Further Reading

The foundational material is treated more broadly in Categories for AGI and more deeply for structural diagnosis and repair in the LINCS book [ Mahadevan , 2026b , 2026e ] . For categories, sketches, and functorial models, see Barr and Wells [ 1999 ] , Lawvere [ 1963 ] , and Ehresmann [ 1968 ] . Mac Lane and Moerdijk develop sheaf and topos semantics [ Mac Lane and Moerdijk , 1992 ] . Cockett and Cruttwell introduce tangent categories as an abstract differential setting, and Leung relates ordinary tangent structure to monoidal theories of Weil algebras [ Cockett and Cruttwell , 2014 , Leung , 2017 , 2018 ] . Burke and MacAdam introduce involution algebroids as a tangent-categorical generalization of Lie algebroids, and MacAdam develops their functorial semantics [ Burke and MacAdam , 2019 , MacAdam , 2023 ] . Mackenzie develops classical double Lie algebroids and distinguishes their tangent, matched-pair, and cotangent Lie-bialgebroid realizations [ Mackenzie , 2000a , 2005 ] . Meinrenken and Pike characterize double compatibility through the commuting differentials of a distinct bigraded cochain Weil algebra [ Meinrenken and Pike , 2021 , Pike , 2020 ] . Gracia-Saz, Jotz Lean, Mackenzie, and Mehta give the equivalent splitting-dependent presentation by matched pairs of 2-term representations up to homotopy [ Gracia-Saz et al. , 2014 ] .