Abstract. 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 the i-essentiality of homological critical values conclude the paper.
Rank invariants of multidimensional persistent homology groups are a parameterized 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
We investigate injectivity in a comma-category C/B using the notion of the "object of sections" S(f ) of a given morphism f : X → B in C. We first obtain that f : X → B is injective in C/B if and only if the morphism 1 X , f : X → X × B is a section in C/B and the object S(f ) of sections of f is injective in C. Using this approach, we study injective objects f with respect to the class of embeddings in the categories ContL/B (AlgL/B) of continuous (algebraic) lattices over B. As a result, we obtain both topological (every fiber of f has maximum and minimum elements and f is open and closed) and algebraic (f is a complete lattice homomorphism) characterizations.
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).
Abstract. The natural pseudo-distance is a similarity measure conceived for the purpose of comparing shapes. In this paper we revisit this pseudo-distance from the point of view of quotients. In particular, we show that the natural pseudo-distance coincides with the quotient pseudo-metric on the space of continuous functions on a compact manifold, endowed with the uniform convergence metric, modulo self-homeomorphisms of the manifold. As applications of this result, the natural pseudo-distance is shown to be actually a metric on a number of function subspaces such as the space of topological embeddings, of isometries, and of simple Morse functions on surfaces.
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