Related papers: Quantitative Homotopy Theory in Topological Data A…
This note extends Quillen's Theorem A to a large class of categories internal to topological spaces. This allows us to show that under a mild condition a fully faithful and essentially surjective functor between such topological categories…
We propose a homology theory for locally compact spaces with ends in which the ends play a special role. The approach is motivated by results for graphs with ends, where it has been highly successful. But it was unclear how the original…
The central problem in computational algebraic topology is the computation of the homotopy groups of a given space, represented as a simplicial set. Algorithms have been found which achieve this, but the running times depend on the size of…
Summarizing topological information from datasets and maps defined on them is a central theme in topological data analysis. \textsf{Mapper}, a tool for such summarization, takes as input both a possibly high dimensional dataset and a map…
Broadly speaking the present is a homotopy complement to the book of Giraud, albeit in a couple of different ways. In the first place there is a representability theorem for maps to a topological champ (a.k.a. stack) and whence an extremely…
We use a classical result of McCord and reduction methods of finite spaces to prove a generalization of Thomason's theorem on homotopy colimits over posets. In particular this allows us to characterize the homotopy colimits of diagrams of…
We develop a homotopy theory of categories enriched in a monoidal model category V. In particular, we deal with homotopy weighted limits and colimits, and homotopy local presentability. The main result, which was known for…
Many of the properties of sectional category, topological complexity and homotopic distance are in fact derived from a small number of basic properties, which, once established, lead to all the others without further recourse to topology.…
This is the first installment of a series of papers whose aim is to lay a foundation for homotopy probability theory by establishing its basic principles and practices. The notion of a homotopy probability space is an enrichment of the…
A taming symplectic structure provides an upper bound on the area of an approximately pseudoholomorphic curve in terms of its homology class. We prove that, conversely, an almost complex manifold with such an area bound admits a taming…
Topological complexity is a numerical homotopy invariant that measures the instability of motion planning in a space. To study the topological complexity of non-simply connected spaces, Costa and Farber introduced a cohomology class whose…
A standard problem in applied topology is how to discover topological invariants of data from a noisy point cloud that approximates it. We consider the case where a sample is drawn from a properly embedded C1-submanifold without boundary in…
We prove that holomorphic maps from an open subset of a complex smooth projective curve to a complex smooth projective rationally simply connected variety can be approximated by algebraic maps for the compact-open topology. This theorem can…
In this article, we construct a cofibrantly generated model structure on the category of spaces stratified over a fixed poset, and show that it is Quillen-equivalent to a category of diagrams of simplicial sets. Then, considering all those…
We develop the theoretical foundations of a generalized Gromov-Hausdorff distance between functions on networks that has recently been applied to various subfields of topological data analysis and optimal transport. These functional…
The notion of homomorphism indistinguishability offers a combinatorial framework for characterizing equivalence relations of graphs, in particular equivalences in counting logics within finite model theory. That is, for certain graph…
Using the notion of contiguity of simplicial maps, we adapt Farber's topological complexity to the realm of simplicial complexes. We show that, for a finite simplicial complex $K$, our discretized concept recovers the topological complexity…
The tautness for a cohomology theory is formulated and studied by various authors. However, the analogous property is not considered for a homology theory. In this paper, we will define and study this very property for the Massey homology…
We compare the cohomology complex defined by Baranovsky and Sazdanovi\'{c}, that is the $E_{1}$ page of a spectral sequence converging to the homology of the configuration space depending on a graph, with the rational model for the…
In his work on singularities, expanders and topology of maps, Gromov showed, using isoperimetric inequalities in graded algebras, that every real valued map on the $n$-torus admits a fibre whose homological size is bounded below by some…