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):

  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 .[2]
  3. 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 left Kan extension.[2]

References

  1. Mac Lane, Ch. V, § 6, Theorem 2.
  2. Mac Lane, Ch. X, § 1, Theorem 2.
  • Saunders Mac Lane (17 April 2013). Categories for the Working Mathematician. Springer Science & Business Media. ISBN 978-1-4757-4721-8.


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