lin-0093

Functorial databases and the adjoint data-migration pattern begin with Spivak [ 2012 ] ; query and constraint semantics via lifting problems are developed by Spivak [ 2014 ] . Tangent categories and their abstract tangent-bundle structure are introduced by Cockett and Cruttwell [ 2014b ] . Involution algebroids provide one route toward Lie-algebroid structure in tangent categories [ Burke and MacAdam , 2019 ] . For the causal semantics of intervention fields and bracket closure, see Mahadevan [ 2026f ] ; for the associated latent-confounded causal-discovery application, see Mahadevan [ 2026b ] . The present chapter combines these lines to define the mathematical interface among tangent instances, bracket enrichment, and categorical queries.