摘要:

This paper presents a number of results concerning sheaves on a topological space, with values in the category BAN of Banach spaces, over K = R or Ø, and linear contractions. After showing that these sheaves are reflective in the corresponding category of presheaves (Proposition 1) and that the resulting reflection is stalk preserving (Proposition 2), we concentrate on theapproximation sheaves, these being BAN-sheaves satisfying a strong patching condition originally due to Auspitz [1]. The interest in these particular sheaves lies in the fact that they are precisely the BAN-sheaves arising as sheaves of continuous sections of the appropriate kind of Banach fibre spaces [1] and thus central to the representation of Banach spaces by continuous sections. Here, we show that the approximation sheaves on any space are characterized as the BAN-presheaves injective relative to certain maps (Proposition 3) and that, for paracompact spaces X, they are exactly those BAN-sheaves S such that each SU, U open in X, admits a suitable C*U-module structure (Proposition 4). Further, we consider the adjointness between the approximation sheaves on a space X and the Banach modules over C*X (Proposition 5) and investigate its special properties for X being Tychonoff (Proposition 6) and Boolean (Proposition 7). We conclude with some observations regarding the failure of the analogues of Swan's Theorem for vector bundles and the Hahn-Banach Theorem in the present context, and some positive facts concerning injectivity for approximation sheaves on Tychonoff spaces.

ISSN:1607-3606    

DOI:10.1080/16073606.1977.9632531

出版商:Taylor & Francis Group    

年代:1977

数据来源: Taylor

摘要:

A concept of normality for nearness spaces is introduced which agrees with the usual normality in the case of topological spaces, is hereditary, and is preserved under the taking of the nearness completion. It is proved that the nearness product of a regular contigual space and a normal nearness space is always normal. The locally fine nearness spaces are studied, particularly in relation to normality conditions.

ISSN:1607-3606    

DOI:10.1080/16073606.1977.9632532

出版商:Taylor & Francis Group    

年代:1977

数据来源: Taylor

摘要:

It is shown that the forgetful functor from the category of contiguity spaces to the category of generalized proximity spaces is topological, and that the right adjoint right inverse of this functor extends the inverse of the forgetful functor from the category of totally bounded uniform spaces to the category of proximity spaces.

ISSN:1607-3606    

DOI:10.1080/16073606.1977.9632533

出版商:Taylor & Francis Group    

年代:1977

数据来源: Taylor

摘要:

This paper is motivated by the search for natural extensions of classical uniform space results to quasi-uniform spaces. As instances of such extensions we restate some theorems of P. Fletcher and W.F. Lindgren [Pacific J. Math. 43 (1971), 619–6311 on transitive quasi-uniformities and of S. Salbany [Thesis, Univ. Cape Town, 1971] on compactification and completion. The theorems as restated describe properties of certain right inverses of the functor which forgets the quasi-uniform structure and retains one induced topology (for Fletcher and Lindgren's work), respectively retains both induced topologies (for Salbany's work). Accordingly we investigate systematically the process by which the right inverses of the forgetful functors can be extended from the classical setting to one of these settings, and from one of these to the other.

ISSN:1607-3606    

DOI:10.1080/16073606.1977.9632534

出版商:Taylor & Francis Group    

年代:1977

数据来源: Taylor

摘要:

A synthesis of notions arising from algebraic geometry, especially those developed by Verdier in Séminaire de Géométric Algébrique IV, and the notion of topological functor (in the sense of G.C.L. Brümmer and R.-E. Hoffmann) is made. In particular, Grothendieck topologies are shown to be topological over the category of categories with pullbacks and pullback preserving functors, and consequences derived.

ISSN:1607-3606    

DOI:10.1080/16073606.1977.9632535

出版商:Taylor & Francis Group    

年代:1977

数据来源: Taylor

摘要:

With the introduction of several new factorisation theorems, this paper is intended to show that previous efforts of the authors [3] [5] and of Strecker [15] to describe the factorisations involving connectedness are incomplete. In Section 1 we give a purely topological construction of such a factorisation, in which the right factor is the class of spreads and the left factor has a certain property hereditarily: crucially, not all members of the left factor need be quotients. Section 2 shows that, given a left factor consisting of onto maps in the category T of topological spaces, then the class of mappings with the relevant properties hereditarily is also a left factor, and the result of section 1 is a particular case of this. Section 3 combines the material in [3] on intrinsic connexion properties with ideas of Preuss (see [1]) on disconnectednesses to yield another range of factorisations, for example, involving the maps with strongly connected fibres; and Section 4 notes some outstánding problems which our work has provoked.

ISSN:1607-3606    

DOI:10.1080/16073606.1977.9632536

出版商:Taylor & Francis Group    

年代:1977

数据来源: Taylor

摘要:

Consider a commuting square of functors TV = GU where G is an algebraic functor over sets (in the sense of Herrlich), and T and U are (regular epi, monosource)—topological and fibre small. Such a square is called a Topological Algebraic Situation (TAS) when the following two conditions are satisfied:

ISSN:1607-3606    

DOI:10.1080/16073606.1977.9632537

出版商:Taylor & Francis Group    

年代:1977

数据来源: Taylor

摘要:

The first part of this paper surveys results and open questions about categories of T1-spaces on which the wallman compactification induces an epireflection. The second part proves results on spaces whose Wallman remainder is Hausdorff.

ISSN:1607-3606    

DOI:10.1080/16073606.1977.9632538

出版商:Taylor & Francis Group    

年代:1977

数据来源: Taylor

摘要:

It is well known that there is a one to one correspondence between idempotent monads in a category and reflective subcategories. In this paper it is examined what replaces the reflective subcategory if the idempotent monad is replaced (a) by a monad and (b) by a symmetric unad. It is shown that in case (a) one obtains the weakly reflective subcategory of objects injective relative to the functor part of the monad. In case (b) one obtains a proto-reflection and it is shown that (for complete categories) the associated orthogonal subcategory is reflective if and only if there exists a free monad associated to the unad.

ISSN:1607-3606    

DOI:10.1080/16073606.1977.9632539

出版商:Taylor & Francis Group    

年代:1977

数据来源: Taylor

摘要:

If a functor U has a left co-unadjoint then U can be factored through a category of semad algebras. An analogue of the Beck monadicity theory is obtained. If R is a ring without a left unit but satisfying R2= R then the category of unitary left R-modules need not be monadic over Set. The forgetful functor has, however, a left co-unadjoint for which a comparison functor is an equivalence of categories. Another example of a semadic functor is obtained by composing the forgetful functor from Abelian groups to Set with the doubling functor. The semi-adjoint situations in the senses of Medvedev and Davis are examined.

ISSN:1607-3606    

DOI:10.1080/16073606.1977.9632540

出版商:Taylor & Francis Group    

年代:1977

数据来源: Taylor