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 $X,Y$:

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

Examples

• Just as closed functors between closed monoidal categories are in bijective correspondence with (lax) monoidal functors, closed natural transformations between such are in bijective corrrespondence with monoidal transformations.

• The same is true for transformations between closed unital multicategories, regarded as closed categories.

References

• Samuel Eilenberg, G. Max Kelly, §I.4 of: Closed Categories, in: Proceedings of the Conference on Categorical Algebra - La Jolla 1965, Springer (1966) 421-562 [doi:10.1007/978-3-642-99902-4]