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"). 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.

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.

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

External links

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