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.[3][4]<ref>Medvedev 1975, p. 675
Moreover, when this is the case then a left adjoint of G can be computed using the right Kan extension.[2]
References[edit]
- ^ Mac Lane 2013, Ch. V, § 6, Theorem 2.
- ^ a b Mac Lane 2013, Ch. X, § 1, Theorem 2.
- ^ Mac Lane 2013, Ch. X, § 7, Theorem 2.
- ^ Kelly 1982, Theorem 4.81
Bibliography[edit]
- Mac Lane, Saunders (17 April 2013). Categories for the Working Mathematician. Springer Science & Business Media. ISBN 978-1-4757-4721-8.
- Borceux, Francis (1994). "Adjoint functors". Handbook of Categorical Algebra. pp. 96–131. doi:10.1017/CBO9780511525858.005. ISBN 978-0-521-44178-0.
- Leinster, Tom (2014), Basic Category Theory, arXiv:1612.09375, doi:10.1017/CBO9781107360068, ISBN 978-1-107-04424-1
- Freyd, Peter (2003). "Abelian categories" (PDF). Reprints in Theory and Applications of Categories (3): 23–164.
- Kelly, Gregory Maxwell (1982), Basic concepts of enriched category theory (PDF), London Mathematical Society Lecture Note Series, vol. 64, Cambridge University Press, Cambridge-New York, ISBN 0-521-28702-2, MR 0651714
- Ulmer, Friedrich (1971). "The adjoint functor theorem and the Yoneda embedding". Illinois Journal of Mathematics. 15 (3). doi:10.1215/ijm/1256052605.
- Medvedev, M. Ya. (1975). "Semiadjoint functors and Kan extensions". Siberian Mathematical Journal. 15 (4): 674–676. doi:10.1007/BF00967444.
External link[edit]
- Porst, Hans-E. (2023). "The history of the General Adjoint Functor Theorem". arXiv:2310.19528 [math.CT].