Full Reflective Subcategory Research Articles

Some free constructions in realizability and proof theory

Some old and new constructions of free categories with good properties (regularity, exactness, etc.) are investigated, consistently showing their role in proof theory and in realizability theory, and in particular in the construction of the “effective topos” of M. Hyland. The subject of “small complete categories” is discussed, with a proposed new definition of what “complete” should mean for a full reflective subcategory of a topos.

Open localizations

An essential localization of a finitely complete category C is a full reflective subcategory D whose reflection has itself a left adjoint. D can then be seen as a C -indexed category and the localization is open when the adjunctions are indexed ones. We give equivalent conditions for the openness of a localization and prove that, with good conditions on C , open localizations constitute a locale.

The Dedekind-MacNeille completion as a reflector

We introduce a special type of order-preserving maps between quasiordered sets, the so-called cut-stable maps. These form the largest morphism class such that the corresponding category of quasiordered sets contains the category of complete lattices and complete homomorphisms as a full reflective subcategory, the reflector being given by the Dedekind-MacNeille completion (alias normal completion or completion by cuts). Suitable restriction of the object class leads to the category of separated quasiordered sets and its full reflective subcategory of completely distributive lattices. Similar reflections are obtained for continuous lattices, algebraic lattices, etc.

Completion of L-fuzzy relations

It is shown that the category L ̃ of complete L-similarities on L-sets is a full reflective subcategory of R (L) ( L-fuzzy graphs); L ̃ is equivalent to Lh (L)(sheaves on L). Connections with other known “fuzzy” categories are also studied.

Triples on Reflective Subcategories of Functor Categories

We show that if $\mathcal {S}$ is a cocontinuous triple on a full reflective subcategory of a functor category then the category of $\mathcal {S}$-algebras is again a full reflective subcategory of a functor category.

Triples on reflective subcategories of functor categories

We show that if S \mathcal {S} is a cocontinuous triple on a full reflective subcategory of a functor category then the category of S \mathcal {S} -algebras is again a full reflective subcategory of a functor category.

GFB-NBS strain tester for rubber and rubber-like compounds

Computational effects can be modelled by observationally-induced algebras, which are algebras whose structure is completely determined by a chosen computational prototype. We show that the category of continuous maps between topological spaces supports a free observationally-induced algebra construction for arbitrary pre-chosen prototype, and give a characterisation of the free algebras as subalgebras of certain powers of these prototypes.Moreover, we present observationally-induced lower and upper powerspace constructions in the category of topological spaces. Our lower powerspace construction is for all topological spaces given by its non-empty closed subsets equipped with the lower Vietoris topology. Dually, our upper powerdomain construction is for a wide class of topological spaces given by the space of proper open filters of its topology equipped with the upper Vietoris topology. Thus, both constructions generalise the classical construction on continuous dcpos, and unify abstract and concrete characterisations of powerdomains on a broader scale.Finally, we show that with a small adjustment of the definitions, observationally-induced algebras form the smallest full reflective subcategory of the category of algebras for the corresponding signature, which contains the computational prototype and is complete and closed under isomorphism.