Formal criteria for adjoint functors

Last updated April 24, 2024

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 $G:{\mathcal {B}}\to {\mathcal {A}}$ 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 ${\mathcal {B}}$ .^[2]
The right Kan extension $G_{!}1_{\mathcal {B}}$ of the identity functor $1_{\mathcal {B}}$ along G exists and is preserved by G.^[3]

Moreover, when this is the case then a left adjoint of G can be computed using the right Kan extension.^[2]

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References

↑ Mac Lane 2013, Ch. V, § 6, Theorem 2.
1 2 Mac Lane 2013, Ch. X, § 1, Theorem 2.
↑ Mac Lane 2013, Ch. X, § 7, Theorem 2.

Mac Lane, Saunders (17 April 2013). Categories for the Working Mathematician. Springer Science & Business Media. ISBN 978-1-4757-4721-8.
"Adjoint functors". Handbook of Categorical Algebra. 1994. pp. 96–131. doi:10.1017/CBO9780511525858.005. ISBN 978-0-521-44178-0.
Leinster, Tom (2014), Basic Category Theory, arXiv: 1612.09375 , doi:10.1017/CBO9781107360068, ISBN 978-1-107-04424-1
Porst, Hans-E. (2023). "The history of the General Adjoint Functor Theorem". arXiv: 2310.19528 [math.CT].
Freyd, Peter (2003). "Abelian categories" (PDF). Reprints in Theory and Applications of Categories (3): 23–164.

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[1] Mac Lane 2013, Ch. V, § 6, Theorem 2.

[Kan-2] 1 2 Mac Lane 2013, Ch. X, § 1, Theorem 2.

[3] Mac Lane 2013, Ch. X, § 7, Theorem 2.

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