Topic
Exponential object
About: Exponential object is a research topic. Over the lifetime, 17 publications have been published within this topic receiving 70 citations. The topic is also known as: map object.
Papers
TL;DR: This paper uses the full subcategory of overt discrete objects of ASD to translate computable bases for classical spaces into objects in the ASD calculus, and shows this subcategory to be equivalent to a notion of computable basis for locally compact sober spaces or locales.
Abstract: ASD (Abstract Stone Duality) is a re-axiomatisation of general topology in
which the topology on a space is treated, not as an infinitary lattice, but as
an exponential object of the same category as the original space, with an
associated lambda-calculus. In this paper, this is shown to be equivalent to a
notion of computable basis for locally compact sober spaces or locales,
involving a family of open subspaces and accompanying family of compact ones.
This generalises Smyth's effectively given domains and Jung's strong proximity
lattices. Part of the data for a basis is the inclusion relation of compact
subspaces within open ones, which is formulated in locale theory as the
way-below relation on a continuous lattice. The finitary properties of this
relation are characterised here, including the Wilker condition for the cover
of a compact space by two open ones. The real line is used as a running
example, being closely related to Scott's domain of intervals. ASD does not use
the category of sets, but the full subcategory of overt discrete objects plays
this role; it is an arithmetic universe (pretopos with lists). In particular,
we use this subcategory to translate computable bases for classical spaces into
objects in the ASD calculus.
18 citations
TL;DR: This note offers an elementary approach which applies to quotient-reflective subcategories as well and includes a natural generalization of the compact-open topology on function spaces.
Abstract: In 1970, Day and Kelly characterized exponential spaces by a condition (C). Eight years later, Hofmann and Lawson pointed out that this is equivalent to quasi-local compactness, i.e. every neighborhood V of a point contains a smaller one W such that any open cover of V admits a finite subcover of W. These characterizations work with topologies on topologies and may be felt to be not really elementary. This note instead offers an elementary approach which applies to quotient-reflective subcategories as well and includes a natural generalization of the compact-open topology on function spaces.
12 citations
TL;DR: In this paper, it was shown that the category of all compactly generated spaces is not cartesian closed (here a compact space need not be Hausdorff) and the same result holds for the category Haus of Haus-dorff spaces.
11 citations
TL;DR: This paper uses the full subcategory of overt discrete objects of ASD to translate computable bases for classical spaces into objects in the ASD calculus, and shows this subcategory to be equivalent to a notion of computable basis for locally compact sober spaces or locales.
Abstract: ASD (Abstract Stone Duality) is a re-axiomatisation of general topology in which the topology on a space is treated, not as an infinitary lattice, but as an exponential object of the same category as the original space, with an associated lambda-calculus. In this paper, this is shown to be equivalent to a notion of computable basis for locally compact sober spaces or locales, involving a family of open subspaces and accompanying family of compact ones. This generalises Smyth's effectively given domains and Jung's strong proximity lattices. Part of the data for a basis is the inclusion relation of compact subspaces within open ones, which is formulated in locale theory as the way-below relation on a continuous lattice. The finitary properties of this relation are characterised here, including the Wilker condition for the cover of a compact space by two open ones. The real line is used as a running example, being closely related to Scott's domain of intervals. ASD does not use the category of sets, but the full subcategory of overt discrete objects plays this role; it is an arithmetic universe (pretopos with lists). In particular, we use this subcategory to translate computable bases for classical spaces into objects in the ASD calculus.
6 citations
TL;DR: It is shown that the subconstruct Fing of Prtop, consisting of all finitely generated pretopological spaces, is the largest Cartesian closed coreflective subconstruct of Pr top, which implies that in any coreflection subconstruct, exponential objects are finitelygenerated.
Abstract: We show that the subconstruct Fing of Prtop, consisting of all finitely generated pretopological spaces, is the largest Cartesian closed coreflective subconstruct of Prtop. This implies that in any coreflective subconstruct of Prtop, exponential objects are finitely generated. Moreover, in any finitely productive, coreflective subconstruct, exponential objects are precisely those objects of the subconstruct that are finitely generated. We give a counterexample showing that without finite productivity the previous result does not hold.
5 citations
Related Topics (5)
Performance Metrics
| Year | Papers |
|---|---|
| 2018 | 1 |
| 2017 | 1 |
| 2012 | 1 |
| 2008 | 1 |
| 2007 | 1 |
| 2006 | 1 |