Further Reading

Leung’s classification gives the precise monoidal relationship between tangent structures and the category of Weil algebras [ Leung , 2017 , 2018 ] . Cockett and Cruttwell develop tangent categories as an axiomatic setting encompassing classical and synthetic differential geometry [ Cockett and Cruttwell , 2014 ] . Leung’s account also suggests viewing \(\mathsf{Weil}_1^{\mathrm{tan}}\) as a monoidal theory for tangent structure, with richer categories of Weil algebras offering probes of higher finite order. Burke and MacAdam’s involution algebroids and MacAdam’s functorial semantics supply the first-order bridge from tangent categories to Lie theory [ Burke and MacAdam , 2019 , MacAdam , 2023 ] . Drinfel’d’s Hamiltonian interpretation of the classical Yang–Baxter equation provides the distinct Poisson–Lie and Lie-bialgebraic lineage [ Drinfel’d , 1983 ] . Piaget’s assimilation–accommodation distinction gives the developmental lineage for separating change inside a maintained schema from change of the schema itself [ Piaget , 1952 , 1985 ] . Drescher supplies the direct computational precedent through predictive schema refinement and synthetic item construction [ Drescher , 1986 , 1991 ] . Mackenzie develops classical double Lie algebroids; Meinrenken and Pike construct their distinct bigraded cochain Weil algebras and characterize double compatibility by commuting horizontal and vertical differentials [ Mackenzie , 2005 , Meinrenken and Pike , 2021 , Pike , 2020 ] . Cockett, Cruttwell, and Lemay develop differential equations, complete vector fields, flows, and commuting dynamics in tangent categories, with curve objects supplying the abstract parameter of evolution [ Cockett et al. , 2021 ] .