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 — Let [math]\displaystyle{ G: \mathcal{B} \to \mathcal{A} }[/math] be a functor between categories such that [math]\displaystyle{ \mathcal{B} }[/math] is complete. Then the following are equivalent (for simplicity ignoring the set-theoretic issues):

1. G has a left adjoint.
2. [math]\displaystyle{ G }[/math] preserves all limits and for each object x in [math]\displaystyle{ \mathcal{A} }[/math], there exist a set I and an I-indexed family of morphisms [math]\displaystyle{ f_i : x \to G y_i }[/math] such that each morphism [math]\displaystyle{ x \to Gy }[/math] is of the form [math]\displaystyle{ G(y_i \to y) \circ f_i }[/math] for some morphism [math]\displaystyle{ y_i \to y }[/math].

Another criterion is:

Kan criterion for the existence of a left adjoint — Let [math]\displaystyle{ G: \mathcal{B} \to \mathcal{A} }[/math] 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 [math]\displaystyle{ \mathcal{A} }[/math], the limit [math]\displaystyle{ \lim ({(x \downarrow G) \to \mathcal{B}}) }[/math] exists in [math]\displaystyle{ \mathcal{B} }[/math].
3. The right Kan extension [math]\displaystyle{ G_! 1_{\mathcal{B}} }[/math] of the identity functor [math]\displaystyle{ 1_{\mathcal{B}} }[/math] 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.

References

• 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.