Further reading

The axiomatic starting point is the theory of tangent categories developed by Cockett and Cruttwell [ 2014b ] . Differential bundles and fibrations [ Cockett and Cruttwell , 2018 ] , and categorical connections [ Cockett and Cruttwell , 2017 ] , show how much differential geometry can be expressed without choosing coordinates. The graphical proof of the Jacobi identity by Cockett and Cruttwell [ 2015 ] shows how 2-categorical diagrams can turn long composites of tangent structure maps into local coherence rewrites. Leung [ 2017c ] relates tangent structures to graph-presented Weil algebras and is the natural next source for the multi-infinitesimal constructions used later in the book.

For the geometry used to select intrinsic repair directions, see the original natural-gradient construction of Amari [ 1998 ] and the broader information-geometric treatment in Amari [ 2016 ] . These sources also clarify why a metric is additional statistical structure on a tangent model rather than a consequence of tangent structure alone.

For reverse transport, compare reverse derivative categories [ Cockett et al. , 2020 ] with reverse tangent categories [ Cruttwell and Lemay , 2024 ] . The former abstracts reverse differentiation; the latter is closer to transporting cotangent information through an existing tangent structure. Standard differential geometry, including flows, brackets, distributions, and Frobenius integrability, can be reviewed in Lee [ 2012 ] . These sources help separate the generic tangent lift from the application-specific interaction signatures that LINCS calls INC.