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

  1. G has a left adjoint.
  2. G {\displaystyle G} preserves all limits and for each object x in A {\displaystyle {\mathcal {A}}} , there exist a set I and an I-indexed family of morphisms f i : x G y i {\displaystyle f_{i}:x\to Gy_{i}} such that each morphism x G y {\displaystyle x\to Gy} is of the form G ( y i y ) f i {\displaystyle G(y_{i}\to y)\circ f_{i}} for some morphism y i y {\displaystyle y_{i}\to y} .

Another criterion is:

Kan criterion for the existence of a left adjoint — Let G : B A {\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 A {\displaystyle {\mathcal {A}}} , the limit lim ( ( x G ) B ) {\displaystyle \lim({(x\downarrow G)\to {\mathcal {B}}})} exists in B {\displaystyle {\mathcal {B}}} .[2]
  3. The right Kan extension G ! 1 B {\displaystyle G_{!}1_{\mathcal {B}}} of the identity functor 1 B {\displaystyle 1_{\mathcal {B}}} along G exists and is preserved by G.[3][4][5]

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

See also

  • Anafunctor

References

  1. ^ Mac Lane 2013, Ch. V, § 6, Theorem 2.
  2. ^ a b Mac Lane 2013, Ch. X, § 1, Theorem 2.
  3. ^ Mac Lane 2013, Ch. X, § 7, Theorem 2.
  4. ^ Kelly 1982, Theorem 4.81
  5. ^ 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)

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.
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