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"^: 284). For example, an algebraic theory, as invented by William Lawvere, is an example of a doctrine.^: 289 The concept of doctrines was invented by Lawvere as part of his work on algebraic theories.^: 12 The name is based on a suggestion by Jon Beck.

A doctrine can be defined in several ways:

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".
As cartesian double theories, as logics, or as a class of limits.

References

1 2 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.
↑ 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.{{cite journal}}: CS1 maint: periodical has ISBN (link)
1 2 Marquis, Jean-Pierre, and Gonzalo Reyes. "The history of categorical logic: 1963-1977." (2004).
↑ 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.
1 2 "doctrine in nLab". ncatlab.org. Retrieved 2025-04-09.

External links

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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[KochReyes-1993-1] 1 2 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.{{cite journal}}: CS1 maint: periodical has ISBN (link)

[MarquisReyes-3] 1 2 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.

[auto-5] 1 2 "doctrine in nLab". ncatlab.org. Retrieved 2025-04-09.

Doctrine (mathematics)

References

Further reading

See also

External links