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Chris Kapulkin

Publications and source records attributed to Chris Kapulkin.

At least 19 recordsLinked to original sources
arXiv2026

Discrete homology computations by reduction to zero differentials

We develop a new algorithm for computing (persistent) discrete homology of graphs using reduction to zero differentials and active enumeration. This allows us to compute the fourth homology group of the Greene sphere, along with several previously unknown groups. We also show that persistent discrete homology computes faster than simplicial homology of Vietoris-Rips complex in the high-noise non-metric settings, making it a better choice for noisy data sets.

cs.CG↗
arXiv2026

Stability of persistent path homology of path complexes

We show stability of persistent path homology of path complexes. As a consequence, we deduce the stability of persistent path homology of hypergraphs and of sequence hypergraphs, and recover the known stability result for digraphs, originally due to Chowdhury and M\'emoli.

math.AT↗
arXiv2026

RedZeD: Computing persistent homology by Reduction to Zero Differentials

We introduce a new algorithm for computing persistent homology of Vietoris--Rips filtrations, which in many cases offers a considerable improvement both in terms of time and memory over the existing implementations of the persistence pairing algorithm. The key innovation, called active enumeration, is made possible by a new theoretical framework of Reduction to Zero Differentials (hence RedZeD) in which to view persistent homology.

cs.CG↗
arXiv2026

Towards fast computation of higher discrete homology

We develop a new algorithm for computing the second discrete homology group of a graph which is much faster when compared to existing algorithms. To do so, we identify five basic shapes, which are quotient graphs of the 3-cube with the property that the injective maps from them detect all possible 2-boundaries in the singular chain complex computing discrete homology.

cs.CG↗
arXiv2026

Discrete homotopy hypothesis for n-types

We show that discrete and classical homotopy theories are equivalent after localizing at n-equivalences for any non-negative integer n. By constructing an explicit homotopy inverse to the graph nerve functor associating an n-fibrant cubical set to a graph, we are also able to give explicit computations of several previously unknown discrete homotopy groups of boundaries of cubes and suspensions of cycles.

math.AT↗
arXiv2025

Yet another cubical type theory, but via a semantic approach

We propose a new cubical type theory, termed (self-deprecatingly) the naive cubical type theory, and study its semantics using the universe category framework, which is similar to Uemura's categories with representable morphisms. In particular, we show that this new type theory admits an interpretation in a wide variety of settings, including simplicial sets and cartesian cubical sets.

cs.LO↗
arXiv2025

(Pointed) Univalence in Universe Category Models of Type Theory

We provide a formulation of the univalence axiom in a universe category model of dependent type theory that is convenient to verify in homotopy-theoretic settings. We further develop a strengthening of the univalence axiom, called pointed univalence, that is both computationally desirable and semantically natural, and verify its closure under Artin-Wraith gluing and formation of inverse diagrams.

cs.LO↗
arXiv2025

A Toolkit for Structured Lifts

We develop a general framework for working with structured lifting problems, establishing closure and uniqueness properties of their solutions. In a subsequent paper, we apply these results to axiomatize computation rules of cubical type theory.

math.CT↗
arXiv2025

Fast computation of the first discrete homology group

We present a new algorithm for computing the first discrete homology group of a graph. By testing the algorithm on different data sets of random graphs, we find that it significantly outperforms other known algorithms.

cs.CG↗
arXiv2025

Cubical models of $\infty$-presheaves and the Bousfield-Kan formula

We construct the covariant and the cocartesian model structures on the slice categories of cubical sets and marked cubical sets, respectively. As an application, we derive a version of the Bousfield-Kan formula for arbitrary cofibrantly generated monoidal model categories satisfying Muro's axiom.

math.AT↗
arXiv2025

Derived mapping spaces of $\infty$-categories

We prove the Derived Mapping Space Lemma, which generalizes the central theorem of Cisinski's work on calculus of fractions for $\infty$-categories, and allows us to provide a unified framework for analyzing mapping spaces in localizations of ($\infty$-)categories. As an application, we give a sufficient condition for when a cubical or simplicial category is the localization of its underlying category at homotopy equivalences.

math.AT↗
arXiv2025

Topological data analysis using persistent discrete homology

We propose persistent discrete homology as a tool for topological data analysis and discuss its advantages over the existing methods. In particular, we provide empirical evidence that persistent discrete homology is more noise-resistant than persistent homology of the Vietoris-Rips complex for data coming from non-metric settings.

math.AT↗
arXiv2025

Categorical foundations of discrete dynamical systems

We develop categorical foundations of discrete dynamical systems, aimed at understanding how the structure of the system affects its dynamics. We introduce the notion of cycle sets to analyze attractors of a system, and use this to generalize multiple decomposition theorems of Kadelka, Veliz-Cuba, Murrugarra, and the last two authors from Boolean networks to arbitrary discrete dynamical systems.

math.DS↗
arXiv2025

Pushforwards in Inverse Homotopical Diagrams

We establish a sufficient condition for the category of homotopical inverse diagrams to be closed under pushforward inside the category of inverse diagrams in a fibration category.

math.CT↗
arXiv2024

Logical Structure on Inverse Functor Categories

Inspired by recent work on the categorical semantics of dependent type theories, we investigate the following question: When is logical structure (crucially, dependent-product and subobject-classifier structure) induced from a category to categories of diagrams in it? Our work offers several answers, providing a variety of conditions on both the category itself and the indexing category of diagrams. Additionally, motivated by homotopical considerations, we investigate the case when the indexing category is equipped with a class of weak equivalences and study conditions under which the localization map induces a structure-preserving functor between presheaf categories.

math.CT↗
arXiv2024

Faster computations of discrete homology

Machine computation of the discrete homology of graphs has stopped at degree two. We present an algorithm that reaches degree four. It generates the singular cubes inductively, pairing cubes one degree down instead of filtering all set maps; quotients the chain modules by the hyperoctahedral group action, over a field of sufficiently large characteristic; and shrinks the graph beforehand using homotopy invariance. The fourth homology group of the five-cycle, previously beyond the reach of machine computation, is computed in under two days.

cs.CG↗
arXiv2024

Homotopy $n$-types of cubical sets and graphs

We give a new construction of the model structure on the category of simplicial sets for homotopy $n$-types, originally due to Elvira-Donazar and Hernandez-Paricio, using a right transfer along the coskeleton functor. We observe that an analogous model structure can be constructed on the category of cubical sets, and use it to equip the category of (simple) graphs with a fibration category structure whose weak equivalences are discrete $n$-equivalences.

math.CT↗