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 $G : ℬ \to 𝒜$ be a functor between categories such that $ℬ$ 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 $𝒜$ , there exist a set I and an I-indexed family of morphisms $f_{i} : x \to G y_{i}$ such that each morphism $x \to G y$ 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 $G : ℬ \to 𝒜$ be a functor between categories. Then the following are equivalent.

G has a left adjoint.
G preserves limits and, for each object x in $𝒜$ , the limit $\lim ((x ↓ G) \to ℬ)$ exists in $ℬ$ .^[3]
The right Kan extension $G_{!} 1_{ℬ}$ of the identity functor $1_{ℬ}$ 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]

References

↑ Freyd 2003, Chapter 3. (pp.84–)
↑ Mac Lane 2013, Ch. V, § 6, Theorem 2.
↑ ^3.0 ^3.1 Mac Lane 2013, Ch. X, § 1, Theorem 2.
↑ Mac Lane 2013, Ch. X, § 7, Theorem 2.
↑ Kelly 1982, Theorem 4.81
↑ 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. https://books.google.com/books?id=gfI-BAAAQBAJ.
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, doi:10.1017/CBO9781107360068, ISBN 978-1-107-04424-1
Freyd, Peter (2003). "Abelian categories". Reprints in Theory and Applications of Categories (3): 23–164. http://www.tac.mta.ca/tac/reprints/articles/3/tr3.pdf.
Kelly, Gregory Maxwell (1982), Basic concepts of enriched category theory, London Mathematical Society Lecture Note Series, 64, Cambridge University Press, Cambridge-New York, ISBN 0-521-28702-2, http://www.tac.mta.ca/tac/reprints/articles/10/tr10.pdf
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. 106. 3.3. Case study of current category theory: specific illustrations.. pp. 201–247. doi:10.1007/BFb0059148. ISBN 978-3-540-04625-7.
Lane, Saunders Mac (1969). "Foundations for categories and sets". Category Theory, Homology Theory and their Applications II. Lecture Notes in Mathematics. 92. V THE ADJOINT FUNCTOR THEOREM. pp. 146–164. doi:10.1007/BFb0080770. ISBN 978-3-540-04611-0.
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. 661. pp. 1–125. doi:10.1007/BFb0061361. ISBN 978-3-540-08914-8. https://books.google.com/books?id=5X98CwAAQBAJ&pg=PA94.

External links

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". https://ncatlab.org/nlab/show/adjoint+functor+theorem.
Jean Goubault-Larrecq. "Adjoint Functor Theorems: GAFT and SAFT". https://projects.lsv.ens-cachan.fr/topology/?page_id=719.
"solution set condition". https://ncatlab.org/nlab/show/solution+set+condition.

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[1] Freyd 2003, Chapter 3. (pp.84–)

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

[Kan-3] 3.0 ^3.1 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

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Formal criteria for adjoint functors

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