nLab
strong adjoint functor

Contents

Definition

For CC a cartesian closed category and L,R:CCL,R : C \to C two endofunctors, they are called strong adjoints to each other if there is a natural isomorphism

[LX,A][X,RA] [L X, A] \simeq [X, R A]

for all objects X,ACX,A \in C and for [,][-,-] the internal hom.

Properties

Notice that for ** the terminal object of CC we have that the global points of the internal hom give the external hom set

Γ[X,A]:=C(*,[X,A])C(X,A). \Gamma [X,A] := C(*, [X, A]) \simeq C(X,A) \,.

Therefore strongly adjoint functors are in particular adjoint functors in the ordinary sense.

References

For instance appendix 6 of

Created on December 7, 2011 19:45:19 by Urs Schreiber (131.174.40.86)