6 Infinitesimal Categorical Databases

Tangent and Bracket-Enriched CSQL

Categorical databases turn a schema into a category and a database instance into a functor. This makes tables, foreign keys, path equations, queries, and data migration part of one compositional language [ Spivak , 2012 , 2014 ] . The original Democritus construction used this language to turn local causal graphs extracted from discourse into a queryable CSQL atlas [ Mahadevan , 2025d ] . Chapter 5 changes the local causal object. A smooth intervention protocol now produces vector fields, and Lie brackets test whether the visible intervention directions close.

This chapter asks what database object should retain that infinitesimal information. The answer has three layers. First, a database instance valued in a tangent category has a pointwise tangent lift. Second, a constrained database sketch admits that lift only when its declared constraints are tangent-stable. Third, Lie brackets require vector fields and a bracket operation in addition to the tangent functor. The resulting object is a bracket-enriched infinitesimal database: a queryable instance that stores generators, brackets, closure witnesses, quotient classes, and provenance without collapsing them into an ordinary graph.

¹1. The construction is not obtained by attaching derivatives to columns after the fact. The schema must declare which infinitesimal objects exist, how they migrate, and which bracket or closure relations are intended. ↩