Formal criteria for adjoint functors
In category theory, a branch of mathematics, the formal criteria for adjoint functors are criteria for the existence of a left or right adjoint of a given functor. One criterion is the following, which first appeared in Peter J. Freyd's 1964 book Abelian Categories, an Introduction to the Theory of Functors:
Freyd's adjoint functor theorem[1] — Let be a functor between categories such that is complete. Then the following are equivalent (for simplicity ignoring the set-theoretic issues):
- G has a left adjoint.
- preserves all limits and for each object x in , there exist a set I and an I-indexed family of morphisms such that each morphism is of the form for some morphism .
Another criterion is:
Kan criterion for the existence of a left adjoint — Let be a functor between categories. Then the following are equivalent.
- G has a left adjoint.
- G preserves limits and, for each object x in , the limit exists in .[2]
- The right Kan extension of the identity functor along G exists and is preserved by G.
Moreover, when this is the case then a left adjoint of G can be computed using the right Kan extension.[2]
References
- ↑ Mac Lane 2013, Ch. V, § 6, Theorem 2.
- ↑ 2.0 2.1 Mac Lane 2013, Ch. X, § 1, Theorem 2.
- Mac Lane, Saunders (17 April 2013). Categories for the Working Mathematician. Springer Science & Business Media. ISBN 978-1-4757-4721-8. https://books.google.com/books?id=gfI-BAAAQBAJ.
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