arXiv subjects

Jana K. Nickel

Publications and source records attributed to Jana K. Nickel.

4 recordsLinked to original sources

arXiv2026

A model structure for cartesian 2-fibrations

Cartesian 2-fibrations provide a way to understand indexed categories, but their classical ``straightening'' construction requires several layers of weak coherence data. This paper develops a homotopical framework that replaces much of this bookkeeping with a fully strict model. By using marked 2-categories to record the cartesian morphisms and 2-cells, we construct a model structure whose fibrant objects are precisely the cartesian 2-fibrations over a fixed 2-category $\mathcal{C}$. We then show that the marked Grothendieck construction identifies these 2-fibrations, up to weak equivalence, with strict 2-functors from $\mathcal{C}$ into $2\mathrm{Cat}$. As an additional contribution, we construct localizations of 2-categories that simultaneously invert selected morphisms and 2-cells.

math.CT↗

arXiv2026

Time-Varying Data as Sheaves: an Invitation to Narratives

Modern science and engineering increasingly rely on time-varying data, yet the mathematical tools used to model temporal phenomena are often developed within separate disciplines, obscuring common principles and limiting the transfer of ideas across fields. This chapter presents the theory of narratives, an abstract framework for time-varying objects of any mathematical kind that supports both theoretical investigations and applications. To illustrate this perspective, the chapter develops three vignettes, each illustrating a different research direction. The first addresses a general concern: What information loss can occur when switching between different representations of temporal data? The second concerns structural and algorithmic approaches: How can we systematically decompose time-varying data into simple pieces and obtain invariants describing its structural complexity? The third is an application to control theory: How can we model multi-agent systems with switching communication topologies? More important than any individual vignette, the central message of this invitation is that a suitable abstract perspective can organize and guide research across remarkably diverse mathematical and scientific domains.

math.CT↗

arXiv2026

Decomposing time-varying data into simple pieces: structured decompositions of narratives

Graphs that change over time arise throughout applications, but there is no single standard way to decompose them into smaller pieces. In this paper, we propose a systematic categorical method for doing so. The main idea is to combine structured decompositions, which generalize graph decompositions, such as tree-decompositions, with persistent narratives, which model time-varying data as diagrams. We prove that, under suitable categorical hypotheses, any static theory of decompositions can be lifted to a corresponding temporal theory. As case studies, we apply this construction to time-varying graphs and recover natural temporal analogues of ordinary tree-width, complemented tree-width, and the tree-independence number.

math.CT↗

arXiv2025

Disjoint $X$-paths in bidirected graphs

Let $B$ be a bidirected multigraph with signing $σ$, let $X$ be a set of vertices in $B$, and let $k$ be a non-negative integer. For any pair of vertex sets $S,T\subset V(B)$ satisfying $X\cap S = X\cap T$, we denote by $B_{S,T}$ the multigraph with the same vertex set as $B$ and with edge set consisting of those edges $e$ of $B$ each of whose endvertices $v$ satisfies $v\notin S\cup T$ or $v\in S\setminus T$, $σ(v,e)=-$ or $v\in T\setminus S$, $σ(v,e)=+$. We prove that $B$ admits a set of $k$ pairwise disjoint $X$-paths if and only if for any $S,T\subseteq V(B)$ with $X\cap S = X\cap T$, the inequality $\left\lvert S\cap T \right\rvert +\sum \lfloor \tfrac{1}{2} \left\lvert V(C)\cap (X\cup S\cup T) \right\rvert \rfloor \geq k$ holds where the sum is indexed by the components of $B_{S,T}$. This result is a generalization of a result of Gallai from undirected graphs to bidirected ones. Furthermore, we will deduce from this a kind of an Erdős-Pósa property for $X$-paths in bidirected multigraphs.

math.CO↗