有向同伦、截面不变量与函子数据库
Directed Homotopy, Sectional Invariants, and Functorial Databases
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中文总结 AI 辅助
研究小范畴上的函子数据库，将自然变换视为有向同伦，引入有向纤维化等概念，定义有向Lusternik - Schnirelmann范畴和截面范畴，通过Grothendieck opfibration模型计算有向截面范畴，研究其在多种操作下的性质及刻画有限连通无环模式初始对象。
中文摘要 AI 辅助
小范畴上的数据库实例可表示为集值函子或离散opfibration。其截面对应全局一致的记录选择。若无全局截面，通过能做出一致选择的子范畴最小数量衡量全局一致性失败情况。将自然变换视为有向同伦，引入左右有向纤维化并与Grothendieck opfibration和纤维化关联。定义Lusternik - Schnirelmann范畴和截面范畴的有向版本并确立其不变性和比较性质。每个函子都有Grothendieck opfibration模型，通过严格局部截面计算有向截面范畴，还有从逗号范畴连通分支得到的规范离散逼近。对于函子数据库，研究分解、迭代和数据迁移下的有向截面范畴，通过对象非空数据库全局截面的存在性刻画有限连通无环模式的初始对象。
英文摘要
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.
