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

  • The following diagram commutes:

    I D F 0 F(I C) G 0 α G(I C)\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,YX,Y:

    F([X,Y]) F^ [F(X),F(Y)] [1,α] α [F(X),G(Y)] [α,1] G([X,Y]) G^ [G(X),G(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)]}

Examples

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)