Further reading

Lawvere’s functorial semantics makes models into structure-preserving functors from an algebraic theory [ Lawvere , 1963 ] . For sketches as more general presentations of structured models, useful entry points are Barr and Wells [ 1999 ] , Makkai and Paré [ 1989 ] , and Adámek and Rosický [ 1994 ] . Barr and Wells [ 1992 ] is especially valuable because it identifies what ordinary sketches cannot present; it motivates the book’s repeated care about enriched, higher, fibrational, and homotopical extensions.

Categorical database theory supplies concrete examples of the same declarative stance. Functorial data migration makes schemas and instances compositional [ Spivak , 2012 ] , while lifting problems express queries and constraints [ Spivak , 2014 ] . Readers coming from logic may also consult Mac Lane and Moerdijk [ 1992 ] for the relation among sites, local models, and global semantics. LINCS reuses these presentational ideas but adds an obstruction object, tangent transport, a repair language, and an empirical admission decision.