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

References

1. Freyd 2003, Chapter 3. (pp.84–)
2. Mac Lane 2013, Ch. V, § 6, Theorem 2.
3. Mac Lane 2013, Ch. X, § 1, Theorem 2.
4. Mac Lane 2013, Ch. X, § 7, Theorem 2.
5. Kelly 1982, Theorem 4.81
6. Medvedev 1975, p. 675

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.
• Feferman, Solomon; Kreisel, G. (1969). "Set-Theoretical foundations of category theory". Reports of the Midwest Category Seminar III. Lecture Notes in Mathematics. Vol. 106. 3.3. Case study of current category theory: specific illustrations. pp. 201–247. doi:10.1007/BFb0059148. ISBN 978-3-540-04625-7.{{cite book}}: CS1 maint: location (link) CS1 maint: location missing publisher (link)
• Lane, Saunders Mac (1969). "Foundations for categories and sets". Category Theory, Homology Theory and their Applications II. Lecture Notes in Mathematics. Vol. 92. V THE ADJOINT FUNCTOR THEOREM. pp. 146–164. doi:10.1007/BFb0080770. ISBN 978-3-540-04611-0.{{cite book}}: CS1 maint: location missing publisher (link)
• Paré, Robert; Schumacher, Dietmar (1978). "Abstract families and the adjoint functor theorems, ch. IV The adjoint functor theorems". Indexed Categories and Their Applications. Lecture Notes in Mathematics. Vol. 661. pp. 1–125. doi:10.1007/BFb0061361. ISBN 978-3-540-08914-8.

Further reading

• Fausk, H.; Hu, P.; May, J. P. (2003). "Isomorphisms between left and right adjoints". Theory and Applications of Categories. 11: 107–131. arXiv:math/0206079.

External link

• Porst, Hans-E. (2023). "The history of the General Adjoint Functor Theorem". arXiv:2310.19528 [math.CT].
• Lehner, Marina (Adviser: Emily, Riehl) (2014). “All Concepts are Kan Extensions” Kan Extensions as the Most Universal of the Universal Constructions (PDF) (cenior thesis). Harvard College.{{cite thesis}}: CS1 maint: multiple names: authors list (link)
• "adjoint functor theorem". ncatlab.org.
• Jean Goubault-Larrecq. "Adjoint Functor Theorems: GAFT and SAFT". Non-Hausdorff Topology and Domain Theory: Electronic supplements to the book.
• "solution set condition". ncatlab.org.