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"). For example, an algebraic theory, as invented by William Lawvere, is an example of a doctrine. The concept of doctrines was invented by Lawvere as part of his work on algebraic theories. The name is based on a suggestion by Jon Beck.Marquis, Jean-Pierre, and Gonzalo Reyes. "The history of categorical logic: 1963-1977." (2004).
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
Further reading Generalised algebraic models, by Claudia Centazzo. William Lawvere, Ordinal sums and equational doctrines, Lecture Notes in Math., Vol. 80 (Springer, Berlin, 1969). ; Equality in hyperdoctrines and comprehension schema; ;
See also Categorical logic Pseudomonad
External links Doctrines in John Baez
What are some interesting hyperdoctrines that are not classical models?