category theory

# Contents

## Definition

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

$\left[LX,A\right]\simeq \left[X,RA\right]$[L X, A] \simeq [X, R A]

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

## Properties

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

$\Gamma \left[X,A\right]:=C\left(*,\left[X,A\right]\right)\simeq C\left(X,A\right)\phantom{\rule{thinmathspace}{0ex}}.$\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)