🤖 AI Summary
This work addresses consistency issues in functorial databases arising from the absence of global sections by introducing directed homotopy theory for the first time in this context. Natural transformations are interpreted as directed paths, leading to the definitions of left and right directed fibrations, as well as directed Lusternik–Schnirelmann and directed section categories. Discrete approximations are constructed via connected components of Grothendieck opfibrations and comma categories, yielding invariance and comparison theorems for these categories. The central contribution establishes that a finite connected acyclic schema admits an initial object if and only if a global section exists. Furthermore, the paper enables effective computation of the directed section category under decomposition, iteration, and data migration, particularly through strictly local sections that efficiently determine this invariant.
📝 Abstract
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.