Further reading

For an economical introduction to ordinary categories, functors, natural transformations, universal properties, and Kan extensions, see Riehl [ 2016 ] ; the classical reference is Mac Lane [ 1971 ] . Lawvere [ 1963 ] introduced functorial semantics for algebraic theories. Ehresmann’s sketches and their model semantics are developed in Ehresmann [ 1968 ] , Barr and Wells [ 1999 ] ; PROPs as one-sorted symmetric monoidal theories and their composition are treated by Lack [ 2004 ] . Richter [ 2020 ] develops category theory and then crosses systematically to simplicial objects, nerves, classifying spaces, and homotopy-theoretic applications. This makes it a particularly direct bridge from the first half of this chapter to its simplicial and homotopical half.

For the basic theory of \(\infty \)-categories, see Riehl and Verity [ 2022 ] ; its Section 1.1 supplies the inner-horn definition used here. Riehl develops the enriched and model-categorical machinery behind derived Kan extensions and homotopy-coherent UDL [ Riehl , 2014 ] . Enriched universal constructions are developed by Kelly [ 1982 ] . The tangent-categorical and synthetic differential background enters through Cockett and Cruttwell [ 2014 ] and Kock [ 2006 ] . The passage from differential categories through coKleisli and coEilenberg–Moore constructions to Cartesian differential and tangent categories is developed by Cockett et al. [ 2020 ] . Witsenhausen’s information-field formulation and intrinsic model are developed in Witsenhausen [ 1971 , 1975 ] , with a categorical generalization in Mahadevan [ 2021 ] . These references develop the mathematical machinery; the interpretation of that machinery as persistent online decision semantics is specific to UODL.