4
$\begingroup$

Given categories $\mathcal{C}$ and $\mathcal{D}$ and functors $F,G : \mathcal{C} \rightarrow \mathcal{D}$, can we somehow turn a natural transformation $$\nu : F \Rightarrow G : \mathcal{C} \rightarrow \mathcal{D}$$ into just another functor $$N : 2.\mathcal{C} \rightarrow \mathcal{D}?$$

I'm using $2.\mathcal{C}$ loosely as an informal notation intended to denote a category obtained by taking the coproduct $\mathcal{C} \uplus \mathcal{C}$ and then adjoining some more arrows.

$\endgroup$
8
$\begingroup$

Yes. A natural transformation between functors $\mathcal{C} \to \mathcal{D}$ can be regarded as a functor $\mathbb{2} \times \mathcal{C} \to \mathcal{D}$, or as a functor $\mathcal{C} \to [\mathbb{2}, \mathcal{D}]$. Thus a natural transformation is like a homotopy.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.