# Cartesian closed coreflective hulls

TL;DR: In this article, conditions are established under which a given class of objects in a topological category will have a Cartesian closed coreflective hull, and the main theorem is used to discover new Cartesian-closed topological categories and to unify a diversity of known special results.

Abstract: Conditions are established under which a given class of objects in a topological category will have a Cartesian closed coreflective hull. The main theorem is used to discover new Cartesian closed topological categories and to unify a diversity of known special results. It also provides a mild criterion for the existence of Cartesian closed topological hulls.

##### Citations

More filters

••

TL;DR: In this paper, the authors discuss the properties of topological hulls and provide several illuminating examples, including concrete quasitopos hull and Wyler completion, for structured sets.

32 citations

••

01 Sep 1986TL;DR: The compact-open topology for function spaces is usually attributed to R. H. Fox in 1945 as mentioned in this paper, but there is no earlier publication to attribute it to. But it is clear from Fox's paper that the idea of the compact-Open topology and its notable success in locally compact spaces were already familiar.

Abstract: The compact–open topology for function spaces is usually attributed to R. H. Fox in 1945 [16]; and indeed, there is no earlier publication to attribute it to. But it is clear from Fox's paper that the idea of the compact–open topology, and its notable success in locally compact spaces, were already familiar. The topology of course goes back to Riemann; and to generalize to locally compact spaces needs only a definition or two. The actual contributions of Fox were (1) to formulate the partial result, and the problem of extending it, clearly and categorically; (2) to show that in separable metric spaces there is no extension beyond locally compact spaces; (3) to anticipate, partially and somewhat awkwardly, the idea of changing the category so as to save the functorial equation. (Scholarly reservations: Fox attributes the question to Hurewicz, and doubtless Hurewicz had a share in (1). As for (2), when Fox's paper was published R. Arens was completing a dissertation which gave a more general result [1] – though worse formulated.)

31 citations

••

TL;DR: In this article, a unified theory of function spaces Mj^(Y, Z) with set-open topologies was developed, the sets in question being the continuous images of selected classes of topological spaces.

Abstract: This paper develops a unified theory of function spaces Mj^(Y, Z) with set-open topologies, the sets in question being the continuous images of selected classes of topological spaces ^ We prove that at least five of these function spaces are distinct and have corresponding exponential homeomorphisms θ: M^ (X, MSΛY, Z)) ~ M^ (X XS/Y, Z) for suitably retopologized product spaces I X y F , Singleton spaces are normally identities with respect to these products and so we have determined four distinct monoidal closed structures for the category of all spaces. Conditions for the category of spaces generated by sf, i.e., the coreflective hull of J^, to be cartesian closed and/or convenient are given. One result asserts that the category of sequential spaces is the smallest convenient category.

29 citations

••

TL;DR: This paper discusses several open problems in categorical topology that deal with full subconstructs or superconstructs of the constructTop of topological spaces and continuous maps.

Abstract: In this paper, a pendant to a recent survey paper, the authors discuss several open problems in categorical topology. The emphasis is on topology-oriented problems rather than on more general category-oriented ones. In fact, most problems deal with full subconstructs or superconstructs of the constructTop of topological spaces and continuous maps.

28 citations

••

TL;DR: In this paper, it is shown that reduction to finally and initially dense classes is possible for epireflective subcategories of the category of limit spaces containing a finite non-indiscrete space.

Abstract: Generalizing results of Herrlich and Nel, the author characterizes by means of smallest proper structures those objects X of an initially structured category for which X x—has a right adjoint, and describes the corresponding function spaces. It is shown that reduction to finally and initially dense classes is possible. The results are applied to epireflective subcategories of the category of limit spaces containing a finite non-indiscrete space, in particular to epireflective subcategories of TOP.

27 citations

##### References

More filters

•

01 Jan 1971

TL;DR: In this article, the authors present a table of abstractions for categories, including Axioms for Categories, Functors, Natural Transformations, and Adjoints for Preorders.

Abstract: I. Categories, Functors and Natural Transformations.- 1. Axioms for Categories.- 2. Categories.- 3. Functors.- 4. Natural Transformations.- 5. Monics, Epis, and Zeros.- 6. Foundations.- 7. Large Categories.- 8. Hom-sets.- II. Constructions on Categories.- 1. Duality.- 2. Contravariance and Opposites.- 3. Products of Categories.- 4. Functor Categories.- 5. The Category of All Categories.- 6. Comma Categories.- 7. Graphs and Free Categories.- 8. Quotient Categories.- III. Universals and Limits.- 1. Universal Arrows.- 2. The Yoneda Lemma.- 3. Coproducts and Colimits.- 4. Products and Limits.- 5. Categories with Finite Products.- 6. Groups in Categories.- IV. Adjoints.- 1. Adjunctions.- 2. Examples of Adjoints.- 3. Reflective Subcategories.- 4. Equivalence of Categories.- 5. Adjoints for Preorders.- 6. Cartesian Closed Categories.- 7. Transformations of Adjoints.- 8. Composition of Adjoints.- V. Limits.- 1. Creation of Limits.- 2. Limits by Products and Equalizers.- 3. Limits with Parameters.- 4. Preservation of Limits.- 5. Adjoints on Limits.- 6. Freyd's Adjoint Functor Theorem.- 7. Subobjects and Generators.- 8. The Special Adjoint Functor Theorem.- 9. Adjoints in Topology.- VI. Monads and Algebras.- 1. Monads in a Category.- 2. Algebras for a Monad.- 3. The Comparison with Algebras.- 4. Words and Free Semigroups.- 5. Free Algebras for a Monad.- 6. Split Coequalizers.- 7. Beck's Theorem.- 8. Algebras are T-algebras.- 9. Compact Hausdorff Spaces.- VII. Monoids.- 1. Monoidal Categories.- 2. Coherence.- 3. Monoids.- 4. Actions.- 5. The Simplicial Category.- 6. Monads and Homology.- 7. Closed Categories.- 8. Compactly Generated Spaces.- 9. Loops and Suspensions.- VIII. Abelian Categories.- 1. Kernels and Cokernels.- 2. Additive Categories.- 3. Abelian Categories.- 4. Diagram Lemmas.- IX. Special Limits.- 1. Filtered Limits.- 2. Interchange of Limits.- 3. Final Functors.- 4. Diagonal Naturality.- 5. Ends.- 6. Coends.- 7. Ends with Parameters.- 8. Iterated Ends and Limits.- X. Kan Extensions.- 1. Adjoints and Limits.- 2. Weak Universality.- 3. The Kan Extension.- 4. Kan Extensions as Coends.- 5. Pointwise Kan Extensions.- 6. Density.- 7. All Concepts are Kan Extensions.- Table of Terminology.

9,254 citations

••

458 citations

••

138 citations

••

86 citations

••

01 Feb 1977

TL;DR: In this article, it is shown that if a concrete category 9 admits embedding as a full finitely productive subcategory of a cartesian closed topological (CCT) category, then W admits such embedding into a smallest CCT category, its CCT hull.

Abstract: ABsTRAcr. It is shown in this paper that if a concrete category 9 admits embedding as a full finitely productive subcategory of a cartesian closed topological (CCT) category, then W admits such embedding into a smallest CCT category, its CCT hull. This hull is characterized internally by means of density properties and externally by means of a universal property. The problem is posed of whether every topological category has a CCT hull.

49 citations