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, an Introduction to the Theory of Functors:

Freyd's adjoint functor theorem[1] — Let G:𝒜 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. G preserves all limits and for each object x in 𝒜, there exist a set I and an I-indexed family of morphisms fi:xGyi such that each morphism xGy is of the form G(yiy)fi for some morphism yiy.

Another criterion is:

Kan criterion for the existence of a left adjoint — Let G:𝒜 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 𝒜, the limit lim((xG)) exists in .[2]
  3. The right Kan extension G!1 of the identity functor 1 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

  1. Mac Lane 2013, Ch. V, § 6, Theorem 2.
  2. 2.0 2.1 Mac Lane 2013, Ch. X, § 1, Theorem 2.