Results 1  10
of
23
An Algebraic Construction of Predicate Transformers
 Science of Computer Programming
, 1994
"... . In this paper we present an algebraic construction of monotonic predicate transformers, using a categorical construction which is similar to the algebraic construction of the integers from the natural numbers. When applied to the category of sets and total functions once, it yields a category isom ..."
Abstract

Cited by 20 (1 self)
 Add to MetaCart
. In this paper we present an algebraic construction of monotonic predicate transformers, using a categorical construction which is similar to the algebraic construction of the integers from the natural numbers. When applied to the category of sets and total functions once, it yields a category isomorphic to the category of sets and relations; a second application yields a category isomorphic to the category of monotonic predicate transformers. This hierarchy cannot be extended further: the category of total functions is not itself an instance of the categorical construction, and can only be extended by it twice. 1 Introduction Predicate transformers were introduced originally by Dijkstra [8] in order to provide an elegant semantics for his programming language. Their strength lies in the fact that they can be used to model nondeterministic and nonterminating behaviour in terms of total functions, rather than relations. Not all monotonic predicate transformers represent programs in ...
One Setting for All: Metric, Topology, Uniformity, Approach Structure
"... For a complete lattice V which, as a category, is monoidal closed, and for a suitable Setmonad T we consider (T, V)algebras and introduce (T, V)proalgebras, in generalization of Lawvere's presentation of metric spaces and Barr's presentation of topological spaces. In this laxalgebraic setting, u ..."
Abstract

Cited by 18 (10 self)
 Add to MetaCart
For a complete lattice V which, as a category, is monoidal closed, and for a suitable Setmonad T we consider (T, V)algebras and introduce (T, V)proalgebras, in generalization of Lawvere's presentation of metric spaces and Barr's presentation of topological spaces. In this laxalgebraic setting, uniform spaces appear as proalgebras. Since the corresponding categories behave functorially both in T and in V, one establishes a network of functors at the general level which describe the basic connections between the structures mentioned by the title. Categories of (T, V)algebras and of (T, V)proalgebras turn out to be topological over Set.
Between Functions and Relations in Calculating Programs
, 1992
"... This thesis is about the calculational approach to programming, in which one derives programs from specifications. One such calculational paradigm is Ruby, the relational calculus developed by Jones and Sheeran for describing and designing circuits. We identify two shortcomings with derivations made ..."
Abstract

Cited by 15 (4 self)
 Add to MetaCart
This thesis is about the calculational approach to programming, in which one derives programs from specifications. One such calculational paradigm is Ruby, the relational calculus developed by Jones and Sheeran for describing and designing circuits. We identify two shortcomings with derivations made using Ruby. The first is that the notion of a program being an implementation of a specification has never been made precise. The second is to do with types. Fundamental to the use of type information in deriving programs is the idea of having types as special kinds of programs. In Ruby, types are partial equivalence relations (pers). Unfortunately, manipulating some formulae involving types has proved difficult within Ruby. In particular, the preconditions of the `induction' laws that are much used within program derivation often work out to be assertions about types; such assertions have typically been verified either by informal arguments or by using predicate calculus, rather than by ap...
Metric, Topology and Multicategory  A Common Approach
 J. Pure Appl. Algebra
, 2001
"... For a symmetric monoidalclosed category V and a suitable monad T on the category of sets, we introduce the notion of reflexive and transitive (T , V)algebra and show that various old and new structures are instances of such algebras. Lawvere's presentation of a metric space as a Vcategory is incl ..."
Abstract

Cited by 12 (7 self)
 Add to MetaCart
For a symmetric monoidalclosed category V and a suitable monad T on the category of sets, we introduce the notion of reflexive and transitive (T , V)algebra and show that various old and new structures are instances of such algebras. Lawvere's presentation of a metric space as a Vcategory is included in our setting, via the BettiCarboniStreetWalters interpretation of a Vcategory as a monad in the bicategory of Vmatrices, and so are Barr's presentation of topological spaces as lax algebras, Lowen's approach spaces, and Lambek's multicategories, which enjoy renewed interest in the study of ncategories. As a further example, we introduce a new structure called ultracategory which simultaneously generalizes the notions of topological space and of category.
The Convergence Approach to Exponentiable Maps
 352 MARIA MANUEL CLEMENTINO, DIRK HOFMANN AND WALTER
, 2000
"... Exponentiable maps in the category Top of topological spaces are characterized by an easy ultrafilterinterpolation property, in generalization of a recent result by Pisani for spaces. From this characterization we deduce that perfect (= proper and separated) maps are exponentiable, generalizing the ..."
Abstract

