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.