Results

**11 - 19**of**19**### Bifibrational functorial semantics of parametric polymorphism

"... Reynolds ’ theory of parametric polymorphism captures the invariance of polymorphically typed programs under change of data representation. Reflexive graph categories and fibrations are both known to give a categorical understanding of parametric polymorphism. This paper contributes further to this ..."

Abstract
- Add to MetaCart

(Show Context)
Reynolds ’ theory of parametric polymorphism captures the invariance of polymorphically typed programs under change of data representation. Reflexive graph categories and fibrations are both known to give a categorical understanding of parametric polymorphism. This paper contributes further to this categorical perspective on parametricity by showing the relevance of bifibrations. Using bifibrations, it develops a framework for models of System F that are parametric, in that they verify the Identity Extension Lemma and Reynolds ’ Abstraction Theorem. We also prove that our models satisfy expected properties, such as the existence of initial algebras and final coalgebras, and that parametricity implies dinaturality.

### Contents

, 2006

"... Hardy and BMO spaces associated to divergence form elliptic operators ..."

(Show Context)
### AT DALHOUSIE UNIVERSITY

, 2005

"... Permission is herewith granted to Dalhousie University to circulate and to have copied for non-commercial purposes, at its discretion, the above title upon the request of individuals or institutions. ..."

Abstract
- Add to MetaCart

(Show Context)
Permission is herewith granted to Dalhousie University to circulate and to have copied for non-commercial purposes, at its discretion, the above title upon the request of individuals or institutions.

### Abstract MFPS XX1 Preliminary Version Towards “dynamic domains”: totally continuous cocomplete Q-categories

"... It is common practice in both theoretical computer science and theoretical physics to describe the (static) logic of a system by means of a complete lattice. When formalizing the dynamics of such a system, the updates of that system organize themselves quite naturally in a quantale, or more generall ..."

Abstract
- Add to MetaCart

It is common practice in both theoretical computer science and theoretical physics to describe the (static) logic of a system by means of a complete lattice. When formalizing the dynamics of such a system, the updates of that system organize themselves quite naturally in a quantale, or more generally, a quantaloid. In fact, we are lead to consider cocomplete quantaloid-enriched categories as fundamental mathematical structure for a dynamic logic common to both computer science and physics. Here we explain the theory of totally continuous cocomplete categories as generalization of the well-known theory of totally continuous suplattices. That is to say, we undertake some first steps towards a theory of “dynamic domains”.

### DISTRIBUTIVE ADJOINT STRINGS

"... ABSTRACT. For an adjoint string V a W a X a Y: B;! C, with Y fully faithful, it is frequently, but not always, the case that the composite VY underlies an idempotent monad. When it does, we call the string distributive. We also study shorter and longer `distributive ' adjoint strings and how to ..."

Abstract
- Add to MetaCart

(Show Context)
ABSTRACT. For an adjoint string V a W a X a Y: B;! C, with Y fully faithful, it is frequently, but not always, the case that the composite VY underlies an idempotent monad. When it does, we call the string distributive. We also study shorter and longer `distributive ' adjoint strings and how to generate them. These provide a new construction of the simplicial 2-category,. 1.

### Many-Valued Complete Distributivity ∗

, 2006

"... Suppose (Ω, ∗,I) is a commutative, unital quantale. Categories enriched over Ω can be studied as generalized, or many-valued, ordered structures. Because many concepts, such as complete distributivity, in lattice theory can be characterized by existence of certain adjunctions, they can be reformulat ..."

Abstract
- Add to MetaCart

(Show Context)
Suppose (Ω, ∗,I) is a commutative, unital quantale. Categories enriched over Ω can be studied as generalized, or many-valued, ordered structures. Because many concepts, such as complete distributivity, in lattice theory can be characterized by existence of certain adjunctions, they can be reformulated in the many-valued setting in terms of categorical postulations. So, it is possible, by aid of categorical machineries, to establish theories of manyvalued complete lattices, many-valued completely distributive lattices, and so on. This paper presents a systematical investigation of many-valued complete distributivity, including the topics: (1) subalgebras and quotient algebras of many-valued completely distributive lattices; (2) categories of (left adjoint) functors; and (3) the relationship between many-valued complete distributivity and properties of the quantale Ω. The results show that enriched category theory is a very useful tool in the study of many-valued versions of orderrelated mathematical entities.

### PARTIAL-SUP LATTICES

"... Abstract. The study of sup lattices teaches us the important distinction between the algebraic part of the structure (in this case suprema) and the coincidental part of the structure (in this case infima). While a sup lattice happens to have all infima, only the suprema are part of the algebraic str ..."

Abstract
- Add to MetaCart

Abstract. The study of sup lattices teaches us the important distinction between the algebraic part of the structure (in this case suprema) and the coincidental part of the structure (in this case infima). While a sup lattice happens to have all infima, only the suprema are part of the algebraic structure. Extending this idea, we look at posets that happen to have all suprema (and therefore all infima), but we will only declare some of them to be part of the algebraic structure (which we will call joins). We find that a lot of the theory of complete distributivity for sup lattices can be extended to this context. There are a lot of natural examples of completely join-distributive partial lattice complete partial orders, including for example, the lattice of all equivalence relations on a set X, and the lattice of all subgroups of a group G. In both cases we define the join operation as union. This is a partial operation, because for example, the union of subgroups of a group is not necessarily a subgroup. However, sometimes it is, and keeping track of this can help with topics such as the inclusion-exclusion principle. Another motivation for the study of sup lattices is as a simplified model for the study of presheaf categories. The construction of downsets is a form of the Yoneda embedding, and the study of downset lattices can be a useful guide for the study of presheaf cate-gories. In this context, partial lattices can be viewed as a simplified model for the study of sheaf categories.