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

Freyd's adjoint functor theorem^[2] — Let ${\displaystyle G:{\mathcal {B}}\to {\mathcal {A}}}$ be a functor between categories such that ${\displaystyle {\mathcal {B}}}$ is complete. Then the following are equivalent (for simplicity ignoring the set-theoretic issues):

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

Another criterion is:

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

• Anafunctor