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 be a functor between categories such that is complete. Then the following are equivalent (for simplicity ignoring the set-theoretic issues):
G has a left adjoint.
preserves all limits and for each object x in , there exist a set I and an I-indexed family of morphisms such that each morphism is of the form for some morphism .
Another criterion is:
Kan criterion for the existence of a left adjoint — Let be a functor between categories. Then the following are equivalent.
G has a left adjoint.
G preserves
limits and, for each object x in , the limit exists in .[2]
The right
Kan extension of the identity functor 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]
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.
ISBN978-3-540-04625-7.{{
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.
ISBN978-3-540-04611-0.{{
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].