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One of the central notions to emerge from the study of persistent homology is that of interleaving distance. It has found recent applications in symplectic and contact geometry, sheaf theory, computational geometry, and phylogenetics. Here we present a general study of this topic. We define interleaving of functors with common codomain as solutions to an extension problem. In order to define interleaving distance in this setting we are led to categorical generalizations of Hausdorff distance, Gromov-Hausdorff distance, and the space of metric spaces. We obtain comparisons with previous notions of interleaving via the study of future equivalences. As an application we recover a definition of shift equivalences of discrete dynamical systems.
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We examine topological properties of pointed metric measure spaces $(Y, p)$ that can be realized as the pointed Gromov-Hausdorff limit of a sequence of complete, Riemannian manifolds ${(M^n_i, p_i)}_{i=1}^{infty}$ with nonnegative Ricci curvature. Cheeger and Colding cite{ChCoI} showed that given such a sequence of Riemannian manifolds it is possible to define a measure $ u$ on the limit space $(Y, p)$. In the current work, we generalize previous results of the author to examine the relationship between the topology of $(Y, p)$ and the volume growth of $ u$. In particular, we prove a Abresch-Gromoll type excess estimate for triangles formed by limiting geodesics in the limit space. Assuming explicit volume growth lower bounds in the limit, we show that if $lim_{r to infty} frac{ u(B_p(r))}{omega_n r^n} > alpha(k,n)$, then the $k$-th group of $(Y,p)$ is trivial. The constants $alpha(k,n)$ are explicit and depend only on $n$, the dimension of the manifolds ${(M^n_i, p_i)}$, and $k$, the dimension of the homotopy in $(Y,p)$.
In many real-world applications data come as discrete metric spaces sampled around 1-dimensional filamentary structures that can be seen as metric graphs. In this paper we address the metric reconstruction problem of such filamentary structures from data sampled around them. We prove that they can be approximated, with respect to the Gromov-Hausdorff distance by well-chosen Reeb graphs (and some of their variants) and we provide an efficient and easy to implement algorithm to compute such approximations in almost linear time. We illustrate the performances of our algorithm on a few synthetic and real data sets.
Given two shapes $A$ and $B$ in the plane with Hausdorff distance $1$, is there a shape $S$ with Hausdorff distance $1/2$ to and from $A$ and $B$? The answer is always yes, and depending on convexity of $A$ and/or $B$, $S$ may be convex, connected, or disconnected. We show that our result can be generalised to give an interpolated shape between $A$ and $B$ for any interpolation variable $alpha$ between $0$ and $1$, and prove that the resulting morph has a bounded rate of change with respect to $alpha$. Finally, we explore a generalization of the concept of a Hausdorff middle to more than two input sets. We show how to approximate or compute this middle shape, and that the properties relating to the connectedness of the Hausdorff middle extend from the case with two input sets. We also give bounds on the Hausdorff distance between the middle set and the input.
We describe a category of undirected graphs which comes equipped with a faithful functor into the category of (colored) modular operads. The associated singular functor from modular operads to presheaves is fully faithful, and its essential image can be classified by a Segal condition. This theorem can be used to recover a related statement, due to Andre Joyal and Joachim Kock, concerning a larger category of undirected graphs whose functor to modular operads is not just faithful but also full.
We make precise the analogy between Goodwillies calculus of functors in homotopy theory and the differential calculus of smooth manifolds by introducing a higher-categorical framework of which both theories are examples. That framework is an extension to infinity-categories of the tangent categories of Cockett and Cruttwell (introduced originally by Rosicky). A tangent structure on an infinity-category X consists of an endofunctor on X, which plays the role of the tangent bundle construction, together with various natural transformations that mimic structure possessed by the ordinary tangent bundles of smooth manifolds and that satisfy similar conditions. The tangent bundle functor in Goodwillie calculus is Luries tangent bundle for infinity-categories, introduced to generalize the cotangent complexes of Andre, Quillen and Illusie. We show that Luries construction admits the additional structure maps and satisfies the conditions needed to form a tangent infinity-category, which we refer to as the Goodwillie tangent structure on the infinity-category of infinity-categories. Cockett and Cruttwell (and others) have started to develop various aspects of differential geometry in the abstract context of tangent categories, and we begin to apply those ideas to Goodwillie calculus. For example, we show that the role of Euclidean spaces in the calculus of manifolds is played in Goodwillie calculus by the stable infinity-categories. We also show that Goodwillies n-excisive functors are the direct analogues of n-jets of smooth maps between manifolds; to state that connection precisely, we develop a notion of tangent (infinity, 2)-category and show that Goodwillie calculus is best understood in that context.
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