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4 typo corrected according to GIorgio Mossa's last comment; added 90 characters in body

Disclaimer: this is not an answer to the question as I have no explanation for why people don't introduce natural transformations in the way explained in the question, but I am posting this in order to expand a comment I made. The comment was

this is the starting observation to make for introducing simplicial categories as a model for $\infty$--categories

Moreover, I am not a specialist neither of category theory nor of homotopy theory (and a posteriori of higher categories).

## The $2$-category of categories

The starting point is that the category $Cat$ of categories is actually a $2$-category. For any to objects (i.e. categories) $\mathcal C$ and $\mathcal D$ we have that $Hom_{Cat}(\mathcal C,\mathcal D)$ is itself a category.

This is very transparent when using the definition $$Hom_{Cat}(\mathcal C,\mathcal D):=Hom_{t_{\leq0}(Cat)}(\mathcal C\times\Delta^1,\mathcal D)\,,$$ where $\Delta^1=\Box^1=G^1$ \Delta^1=\Box^1=\mathbb{G}^1$is the arrow category$0\to 1$and$t_{\leq0}(Cat)$is the underlying$1$-category of$Cat$. Remark: In general one can see a$2$-category$\mathcal C$as a simplicial category by replacing the$Hom$-categories by their nerves. In the case of$Cat$, we see that the$Hom$-categories naturally appear as$1$-truncations of simplicial sets (one can replace here "simplicial" by "cubical" of "globular"). ## The$3$-category of$2$-categories Le us now go to natural transformations of (strict)$2$-functors between (strict)$2$-categories. Given two such$2$-functors$F,G:\mathcal C\to\mathcal D$one can see that a natural transformation$F\Rightarrow G$is the same as a$2$-functors $$\phi:\mathcal C\times G^2\to\mathcal \mathbb{G}^2\to\mathcal D$$ such that$\phi(-,0)=F$and$\phi(-,0)=G$, \phi(-,1)=G$, where $G^2$ \mathbb{G}^2$is the$2$-category with two objects$0$and$1$and such that$Hom_{G^2}(0,1)$Hom_{\mathbb{G}^2}(0,1)$ is the arrow category $G^1=(0\to \mathbb{G}^1=(0\to 1)$.

Therefore the "set" of $2$-functors is a naturally a $2$-category.

Remark: as before we can then see any $3$-category as a simplicial/cubical/globular category by replacing the $Hom$-$2$-categories by their (simplicial/cubical/globular) nerves.

In the case of $2-Cat$, we see that the $Hom$-$2$-categories naturally appear as $2$-truncations of globular sets.

