Doctrine (mathematics)

In mathematics, specifically category theory, a doctrine is roughly a system of theories ("categorical analogues of fragments of logical theories which have sufficient category-theoretic structure for their models to be described as functors"^[1]: 284). For example, an algebraic theory, as invented by William Lawvere, is an example of a doctrine.^[1]: 289 The concept of doctrines was invented by Lawvere as part of his work on algebraic theories.^[2][3]: 12 The name is based on a suggestion by Jon Beck.^[3][4]

A doctrine can be defined in several ways:^[5]

• as a 2-monad. This was Lawvere's original approach.
• as a 2-category; the idea is that each object there amounts to a "theory".^[5]
• As cartesian double theories, as logics, or as a class of limits.

• Generalised algebraic models, by Claudia Centazzo.
• William Lawvere, Ordinal sums and equational doctrines, Lecture Notes in Math., Vol. 80 (Springer, Berlin, 1969).
• Lawvere, F William (1975). "Introduction to Part I". Model Theory and Topoi. Lecture Notes in Mathematics. Vol. 445. pp. 3–14. doi:10.1007/BFb0061291. ISBN 978-3-540-07164-8.;
• Equality in hyperdoctrines and comprehension schema;
• Dagnino, Francesco; Rosolini, Giuseppe (2021). "Doctrines, modalities and comonads". Mathematical Structures in Computer Science. 31 (7): 769–798. arXiv:2107.14031. doi:10.1017/S0960129521000207.;
• Emmenegger, Jacopo; Pasquali, Fabio; Rosolini, Giuseppe (2020). "Elementary doctrines as coalgebras". Journal of Pure and Applied Algebra. 224 (12). doi:10.1016/j.jpaa.2020.106445.
• Zöberlein, Volker (1976). "Doctrines on 2-categories". Mathematische Zeitschrift. 148 (3): 267–279. doi:10.1007/BF01214522.

• Categorical logic

• Doctrines in John Baez
• Baez, John (December 31, 2003). "This Week's Finds in Mathematical Physics (Week 200)".
• What are some interesting hyperdoctrines that are not classical models?

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