Weak and Strong Fibrations of Functors

TL;DR

This paper develops a homotopical framework for small categories, extending classical invariants and introducing fibrations, with applications to motion planning and topological complexity.

Contribution

It introduces a new homotopical approach to small categories, defining strong and weak fibrations and extending invariants like sectional category.

Findings

Established properties of fibrations in small categories.

Extended homotopical invariants such as Svarc genus.

Applied framework to categorical motion planning, removing finiteness constraints.

Abstract

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.