Formal criteria for adjoint functors
# 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 $G:{\mathcal {B}}\to {\mathcal {A}}$ be a functor between categories such that ${\mathcal {B}}$ is complete. Then the following are equivalent (for simplicity ignoring the set-theoretic issues):

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

Another criterion is:

Kan criterion for the existence of a left adjoint — Let $G:{\mathcal {B}}\to {\mathcal {A}}$ be a functor between categories. Then the following are equivalent.

* G has a left adjoint.
* G preserves limits and, for each object x in ${\mathcal {A}}$, the limit $\lim({(x\downarrow G)\to {\mathcal {B}}})$ exists in ${\mathcal {B}}$.
* The right Kan extension $G_{!}1_{\mathcal {B}}$ of the identity functor $1_{\mathcal {B}}$ 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.

