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.

## References

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1. ^ a b Kock, A.; Reyes, G.E. (1993). "Doctrines in Categorical Logic". In Barwise, Jon (ed.). Handbook of Mathematical Logic. Elsevier Science Publishers B.V. pp. 283–313\. doi:10.1016/S0049-237X(08)71104-2. ISBN 0444863885.
2. ^ Lawvere, F. William (1969). Eckmann, B. (ed.). "Ordinal sums and equational doctrines". Seminar on Triples and Categorical Homology Theory. Lecture Notes in Mathematics. 80. Berlin: Springer: 141–155\. doi:10.1007/BFb0083085. ISBN 978-3-540-04601-1. Retrieved 9 April 2025.
3. ^ a b Marquis, Jean-Pierre, and Gonzalo Reyes. "The history of categorical logic: 1963-1977." (2004).
4. ^ Lambert, Michael; Patterson, Evan (May 2024). "Cartesian Double Theories: A Double-categorical Framework for Categorical Doctrines". Advances in Mathematics. 444. arXiv:2310.05384. doi:10.1016/j.aim.2024.109630.
5. ^ a b "doctrine in nLab". ncatlab.org. Retrieved 2025-04-09.

## Further reading

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* 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.

## See also

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* Categorical logic

## External links

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* Doctrines in John Baez
* What are some interesting hyperdoctrines that are not classical models?

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