arXiv subjects

Isaac Carcacía-Campos

Publications and source records attributed to Isaac Carcacía-Campos.

7 recordsLinked to original sources

arXiv2026

Reducing and Classifying Fiber Bundles over Small Categories

A fiber bundle over a small category is a locally constant family of categories whose transition functors encode how a fixed fiber is transported over the base. Its Grothendieck construction assembles this data into a category over the base, while its global behavior is governed by monodromy. For finite acyclic categories, relative beat objects provide reductions of the total category over the fixed base, leading to relative cores that exist and are unique up to isomorphism. Monodromy classifies categorical fiber bundles by non-abelian cohomology and describes their strict gauge groups and sections. We also prove a fundamental-groupoid version of Quillen's Theorem~A, which gives reductions of the base that are more general than beat reductions and preserve the classification of bundles with fixed fiber. The results are illustrated by explicit finite examples.

math.CT↗

arXiv2026

Motion planning and topological complexity for diagrams of spaces

A diagram of spaces describes a system of interacting state spaces. Motion planning in such a system is not merely an objectwise problem since the selected paths must form a natural family. This leads to a notion of topological complexity for diagrams, defined as the sectional category of their endpoint evaluation map. The role of points in this setting is played by orbits, namely diagrams whose colimit is a point. Different choices of admissible orbit shapes give rise to different levels of categorical compression. We consider the families of representable, discrete and arbitrary topological orbits, and the corresponding orbit-relative Lusternik--Schnirelmann categories. Their relationship with sectional category provides upper bounds for the topological complexity of a diagram, while its values and suitable inverse limits provide lower bounds. For diagrams indexed by a finite discrete group, the construction recovers equivariant LS-category, equivariant sectional category and equivariant topological complexity.

math.AT↗

arXiv2026

Vector fields, initial scaffolds and database reduction

Reduction replaces a mathematical object with a simpler model that retains the relevant information. We introduce left and right vector fields on small categories as tools for reducing finite acyclic categories while preserving their directed homotopical information. We relate these fields to directed deformation retracts and beat-object reductions, and show that right vector-field reductions preserve the directed sectional category of right directed fibrations and the global sections of functorial databases. We also extend initial scaffolds from posets to acyclic categories. These provide smaller indexing categories that preserve limits and, in particular, globally coherent selections in databases. Finally, we prove that initial scaffolds are preserved by right directed deformation retracts and hence by right vector-field reductions.

math.CT↗

arXiv2026

Directed Homotopy, Sectional Invariants, and Functorial Databases

A database instance on a small category may be represented as a set-valued functor or, equivalently, as a discrete opfibration. Its sections correspond to globally coherent choices of records. When no global section exists, we measure the failure of global coherence by the minimum number of subcategories on which coherent choices can be made. Regarding natural transformations as directed homotopies, we introduce right and left directed fibrations and relate them to Grothendieck opfibrations and fibrations. We define directed versions of Lusternik-Schnirelmann category and sectional category and establish their invariance and comparison properties. Every functor admits a Grothendieck opfibration model on which directed sectional category is computed by strict local sections, together with a canonical discrete approximation obtained from connected components of comma categories. For functorial databases, we study directed sectional category under decomposition, iteration, and data migration, and characterize initial objects of finite connected acyclic schemas through the existence of global sections of objectwise non-empty databases.

math.CT↗

arXiv2026

Subword representations and weak hypercube dimension for acyclic categories

We introduce a categorical analogue of weak hypercube representations of finite posets by means of faithful embeddings into categories of subwords of finite words. For finite acyclic categories, we characterize those admitting such a weak subword representation: they are precisely the monic categories whose hom-sets carry a left-compatible local total order. The proof is constructive and gives an explicit word representation. We also introduce a query game for categories, generalizing a Boolean query game for posets, and show how winning sets produce explicit word representations and hence upper bounds for the weak word dimension.

math.CO↗

arXiv2026

Weak and Strong Fibrations of Functors

We develop a homotopical framework for small categories that extends classical invarints of algebraic topology to the categorical setting. Our approach is based on the construction of genuine path category, obtained trough a localization procedure, which allows us to define strong and weak fibrations for functor. We establish their basic properties, introduce a fibrant replacement for functors, and extend homotopical invariants such as the Svarc genus and sectional category to small categories. Finally, we apply this framework to motion planning in small categories, providing categorical analogues of Farber's topological complexity while removing finiteness constraints typical of existing approaches.

math.CT↗

arXiv2026

Baues-Wirsching Cohomology and Svarc Genus in Small Categories

We prove that for a bifibration P between small categories, the lenght of the cup product in the kernel of the induced morphism in the Baues-Wirsching cohomology with coefficients in any natural system is a lower bound for the homotopic sectional category (also called Svarc genus). Our results extend classical Svarc type inequalties to the categorical setting and introduce a computationally efficient method via a reduced cochain complex for Baues-Wirching cohomology.

math.CT↗