In
category theory
Category theory is a general theory of mathematical structures and their relations. It was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Category theory ...
, a branch of mathematics, the formal criteria for adjoint functors are criteria for the existence of a left or right
adjoint
In mathematics, the term ''adjoint'' applies in several situations. Several of these share a similar formalism: if ''A'' is adjoint to ''B'', then there is typically some formula of the type
:(''Ax'', ''y'') = (''x'', ''By'').
Specifically, adjoin ...
of a given
functor
In mathematics, specifically category theory, a functor is a Map (mathematics), mapping between Category (mathematics), categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) ar ...
.
One criterion is the following, which first appeared in
Peter J. Freyd's 1964 book ''Abelian Categories,
an Introduction to the Theory of Functors'':
Another criterion is:
See also
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Anafunctor
An anafunctor is a notion introduced by for ordinary categories that is a generalization of functors. In category theory, some statements require the axiom of choice, but the axiom of choice can sometimes be avoided when using an anafunctor. For ...
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Adjoint functors