# nLab closed natural transformation

## Idea

A closed natural transformation is the appropriate sort of natural transformation relating closed functors between closed categories.

## Definition

Let $F,G\colon C\to D$ be closed functors. A closed natural transformation $\alpha\colon F\to G$ is simply an ordinary natural transformation such that:

• The following diagram commutes:

$\array{I_D & \overset{F^0}{\to} & F(I_C)\\ & _{G^0}\searrow & \downarrow^{\alpha}\\ & & G(I_C)}$
• The following diagram commutes for any $X,Y$:

$\array{F([X,Y]) & \overset{\hat{F}}{\to} & [F(X),F(Y)]\\ && \downarrow^{[1,\alpha]}\\ ^{\alpha }\downarrow && [F(X),G(Y)]\\ && \uparrow^{[\alpha,1]}\\ G([X,Y])& \underset{\hat{G}}{\to} & [G(X),G(Y)]}$

## References

• Samuel Eilenberg and Max Kelly, Closed categories. Proc. Conf. Categorical Algebra (La Jolla, Calif., 1965).

Created on May 4, 2010 01:10:11 by Mike Shulman (75.3.120.120)