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):

1. G has a left adjoint.
2. $G$ 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.

1. G has a left adjoint.
2. G preserves limits and, for each object x in ${\mathcal {A}}$, the limit $\lim({(x\downarrow G)\to {\mathcal {B}}})$ exists in ${\mathcal {B}}$.
3. The right Kan extension $G_{!}1_{\mathcal {B}}$ of the identity functor $1_{\mathcal {B}}$ along G exists and is preserved by G.<ref>Medvedev 1975, p. 675

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

Bibliography

• Mac Lane, Saunders (17 April 2013). Categories for the Working Mathematician. Springer Science & Business Media. ISBN 978-1-4757-4721-8.
• Borceux, Francis (1994). "Adjoint functors". Handbook of Categorical Algebra. 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
• Freyd, Peter (2003). "Abelian categories" (PDF). Reprints in Theory and Applications of Categories (3): 23–164.
• Kelly, Gregory Maxwell (1982), Basic concepts of enriched category theory (PDF), London Mathematical Society Lecture Note Series, vol. 64, Cambridge University Press, Cambridge-New York, ISBN 0-521-28702-2, MR 0651714
• Ulmer, Friedrich (1971). "The adjoint functor theorem and the Yoneda embedding". Illinois Journal of Mathematics. 15 (3). doi:10.1215/ijm/1256052605.
• Medvedev, M. Ya. (1975). "Semiadjoint functors and Kan extensions". Siberian Mathematical Journal. 15 (4): 674–676. doi:10.1007/BF00967444.

External link

• Porst, Hans-E. (2023). "The history of the General Adjoint Functor Theorem". arXiv:2310.19528 [math.CT].