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Basic results are obtained concerning Galois connections between collections of closure operators (of various types) and collections consisting of subclasses of (pairs of) morphisms in M for an hE; Mi-category X. In eeect, the \lattice" of closure operators on M is shown to be equivalent to the xed point lattice of the polarity induced by the orthogonality… (More)

- JÜRGEN KOSLOWSKI
- 1997

Given a bicategory, Y , with stable local coequalizers, we construct a bicategory of monads mnd (Y) by using lax functors from the generic 0-cell, 1-cell and 2-cell, respectively, into Y. Any lax functor into Y factors through mnd (Y) and the 1-cells turn out to be the familiar bimodules. The locally ordered bicategory rel and its bicategory of monads both… (More)

- M Ern, J Koslowski, A Melton, G E Strecker
- 1992

We provide the rudiments of the theory of Galois connections (or residuation theory, as it is sometimes called) together with many examples and applications. Galois connections occur in profusion and are well-known to most mathematicians who deal with order theory; they seem to be less known to topologists. However, because of their ubiquity and simplicity,… (More)

Linear bicategories are a generalization of bicategories, in which the one horizontal composition is replaced by two (linked) horizontal compositions. These compositions provide a semantic model for the tensor and par of linear logic: in particular, as composition is fundamentally noncommutative, they provide a suggestive source of models for noncommutative… (More)

- J R B Cockett, J Koslowski, R A G Seely
- 1995

Linear bicategories are a generalization of ordinary bicategories in which there are two horizontal (1-cell) compositions corresponding to the " tensor " and " par " of linear logic. Benabou's notion of a morphism (lax 2-functor) of bicategories may be generalized to linear bicategories, where they are called linear functors. Unfortunately, as for the… (More)

0. INTRODUCTION Closure operators are well-known in topology and order theory. In the setting of an hE; Mi-category for sinks we show that the categorical abstraction of the notion of closure operator is such that closure operators appear as essentially the xed points (i.e., as the Galois-closed members) of a natural Galois connection. We identify a common… (More)

In the quest for an elegant formulation of the notion of " polycategory " we develop a more symmetric counterpart to Burroni's notion of " T-category " , where T is a cartesian monad on a category X with pullbacks. Our approach involves two such monads, S and T , that are linked by a suitable generalization of a distributive law in the sense of Beck. This… (More)

The Galois connection given in 1985 by Pumpl un and RR ohrl between the classes of objects and the classes of morphisms in any category is shown (under ordinary circumstances) to have a \natural" factorization through the system of all idempotent closure operators over the category. Furthermore, each \component" of the factorization is a Galois connection… (More)

We provide a simple direct proof that for a nitary signature and a set of inequalities the free algebra functor on the category of directed complete partial orders (dcpo's) takes continuous, respectively algebraic, dcpo's to free algebras whose underlying dcpo is again continuous, respectively algebraic. 0 Introduction Consider a nitary signature and a set… (More)

R esum e. Nous introduisons des techniques nouvelles pour obtenir, dans un con-texte cat egorique, un analogue compl etement sym etrique de la fermeture reguli ere de Salbany. Pour cette investigation, nous utilisons des correspondances de Galois et des structures de factorisation cribl ee. Nous obtenons des factorisations pour les plus importantes… (More)