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"{{cite book |last1=Kock |first1=A. |last2=Reyes |first2=G.E. |title=Handbook of Mathematical Logic |date=1993 |volume= |pages=283–313 |doi=10.1016/S0049-237X(08)71104-2|isbn=0444863885 |url=https://www.academia.edu/50925989/HANDBOOK_OF_MATHEMATICAL_LOGIC |editor-first= Jon | editor-last = Barwise|publisher = Elsevier Science Publishers B.V.| chapter=Doctrines in Categorical Logic }}{{rp|284}}). For example, an algebraic theory, as invented by William Lawvere, is an example of a doctrine.{{rp|289}}

The concept of doctrines was invented by Lawvere as part of his work on algebraic theories.{{cite journal |last1=Lawvere |first1=F. William |editor1-last=Eckmann |editor1-first=B. |title=Ordinal sums and equational doctrines |journal=Seminar on Triples and Categorical Homology Theory |series=Lecture Notes in Mathematics |date=1969 |volume=80 |pages=141–155 |doi=10.1007/BFb0083085 |url=https://link.springer.com/chapter/10.1007/BFb0083085 |access-date=9 April 2025 |publisher=Springer |location=Berlin |isbn=978-3-540-04601-1 |language=en|url-access=subscription }}{{rp|12}} The name is based on a suggestion by Jon Beck.Marquis, Jean-Pierre, and Gonzalo Reyes. [https://philarchive.org/archive/MARTHO-8 "The history of categorical logic: 1963-1977."] (2004).{{cite journal |last1=Lambert |first1=Michael |last2=Patterson |first2=Evan |title=Cartesian Double Theories: A Double-categorical Framework for Categorical Doctrines |journal=Advances in Mathematics |date=May 2024 |volume=444 |doi=10.1016/j.aim.2024.109630|arxiv=2310.05384 }}

A doctrine can be defined in several ways:{{Cite web |title=doctrine in nLab |url=https://ncatlab.org/nlab/show/doctrine |access-date=2025-04-09 |website=ncatlab.org}}

• 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.