Further Reading

Lawvere introduced functorial semantics for algebraic theories [ Lawvere , 1963 ] . Ehresmann introduced sketches as presentations of categorical structure, and Barr and Wells give a broad account of sketches and their models [ Ehresmann , 1968 , Barr and Wells , 1999 ] . Lack explains PROPs as one-sorted symmetric monoidal theories and their composition [ Lack , 2004 ] . Mac Lane and Moerdijk develop sheaf and topos semantics, while Garner and Lack state the left-exact localization characterization used here [ Mac Lane and Moerdijk , 1992 , Garner and Lack , 2012 ] . Piaget’s accounts of cognitive construction and equilibration motivate the explicitly speculative developmental analogy [ Piaget , 1952 , 1985 ] . That analogy should be read through the theory maps, reduct functors, and model expansions defined above, not as a claim that Piaget’s stages are categorical doctrines. Mac Lane and Riehl develop ordinary Kan extensions, and Kelly supplies the enriched setting needed when DIAL models take values beyond ordinary categories [ Mac Lane , 1998 , Riehl , 2016 , Kelly , 1982 ] .