Powers and Exponential Objects in Initially Structured Categories and Applications to Categories of Limit Spaces

Schwarz, Friedhelm

doi:10.1080/16073606.1983.9632302

Cited by 25 publications

(6 citation statements)

References 21 publications

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“…The facts stated in the above proposition can be found in [11,18,29,30,13]. In these references, it is also shown that the Isbell topology is splitting.…”

Section: Proofmentioning

confidence: 77%

Comparing Cartesian closed categories of (core) compactly generated spaces

Escardó

Lawson

Simpson³

2004

Topology and its Applications

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It is well known that, although the category of topological spaces is not cartesian closed, it possesses many cartesian closed full subcategories, e.g.: (i) compactly generated Hausdorff spaces; (ii) quotients of locally compact Hausdorff spaces, which form a larger category; (iii) quotients of locally compact spaces without separation axiom, which form an even larger one; (iv) quotients of core compact spaces, which is at least as large as the previous; (v) sequential spaces, which are strictly included in (ii); and (vi) quotients of countably based spaces, which are strictly included in the category (v).We give a simple and uniform proof of cartesian closedness for many categories of topological spaces, including (ii)-(v), and implicitly (i), and we also give a self-contained proof that (vi) is cartesian closed. Our main aim, however, is to compare the categories (i)-(vi), and others like them.When restricted to Hausdorff spaces, (ii)-(iv) collapse to (i), and most non-Hausdorff spaces of interest, such as those which occur in domain theory, are already in (ii). Regarding the cartesian closed structure, finite products coincide in (i)-(vi). Function spaces are characterized as coreflections of both the Isbell and natural topologies. In general, the function spaces differ between the categories, but those of (vi) coincide with those in any of the larger categories (ii)-(v). Finally, the topologies of the spaces in the categories (i)-(iv) are analysed in terms of Lawson duality.MSC (2000): 54D50, 54D55, 54C35, 06B35.Lemma 2·1. For X exponentiable, the evaluation mappingProof. The transpose of E is the identity map on [X ⇒ Y ], hence continuous, and thus E is continuous since X is exponentiable.Lemma 2·2. For X exponentiable, the exponential topology on C(X, Y ) is uniquely determined.Proof. Suppose that [X ⇒ Y ] and [X ⇒ Y ] are C(X, Y ) endowed with topologies that satisfy the properties of an exponential. By Lemma 2·1 the evaluation map E : [X ⇒ Y ] × X → Y is continuous, and then the identity function on C(X, Y ) from [X ⇒ Y ] to [X ⇒ Y ] is continuous since the latter is an exponential. Reversing the roles of [X ⇒ Y ] and [X ⇒ Y ] gives continuity of the identity in the reverse direction.In light of the preceding observation we denote by [X ⇒ Y ] the set C(X, Y ) of continuous maps endowed with the exponential topology.Lemma 2·3. For X exponentiable and g ∈ C(Y, Z), the mapis continuous by Lemma 2·1, and thus its transpose, which one sees directly to be [X ⇒ g], is continuous.It follows from the preceding that every exponentiable X gives rise to a functor [X ⇒ ·] : Top → Top defined by Y → [X ⇒ Y ] and g → [X ⇒ g]. One observes directly from the fact that [X ⇒ Y ] is an exponential that this functor is right adjoint to the functor · × X. Lemma 2·4. The product of two exponentiable spaces is exponentiable.Proof. Let X 1 and X 2 be exponentiable spaces. Then for all spaces A, one has bijections. Hence the exponential topology on C(X 1 , [X 2 ⇒ Y ]) induces an exponential topology on C(X 1 × X 2 , Y ).Given a fa...

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“…The facts stated in the above proposition can be found in [11,18,29,30,13]. In these references, it is also shown that the Isbell topology is splitting.…”

Section: Proofmentioning

confidence: 77%

Comparing Cartesian closed categories of (core) compactly generated spaces

Escardó

Lawson

Simpson³

2004

Topology and its Applications

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“…Let us also recall the following, which is a special case of Theorem 3.3 from [8], since it was shown in [7] that UAP is finally dense in PSAP: Theorem 2.1 (Schwarz [8]). For every Y ∈ |UAP| the following assertions are equivalent:…”

Section: Resultsmentioning

confidence: 99%

On the multitude of monoidal closed structures on UAP

Löwen

Sioen

2004

Topology and its Applications

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In this note, we prove that all compact Hausdorff topological spaces are exponential objects in the category UAP of uniform approach spaces and contractions as introduced in R. Lowen, Approach Spaces: the Missing Link in the Topology-Uniformity-Metric Triad, Oxford University Press, 1997. As a consequence, we show that UAP admits at least as many monoidal closed structures as there are infinite cardinals. We also prove that under the assumption that no measurable cardinals exist, there exists a proper conglomerate of these monoidal closed structures on UAP.

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“…It is known [18] that a topology is T -exponential if and only if it is core compact, that is, for every x and each neighborhood O of x there is a neighborhood V of x which is compactoid in O [13]. We shall refine this classical result.…”

Section: Compactoidnessmentioning

confidence: 85%