.Pitts on the lex
completion of a category with products is employed for this purpose.

**Constructing parametric models of polymorphism**
Polymorphism enforces a type discipline on the normalizing terms
of the lambda-calculus and its many extensions intend to improve the
intuition about the behaviour of terms. Parametric polymorphism tries to
extend an algebraic property at the first level of types to the higher
types. But there seems to be still a striking difference between the syntax
of polymorphism and its categorical models: the categorical models have
produced only partial results in the direction of parametricity. The
functorial contra/covariant behaviour of the type constructors has precluded
a neat development of models. In particular, it is not known if there is a
categorical parametric model.
We review Ma and Reynolds's proposal for a semantic expression of
parametricity, and recall the basic notions involved in this. In particular,
their construction of the PL-category of relations over a given PL-category
which is "suitable for parametricity". Then we present a new construction of
a parametric completion for a PL-category: the basic idea is to use
reflexive graphs of categories as introduced by O'Hearn and Tennent. The
main theorem states that the parametric completion of a PL-category which is
suitable for polymorphism satisfies parametricity (the proof uses Pitts's
representation theorem). We discuss the result presenting some instances.
Also, the construction is connected to relators and structors and we shall
suggest possible modifications to include these.
This is joint work with Edmund Robinson.

**Categorie di Maltsev e teoria dei commutatori**
A category is called a Maltsev category if every reflexive relation
is an equivalence; in the case of regular categories (in the sense of Barr)
this property corresponds exactly to commutativity for the composition
of equivalence relations.
In my talk, I will first remind basic examples and properties of Maltsev
categories and then I will study and characterize internal groupoids in this
context. Mainly, I intend to show that it is possible to have a general theory
of commutators, strongly related with the construction of the free groupoid
on a reflexive graph.

**Zermelo-Fraenkel algebras and bisimulation**
In this lecture we will use a general notion of bisimulation
to construct categorical models of ZF set theory from an axiomatic
theory of "small" maps (joint work with A. Joyal).

**Duality and Synthetic Domain Theory**
This is an introductory presentation of the basic aspects of
the Synthetic Theory of Scott Domains. We shall recall some general
results about dualities based on an object Sigma. We then present
some crucial examples where a "Sierpinsky" object Sigma enjoy very
particular properties. These will allow to deduce fundamental results
about a category of "Scott domains" such as the limit-colimit
coincidence, and (a form of) algebraic compactness.

**Evaluation Logic and Synthetic Domain Theory**
Evaluation Logic is an extension of predicate calculus with
*computational types* and *evaluation modalities*. We shall
recall previous attempts to interpret Evaluation Logic in categories
of cpos, and discuss their limitations. Then we will show that, in the
setting of Synthetic Domain Theory, it is possible to obtain an
interpretation of Evaluation Logic starting from an interpretation for
a programming language.

**Co-inductive reasoning about functional programs**
The use of positive inductive definitions pervades the
mathematical and logical foundations of computer science. The dual
concept of co-inductive (or negative inductive) definition is also
important, if somewhat less common. It often arises when constructing
and reasoning about (potentially) infinite objects. Co-inductively
defined equivalence relations (so-called `bisimulations') are commonly
associated with the study of non-deterministic and concurrent
computation. However, in this talk I hope to show that they can be a
useful tool for reasoning about equality of more traditional,
sequential functional programs.

**Toposes as cocomplete categories**
In a recent joint work in collaboration with M. Bunge, we
described the symmetric topos construction on a suitable 2-category A
of cocomplete categories, in a purely `algebraic' way. As a gift, we
were able to give a simple characterization of topos as cocomplete
categories, which is totally internal to cocomplete categories, but
uses the symmetric monad on A. The same of course applies to
sup-lattices and frames too.

**Homotopical algebra and triangulated categories**
This work studies the connections between an abstract setting for
homotopical algebra, based on homotopy kernels and cokernels, and the
well-known Puppe-Verdier notion of triangulated category.
We show that a "right-homotopical" category A (having well-behaved
homotopy cokernels, i.e. mapping cones) has a sort of weak triangulated
structure with regard to the suspension endofunctor S, called S-homotopical
category. If A is "homotopical" and "h-stable" (in a sense related to the
suspension-loop adjunction), also this structure is h-stable, i.e.
satisfies "up to homotopy" the axioms of Verdier for a triangulated
category, excepting the octahedral one, which depends on some further
elementary conditions on the cone endofunctor of A. Every S-homotopical
category can be stabilised, by two universal procedures.

