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Formal criteria for adjoint functors

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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,[1] an Introduction to the Theory of Functors:

Freyd's adjoint functor theorem[2] — Let be a functor between categories such that is complete. Then the following are equivalent (for simplicity ignoring the set-theoretic issues):

  1. G has a left adjoint.
  2. 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.

  1. G has a left adjoint.
  2. G preserves limits and, for each object x in , the limit exists in .[3]
  3. The right Kan extension of the identity functor along G exists and is preserved by G.[4][5][6]

Moreover, when this is the case then a left adjoint of G can be computed using the right Kan extension.[3]

See also

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References

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  1. ^ Freyd 2003, Chapter 3. (pp.84–)
  2. ^ Mac Lane 2013, Ch. V, § 6, Theorem 2.
  3. ^ a b Mac Lane 2013, Ch. X, § 1, Theorem 2.
  4. ^ Mac Lane 2013, Ch. X, § 7, Theorem 2.
  5. ^ Kelly 1982, Theorem 4.81
  6. ^ Medvedev 1975, p. 675

Bibliography

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  • 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.
  • Feferman, Solomon; Kreisel, G. (1969). "Set-Theoretical foundations of category theory". Reports of the Midwest Category Seminar III. Lecture Notes in Mathematics. Vol. 106. 3.3. Case study of current category theory: specific illustrations. pp. 201–247. doi:10.1007/BFb0059148. ISBN 978-3-540-04625-7.{{cite book}}: CS1 maint: location (link) CS1 maint: location missing publisher (link)
  • Lane, Saunders Mac (1969). "Foundations for categories and sets". Category Theory, Homology Theory and their Applications II. Lecture Notes in Mathematics. Vol. 92. V THE ADJOINT FUNCTOR THEOREM. pp. 146–164. doi:10.1007/BFb0080770. ISBN 978-3-540-04611-0.{{cite book}}: CS1 maint: location missing publisher (link)
  • Paré, Robert; Schumacher, Dietmar (1978). "Abstract families and the adjoint functor theorems, ch. IV The adjoint functor theorems". Indexed Categories and Their Applications. Lecture Notes in Mathematics. Vol. 661. pp. 1–125. doi:10.1007/BFb0061361. ISBN 978-3-540-08914-8.

Further reading

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