5 research outputs found
An equational notion of lifting monad
We introduce the notion of an equational lifting monad: a commutative strong monad satisfying one additional equation (valid for monads arising from partial map classifiers). We prove that any equational lifting monad has a representation by a partial map classifier such that the Kleisli category of the former fully embeds in the partial category of the latter. Thus equational lifting monads precisely capture the equational properties of partial maps as induced by partial map classifiers. The representation theorem also provides a tool for transferring non-equational properties of partial map classifiers to equational lifting monads. It is proved using a direct axiomatization of Kleisli categories of equational lifting monads. This axiomatization is of interest in its own right.
Restriction categories III: colimits, partial limits, and extensivity
A restriction category is an abstract formulation for a category of partial
maps, defined in terms of certain specified idempotents called the restriction
idempotents. All categories of partial maps are restriction categories;
conversely, a restriction category is a category of partial maps if and only if
the restriction idempotents split. Restriction categories facilitate reasoning
about partial maps as they have a purely algebraic formulation.
In this paper we consider colimits and limits in restriction categories. As
the notion of restriction category is not self-dual, we should not expect
colimits and limits in restriction categories to behave in the same manner. The
notion of colimit in the restriction context is quite straightforward, but
limits are more delicate. The suitable notion of limit turns out to be a kind
of lax limit, satisfying certain extra properties.
Of particular interest is the behaviour of the coproduct both by itself and
with respect to partial products. We explore various conditions under which the
coproducts are ``extensive'' in the sense that the total category (of the
related partial map category) becomes an extensive category. When partial
limits are present, they become ordinary limits in the total category. Thus,
when the coproducts are extensive we obtain as the total category a lextensive
category. This provides, in particular, a description of the extensive
completion of a distributive category.Comment: 39 page
Complete Axioms for Categorical Fixed-Point Operators
We give an axiomatic treatment of fixed-point operators in categories. A notion of iteration operator is defined, embodying the equational properties of iteration theories. We prove a general completeness theorem for iteration operators, relying on a new, purely syntactic characterisation of the free iteration theory. We then show how iteration operators arise in axiomatic domain theory. One result derives them from the existence of sufficiently many bifree algebras (exploiting the universal property Freyd introduced in his notion of algebraic compactness) . Another result shows that, in the presence of a parameterized natural numbers object and an equational lifting monad, any uniform fixed-point operator is necessarily an iteration operator. 1. Introduction Fixed points play a central role in domain theory. Traditionally, one works with a category such as Cppo, the category of !-continuous functions between !-complete pointed partial orders. This possesses a least-fixed-point oper..
Equational Lifting Monads
We introduce the notion of an equational lifting monad: a commutative strong monad satisfying one additional equation (valid for monads arising from partial map classifiers). We prove that any equational lifting monad has a representation as a partial map classifier such that the Kleisli category of the former fully embeds in the partial category of the latter. Thus equational lifting monads precisely capture the (partial) equational properties of partial map classifiers. The representation theorem also provides a tool for transferring non-equational properties of partial map classifiers to equational lifting monads. 1 Introduction Ever since Moggi's pioneering work [11, 12], the use of strong monads has provided a structural discipline underpinning the categorical approach to denotational semantics. The underlying idea is to make a denotational distinction between the operational notions of value and computation by modelling them in two separate categories. The category of values, C,..