**Objective Number Theory**
The possibility of encoding information about combinatorial objects
into the geometry of complex numbers is often described as one of the
mysteries of analysis. A demystifying tool was provided by S. H. Schanuel's
Como paper (in SLNM 1488) "Negative sets have Euler characteristic and
dimension" in the form of the functorial construction called the Burnside
rig of a distributive category. Schanuel has recently showed that in a finite
separable extension of the category of finite sets, the polynomial equations
satisfied by the objects have in the complex numbers only integral roots, but
that there are many combinatotial categories to which the result applies, for
example directed graphs. But there are also "infinite" examples where
abstract equations can be seen as a reflection of objective isomorphisms:
for example the Euler product formula for the zeta function has behind it
an isomorphism involving cartesian products of free monoids. My calculation
showing that the "space" of binary trees has as Euler characteristic the
complex number which is the primitive sixth root of unity, has recently
been completed by A. Blass: seven-tuples of trees can be precisely coded as
single trees by a very simple bijection, but if "7" is replaced by 5 or any
number not congruent to 1 mod 6, such bijections can only be found in a
category qualitatively more complicated.

**Categorical Completeness Results for the Simply-Typed Lambda Calculus**
Let C be a cartesian closed category (where
ccc = finite products + exponentials, not necessarily finite limits).
The talk concerns conditions on C under which beta-eta conversion is
complete for deducing all the equalities between terms of the
simply-typed lambda calculus (with finite products and an arbitrary
set of base types) which are validated by interpretations of the
calculus in C. There are two natural forms of completeness:
We say that C is COMPLETE (relative to beta-eta conversion) if, for
all identically typed closed terms M, N, it holds that M and N are
beta-eta equivalent iff, for all interpretations [.] of the
lambda-calculus in C, [M] = [N]. (Categorically, C is complete if the
class of ccc-functors from the free ccc, generated by the set of base
types, to C is collectively faithful.)
We say that C has a COMPLETE INTERPRETATION if there exists an
interpretation [.] of the lambda-calculus in C such that, for all
identically typed closed terms M, N, it holds that M and N are
beta-eta equivalent iff [M] = [N]. (Categorically, C has a complete
interpretation if there exists a faithful functor from the free ccc
to C.)
Examples. An interesting example of a CCC that has a complete
interpretation is the category of sets (this is essentially Friedman's
completeness theorem). Clearly any C that has a complete
interpretation is also complete. The converse does not hold. For
example, the category of finite sets is complete but has no complete
interpretation. Further, not every ccc is complete. Trivially, any
preorder that is cartesian closed (when viewed as a category) is not
complete.
In the talk I shall prove the following theorems, which give necessary
and sufficient conditions for the two forms of completeness to hold.
Theorem 1. C is complete if and only if it is not a preorder.
Theorem 2. C has a complete interpretation if and only if it
contains an endomorphism all of whose iterates are distinct.

**Separable Objects and Coverings in Algebra and Geometry**
The general theory of separable objects in `good'
categories will be discussed. This theory includes the theories
of commutative separable algebras, of decidable objects in
toposes and of covering spaces. The connection with general
Galois theory in categories, and in particular with
Grothendieck's Galois theory, will be considered.

**Distributive adjoint strings**
For a string of adjoint functors V -| W -| X -| Y : B --> C,
with Y fully faithful, it is often but not always the case that VY
underlies an idempotent monad on B. Y has a profunctor right
adjoint Z; it is equivalent to ask whether ZW is an idempotent
(pro)comonad and sense for any `adjoint unity and identity of
opposites'. We call such adjoint strings distributive, and give a
wide variety of examples and a characterization. From a distributive
string of length 3, such as W -| X -| Y, we can construct both
longer and shorter distributive adjoint strings using profunctor
calculus. This is joint work with R. J. Wood.