14 research outputs found
Homotopy theory of diagrams
In this paper we develop homotopy theoretical methods for studying diagrams.
In particular we explain how to construct homotopy colimits and limits in an
arbitrary model category. The key concept we introduce is that of a model
approximation. Our key result says that if a category admits a model
approximation then so does any diagram category with values in this category.
From the homotopy theoretical point of view categories with model
approximations have similar properties to those of model categories. They admit
homotopy categories (localizations with respect to weak equivalences). They
also can be used to construct derived functors by taking the analogs of fibrant
and cofibrant replacements.
A category with weak equivalences can have several useful model
approximations. We take advantage of this possibility and in each situation
choose one that suits our needs. In this way we prove all the fundamental
properties of the homotopy colimit and limit: Fubini Theorem (the homotopy
colimit -respectively limit- commutes with itself), Thomason's theorem about
diagrams indexed by Grothendieck constructions, and cofinality statements.
Since the model approximations we present here consist of certain functors
"indexed by spaces", the key role in all our arguments is played by the
geometric nature of the indexing categories.Comment: 95 pages with inde
Combinatorial presentation of multidimensional persistent homology
A multifiltration is a functor indexed by that maps any
morphism to a monomorphism. The goal of this paper is to describe in an
explicit and combinatorial way the natural -graded -module structure on the homology of a multifiltration of simplicial
complexes. To do that we study multifiltrations of sets and vector spaces. We
prove in particular that the -graded -modules
that can occur as -spans of multifiltrations of sets are the direct sums of
monomial ideals.Comment: 21 pages, 3 figure
Homotopy excision and cellularity
Consider a push-out diagram of spaces C B, construct the homotopy
push-out, and then the homotopy pull-back of the diagram one gets by forgetting
the initial object A. We compare the difference between A and this homotopy
pull-back. This difference is measured in terms of the homotopy fibers of the
original maps. Restricting our attention to the connectivity of these maps, we
recover the classical Blakers-Massey Theorem.Comment: 22 pages, we took special care in this revised version in
distinguishing fiber sets from single fibers, in indicating what we mean by
the loop space on a possibly non-connected and unpointed space, thus
smoothing the expositio
Landscapes of data sets and functoriality of persistent homology
The aim of this article is to describe a new perspective on functoriality of
persistent homology and explain its intrinsic symmetry that is often
overlooked. A data set for us is a finite collection of functions, called
measurements, with a finite domain. Such a data set might contain internal
symmetries which are effectively captured by the action of a set of the domain
endomorphisms. Different choices of the set of endomorphisms encode different
symmetries of the data set. We describe various category structures on such
enriched data sets and prove some of their properties such as decompositions
and morphism formations. We also describe a data structure, based on coloured
directed graphs, which is convenient to encode the mentioned enrichment. We
show that persistent homology preserves only some aspects of these landscapes
of enriched data sets however not all. In other words persistent homology is
not a functor on the entire category of enriched data sets. Nevertheless we
show that persistent homology is functorial locally. We use the concept of
equivariant operators to capture some of the information missed by persistent
homology
Homotopical decompositions of simplicial and Vietoris Rips complexes
Motivated by applications in Topological Data Analysis, we consider
decompositions of a simplicial complex induced by a cover of its vertices. We
study how the homotopy type of such decompositions approximates the homotopy of
the simplicial complex itself. The difference between the simplicial complex
and such an approximation is quantitatively measured by means of the so called
obstruction complexes. Our general machinery is then specialized to clique
complexes, Vietoris-Rips complexes and Vietoris-Rips complexes of metric
gluings. For the latter we give metric conditions which allow to recover the
first and zero-th homology of the gluing from the respective homologies of the
components