37 research outputs found
The contraction property is sufficient to guarantee the uniqueness of fixed points of endofunctors in a category of acomplete metric spaces
In de Bakker and Zucker proposed to use complete metric spaces for the semantic definition of programming languages that allow for concurrency and synchronisation. The use of the tools of metric topology has been advocated by Nivat and his colleagues already in the seventies and metric topology was successfully applied to various problems (12, 13). Recently, the question under which circumstances fixed point equations involving complete metric spaces can be (uniquely) solved has attracted attention, e.g. (1,11). The solution of such equation provides the basis for the semantics of a given language and is hence of practical relevance. In (1), a criterion for the existence of a solution, namely that the respective functor is contracting, is provided. This property together with an additional criterion, namely that the respective functor is hom-contracting, was shown in (1) to guarantee uniqueness. In this paper we show that the contraction property is already sufficient to guarantee the uniqueness
A fixed-point theorem in a category of compact metric spaces
AbstractVarious results appear in the literature for deriving existence and uniqueness of fixed points for endofunctors on categories of complete metric spaces. All these results are proved for contracting functors which satisfy some further requirements, depending on the category in question.Following a new kind of approach, based on the notion of η-isometry, we show that the sole hypothesis of contractivity is enough for proving existence and uniqueness of fixed points for endofunctors on the category of compact metric spaces and embedding-projection pairs
Some comments on CPO-semantics and metric space semantics for imperative languages
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A general theory of self-similarity
A little-known and highly economical characterization of the real interval
[0, 1], essentially due to Freyd, states that the interval is homeomorphic to
two copies of itself glued end to end, and, in a precise sense, is universal as
such. Other familiar spaces have similar universal properties; for example, the
topological simplices Delta^n may be defined as the universal family of spaces
admitting barycentric subdivision. We develop a general theory of such
universal characterizations.
This can also be regarded as a categorification of the theory of simultaneous
linear equations. We study systems of equations in which the variables
represent spaces and each space is equated to a gluing-together of the others.
One seeks the universal family of spaces satisfying the equations. We answer
all the basic questions about such systems, giving an explicit condition
equivalent to the existence of a universal solution, and an explicit
construction of it whenever it does exist.Comment: 81 pages. Supersedes arXiv:math/0411344 and arXiv:math/0411345. To
appear in Advances in Mathematics. Version 2: tiny errors correcte
Way of the dagger
A dagger category is a category equipped with a functorial way of reversing morphisms,
i.e. a contravariant involutive identity-on-objects endofunctor. Dagger categories
with additional structure have been studied under different names in categorical
quantum mechanics, algebraic field theory and homological algebra, amongst others.
In this thesis we study the dagger in its own right and show how basic category theory
adapts to dagger categories.
We develop a notion of a dagger limit that we show is suitable in the following
ways: it subsumes special cases known from the literature; dagger limits are unique
up to unitary isomorphism; a wide class of dagger limits can be built from a small
selection of them; dagger limits of a fixed shape can be phrased as dagger adjoints to
a diagonal functor; dagger limits can be built from ordinary limits in the presence of
polar decomposition; dagger limits commute with dagger colimits in many cases.
Using cofree dagger categories, the theory of dagger limits can be leveraged to
provide an enrichment-free understanding of limit-colimit coincidences in ordinary
category theory. We formalize the concept of an ambilimit, and show that it captures
known cases. As a special case, we show how to define biproducts up to isomorphism
in an arbitrary category without assuming any enrichment. Moreover, the limit-colimit
coincidence from domain theory can be generalized to the unenriched setting and we
show that, under suitable assumptions, a wide class of endofunctors has canonical fixed
points.
The theory of monads on dagger categories works best when all structure respects
the dagger: the monad and adjunctions should preserve the dagger, and the monad and
its algebras should satisfy the so-called Frobenius law. Then any monad resolves as an
adjunction, with extremal solutions given by the categories of Kleisli and Frobenius-
Eilenberg-Moore algebras, which again have a dagger.
We use dagger categories to study reversible computing. Specifically, we model reversible
effects by adapting Hughes’ arrows to dagger arrows and inverse arrows. This
captures several fundamental reversible effects, including serialization and mutable
store computations. Whereas arrows are monoids in the category of profunctors, dagger
arrows are involutive monoids in the category of profunctors, and inverse arrows
satisfy certain additional properties. These semantics inform the design of functional
reversible programs supporting side-effects