Cited by 9 (7 self)
 Add to MetaCart
Exponentiable maps in the category Top of topological spaces are characterized by an easy ultrafilterinterpolation property, in generalization of a recent result by Pisani for spaces. From this characterization we deduce that perfect (= proper and separated) maps are exponentiable, generalizing the classical result for compact Hausdorff spaces. Furthermore, in generalization of the WhiteheadMichael characterization of locally compact Hausdorff spaces, we characterize exponentiable maps of Top between Hausdorff spaces as restrictions of perfect maps to open subspaces.
Canonical and opcanonical lax algebras
 Theory Appl. Categ
, 2005
"... Abstract. The definition of a category of (T, V)algebras, where V is a unital commutative quantale and T is a Setmonad, requires the existence of a certain lax extensionof T. In this article, we present a general construction of such an extension. This leads tothe formation of two categories of ( ..."
Abstract

Cited by 7 (2 self)
 Add to MetaCart
Abstract. The definition of a category of (T, V)algebras, where V is a unital commutative quantale and T is a Setmonad, requires the existence of a certain lax extensionof T. In this article, we present a general construction of such an extension. This leads tothe formation of two categories of (
Exponentiability in Categories of Lax Algebras
, 2003
"... For a complete cartesianclosed category V with coproducts, and for any pointed endofunctor T of the category of sets satisfying a suitable BeckChevalleytype condition, it is shown that the category of lax reflexive (T , V)algebras is a quasitopos. This result encompasses many known and new examp ..."
Abstract

Cited by 6 (2 self)
 Add to MetaCart
For a complete cartesianclosed category V with coproducts, and for any pointed endofunctor T of the category of sets satisfying a suitable BeckChevalleytype condition, it is shown that the category of lax reflexive (T , V)algebras is a quasitopos. This result encompasses many known and new examples of quasitopoi. 1.
Convergence In Exponentiable Spaces
 Theory Appl. Categories
, 1999
"... . Exponentiable spaces are characterized in terms of convergence. More precisely, we prove that a relation R : UX * X between ultrafilters and elements of a set X is the convergence relation for a quasilocallycompact (that is, exponentiable) topology on X if and only if the following conditions ar ..."
Abstract

Cited by 6 (1 self)
 Add to MetaCart
. Exponentiable spaces are characterized in terms of convergence. More precisely, we prove that a relation R : UX * X between ultrafilters and elements of a set X is the convergence relation for a quasilocallycompact (that is, exponentiable) topology on X if and only if the following conditions are satisfied: 1. id ` R ffi j 2. R ffi UR = R ffi ¯ where j : X ! UX and ¯ : U(UX) ! UX are the unit and the multiplication of the ultrafilter monad, and U : Rel ! Rel extends the ultrafilter functor U : Set ! Set to the category of sets and relations. (U ; j; ¯) fails to be a monad on Rel only because j is not a strict natural transformation. So, exponentiable spaces are the lax (with respect to the unit law) algebras for a lax monad on Rel . Strict algebras are exponentiable and T 1 spaces. 1. Introduction In [4] it was implicitly proved that a topological space is exponentiable if and only if its lattice of open sets is a continuous lattice [6, 8], so fixing an important topological pr...
An Australian conspectus of higher categories

, 2004
"... Much Australian work on categories is part of, or relevant to, the development of higher categories and their theory. In this note, I hope to describe some of the origins and achievements of our efforts that they might perchance serve as a guide to the development of aspects of higherdimensional wo ..."
Abstract

Cited by 6 (0 self)
 Add to MetaCart
Much Australian work on categories is part of, or relevant to, the development of higher categories and their theory. In this note, I hope to describe some of the origins and achievements of our efforts that they might perchance serve as a guide to the development of aspects of higherdimensional work. I trust that the somewhat autobiographical style will add interest rather than be a distraction. For so long I have felt rather apologetic when describing how categories might be helpful to other mathematicians; I have often felt even worse when mentioning enriched and higher categories to category theorists. This is not to say that I have doubted the value of our work, rather that I have felt slowed down by the continual pressure to defend it. At last, at this meeting, I feel justified in speaking freely amongst motivated researchers who know the need for the subject is well established. Australian Category Theory has its roots in homology theory: more precisely, in the treatment of the cohomology ring and the Künneth formulas in the book by Hilton and Wylie [HW]. The first edition of the book had a mistake concerning the cohomology ring of a product. The Künneth formulas arise from splittings of the natural short exact sequences
Lawvere completeness in Topology
, 2008
"... It is known since 1973 that Lawvere’s notion of (Cauchy)complete enriched category is meaningful for metric spaces: it captures exactly Cauchycomplete metric spaces. In this paper we introduce the corresponding notion of Lawvere completeness for (Ì, V)categories and show that it has an interestin ..."
Abstract

Cited by 5 (3 self)
 Add to MetaCart
It is known since 1973 that Lawvere’s notion of (Cauchy)complete enriched category is meaningful for metric spaces: it captures exactly Cauchycomplete metric spaces. In this paper we introduce the corresponding notion of Lawvere completeness for (Ì, V)categories and show that it has an interesting meaning for topological spaces and quasiuniform spaces: for the former ones means weak sobriety while for the latter means Cauchy completeness. Further, we show that V has a canonical (Ì, V)category structure which plays a key role: it is Lawverecomplete under reasonable conditions on the setting; permits us to define a Yoneda embedding in the realm of (Ì, V)categories.