# Higher transformations between natural transformations and so on

In category theory, arrows between categories are functors, arrows between functors are natural transformations, so a natural question is to ask what are arrows between natural transformations ?

I've got an idea to generalize that, but I don't know if its any useful or known. Let $F$ and $G$ be two functors from categories $\mathcal{C}$ to $\mathcal{D}$, and let $\eta: F \Rightarrow G$. My idea is to see a natural transformation as a functor from $\mathcal{C}$ to the arrow category of $\mathcal{D}$.

Let $\eta : \mathcal{C} \rightarrow Arr(\mathcal{D})$ be a functor. We say that $\eta$ is a natural transformation between $F$ and $G$ if for all objects $A$, $B$, and arrows $f : A \rightarrow B$ in $\mathcal{C}$, we have the following conditions :

1) $\eta(A) = \eta_A : F(A) \rightarrow G(A)$ (we call it like that even though it is not the component of a natural transformation, it is only an object in the arrow category of $\mathcal{D}$)

2) $\eta$ sends $f$ to $\eta(f) = (F(f), G(f))$ such that $\eta_B \circ F(f) = G(f) \circ \eta_A$ (that is, the square commutes). Such an $\eta$ encodes what a natural transformation is, and it is easily seen what a "natural transformation" between two such functors is. It seems that one could easily generalize that up to infinity.

Does anyone know if such an idea is known, useful or not, or if there's better ?

edit: looks like it's now well defined

• This formulation of natural transformations is well known and apparently frequently rediscovered. To answer your title question, however, there are no higher morphisms. You will find that, after imposing all the relevant boundary conditions, a "transformation" between natural transformations (between the same two functors) must be the identity. – Zhen Lin Feb 11 '15 at 9:10
• ah alright, I'll try to show that, thanks – sure Feb 11 '15 at 9:21
• @ZhenLin Apparently though, Borceux seems to give a definition of what an "arrow between natural transformation" may be and call it a "modification". Everything is in paragraph 7.3 of the first volume of the Handbook. Is there something I am missing? – Marco Vergura Feb 11 '15 at 9:32
• That is in the context of 2-categories. Of course, you can regard any category as a 2-category, but you don't get any non-trivial modifications that way. – Zhen Lin Feb 11 '15 at 9:52
• @ZhenLin Yes, of course, I see the point. Thanks! – Marco Vergura Feb 11 '15 at 10:27