Sandra Mantovani

Injective topological fibre spaces

We investigate injective objects with respect to the class of embeddings in the categories Top/B (Top 0/B )o f (T 0) topological fibre spaces and their relations with exponentiable morphisms. As a result, we obtain a charaterization of such injective objects as retracts of partial products of the three-point space S (S the Sierpinski space for Top0).  2002 Elsevier Science B.V. All rights reserved. MSC: 55R05; 55R70; 54B30; 18G05

Local homeomorphisms as the exponentiable morphisms in compact Hausdorff spaces

Abstract We find that the exponentiable morphisms in the category of compact Hausdorff spaces are exactly the local homeomorphisms.

T0-reflection and injective hulls of fibre spaces

Abstract We give a characterization of injective (with respect to the class of embeddings) topological fibre spaces using their T 0 -reflection, that turns out to be injective itself. We then prove that the existence of an injective hull of ( X , f ) in the category Top / B of topological fibre spaces is equivalent to the existence of an injective hull of its T 0 -reflection ( X 0 , f 0 ) in Top / B 0 (and in the category Top 0 / B 0 of T 0 topological fibre spaces).

A push forward construction and the comprehensive factorization for internal crossed modules

In a semi-abelian category, we give a categorical construction of the push forward of an internal pre-crossed module, generalizing the pushout of a short exact sequence in abelian categories. The main properties of the push forward are discussed. A simplified version is given for action accessible categories, providing examples in the categories of rings and Lie algebras. We show that push forwards can be used to obtain the crossed module version of the comprehensive factorization for internal groupoids.

Regularity of the category of Kelley spaces

We show that the cartesian closed category of compactly generated Hausdorff spaces is regular, but is neither exact, nor locally cartesian closed. In fact we find a coequalizer of an equivalence relation which is not stable under pullback.

Injective Hulls of T0 Topological Fibre Spaces

We investigate the existence of injective hulls (with respect to the class of embeddings) in the categories Top0/B of T0 topological fibre spaces over B. We prove that, if f:A→B has a restriction to the image injective in Top0/f(A), (A,f) has an injective hull in Top0/B if and only if f(A′) is locally closed in B, where A′ denotes the union of non-indiscrete fibres of f.

EPIREFLECTIVE HULLS OF SPACES OF ORDINALS IN To

Abstract A particular class of epireflective subcategories of To is investigated, exactly the epireflective hulls g(S(a) of the spaces S(a) of the ordinals with the “open half lines” topology. The topological structure of the objects of these hulls is studied, also in relation with their sobrification. Furthermore, a bijective correspondence between hulls and classes of cofinality of ordinals is found.

Exponentiable morphisms of domains

Given a map f in the category ω-Cpo of ω-complete posets, exponentiability of f in ω-Cpo easily implies exponentiability of f in the category Pos of posets, while the converse is not true. We find then the extra conditions needed on f exponentiable in Pos to be exponentiable in ω-Cpo, showing the existence of partial products of the two-point ordered set S = {0 < 1} (Theorem 1.8). Using this characterization and the embedding via the Scott topology of ω-Cpo in the category Top of topological spaces, we can compare exponentiability in each setting, obtaining that a morphism in ω-Cpo, exponentiable both in Top and in Pos, is exponentiable also in ω-Cpo. Furthermore we show that the exponentiability in Top and in Pos are independent from each other.

On Fibrations Between Internal Groupoids and Their Normalizations

We characterize fibrations and