Francesca Cagliari

One-dimensional reduction of multidimensional persistent homology

A recent result on size functions is extended to higher homology modules: the persistent homology based on a multidimensional measuring function is reduced to a 1-dimensional one. This leads to a stable distance for multidimensional persistent homology. Some reflections on i-essentiality of homological critical values conclude the paper.

Size Functions from a Categorical Viewpoint

A new categorical approach to size functions is given. Using this point of view, it is shown that size functions of a Morse map, f: M→ℜ can be computed through the 0-dimensional homology. This result is extended to the homology of arbitrary degree in order to obtain new invariants of the shape of the graph of the given map.

Finiteness of rank invariants of multidimensional persistent homology groups

Rank invariants are a parametrized version of Betti numbers of a space multi-filtered by a continuous vector-valued function. In this note we give a sufficient condition for their finiteness. This condition is sharp for spaces embeddable in R^n.

Preservation of topological properties under exponentiation

Abstract We show that, given a subcategory C of Top closed under refinement and products, exponentiations of morphisms in Top and in C do coincide. As a consequence, we find that a map in Haus is exponentiable if and only if it is open and locally perfect.

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).

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 and not totally reflective subcategories of top

Abstract A negative answer to the conjecture made by Dyckhoff in [4] is given. A sufficient condition under which an extremal epireflective subcategory of Top is not totally reflective is proved.