Doctrine (mathematics)
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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.
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