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

1. G has a left adjoint.
2. $G$ preserves all limits, and the following solution set condition is satisfied: 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 $x\to Gy$ 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 $G:{ℬ}\to {𝒜}$ 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 ${𝒜}$, the limit $\lim ((x\downarrow G)\to {ℬ})$ 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.

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

See also

• Anafunctor

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. 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. 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. 

Further reading

• Fausk, H.; Hu, P.; May, J. P. (2003). "Isomorphisms between left and right adjoints". Theory and Applications of Categories 11: 107–131. doi:10.70930/tac/1z9nxv9i. http://eudml.org/doc/123681. 
• Nguyen, Hoang Kim; Raptis, George; Schrade, Christoph (April 2020). "Adjoint functor theorems for ∞‐categories". Journal of the London Mathematical Society 101 (2): 659–681. doi:10.1112/jlms.12282. 
• Arkor, Nathanael; Liberti, Ivan Di; Loregian, and Fosco (5 June 2024). "Adjoint functor theorems for lax-idempotent pseudomonads". Theory and Applications of Categories 41 (20): 667–685. doi:10.70930/tac/o35ik5j5. 
• Street, Ross; Walters, Robert (February 1978). "Yoneda structures on 2-categories". Journal of Algebra 50 (2): 350–379. doi:10.1016/0021-8693(78)90160-6. 
• Lurie, Jacob (2009). "5.5.2 Representable Functors and the Adjoint Functor Theorem". Higher Topos Theory. Princeton University Press. ISBN 978-0-691-14048-3. 

External links

• Porst, Hans-E. (2023). "The history of the General Adjoint Functor Theorem". arXiv:2310.19528 [math.CT].
• Lehner, Marina (May 2014). "All Concepts are Kan Extensions": Kan Extensions as the Most Universal of the Universal Constructions (PDF) (BA thesis). Harvard College.
• "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.