The globe category $G$, \mathbb{G}$, the cubical category$\Box$and the simplicial category$\Delta$are known to be suitable geometric shape to model higher structures. Simplicial sets are good models for (weak)$\infty$-groupoids. It was proved (by Jardine ... with some improvement by Cisinski if I remember well) that cubical sets also provide a model for (weak)$\infty$-groupoids. I don't know any reference but I guess that the same holds for globular sets (which are quite more used by people working with automata). ## The$(n+1)$-category of$n$-categories Let me consider the category$n-Cat$of (strict)$n$-categories. A a natural transformation between (strict)$n$-functor$F,G:\mathcal C\to\mathcal D$can be written as an$n$-functor $$\phi:\mathcal C\times G^n\to\mathcal \mathbb{G}^n\to\mathcal D$$ such that$\phi(-,0)=F$and$\phi(-,0)=G$, \phi(-,1)=G$, where $G^n$ \mathbb{G}^n$is the$n$-category with two objects$0$and$1$and such that$Hom_{G^n}(0,1)$Hom_{\mathbb{G}^n}(0,1)$ is the $(n-1)$-category $G^{n-1}$. \mathbb{G}^{n-1}$. Therefore the "set" of$n$-functors is a naturally a (strict)$n$-category, and thus$n-Cat$is a (strict)$n+1$-category. It also naturally appears as a$n$-truncation of a globular category. ## The advantage of working with simplicial/cubical/globular categories Working directly with simplicial/cubical/globular categories has the following advantages: 1. it does allow to work directly with higher categories without going through an inductive process. 2. it allows to deal with weak$(\infty,1)$-categories, as simplicial/cubical/globular are models for weak$\infty$-groupoids (here$(\infty,1)$stands for "$\infty$-categories such that$n$-arrows for$n\geq2$are weakly invertible"). 3 added 69 characters in body Disclaimer: this is not an answer to the question as I have no explanation for why people don't introduce natural transformations in the way explained in the question, but I am posting this in order to expand a comment I made. The comment was this is the starting observation to make for introducing simplicial categories as a model for$\infty$--categories Moreover, I am not a specialist neither of category theory nor of homotopy theory (and a posteriori of higher categories). ## The$2$-category of categories The starting point is that the category$Cat$of categories is actually a$2$-category. For any to objects (i.e. categories)$\mathcal C$and$\mathcal D$we have that$Hom_{Cat}(\mathcal C,\mathcal D)$is itself a category. This is very transparent when using the definition $$Hom_{Cat}(\mathcal C,\mathcal D):=Hom_{t_{\leq0}(Cat)}(\mathcal C\times\Delta^1,\mathcal D)\,,$$ where$\Delta^1=\Box^1=G^1$is the arrow category$0\to 1$and$t_{\leq0}(Cat)$is the underlying$1$-category of$Cat$. Remark: In general one can see a$2$-category$\mathcal C$as a simplicial category by replacing the$Hom$-categories by their nerves. In the case of$Cat$, we see that the$Hom$-categories naturally appear as$1$-truncations of simplicial sets (one can replace here "simplicial" by "cubical" of "globular"). ## The$3$-category of$2$-categories Le us now go to natural transformations of (strict)$2$-functors between (strict)$2$-categories. Given two such$2$-functors$F,G:\mathcal C\to\mathcal D$one can see that a natural transformation$F\Rightarrow G$is the same as a$2$-functors $$\phi:\mathcal C\times G^2\to\mathcal D$$ such that$\phi(-,0)=F$and$\phi(-,0)=G$, where$G^2$is the$2$-category with two objects$0$and$1$and such that$Hom_{G^2}(0,1)$is the arrow category$G^1=(0\to 1)$. Therefore the "set" of$2$-functors is a naturally a$2$-category. Remark: as before we can then see any$3$-category as a simplicial/cubical/globular category by replacing the$Hom$-$2$-categories by their (simplicial/cubical/globular) nerves. In the case of$2-Cat$, we see that the$Hom$-$2$-categories naturally appear as$2$-truncations of globular sets. ## Simplices, Cubes, and globes The globe category$G$, the cubical category$\Box$and the simplicial category$\Delta$are known to be suitable geometric shape to model higher structures. Simplicial sets are good models for (weak)$\infty$-groupoids. It was proved (by Jardine ... with some improvement by Cisinski if I remember well) that cubical sets also provide a model for (weak)$\infty$-groupoids. I don't know any reference but I guess that the same holds for globular sets (which are quite more used by people working with automata). ## The$(n+1)$-category of$n$-categories Let me consider the category$n-Cat$of (strict)$n$-categories. A a natural transformation between (strict)$n$-functor$F,G:\mathcal C\to\mathcal D$can be written as a functor an$n$-functor $$\phi:\mathcal C\times G^n\to\mathcal D$$ such that$\phi(-,0)=F$and$\phi(-,0)=G$, where$G^n$is the$n$-category with two objects$0$and$1$and such that$Hom_{G^n}(0,1)$is the$(n-1)$-category G^{n-1}$. $G^{n-1}$.

Therefore the "set" of $n$-functors is a naturally a (strict) $n$-category, and thus $n-Cat$ is a (strict) $n+1$-category. It also naturally appears as a $n$-truncation of a globular category.

## The advantage of working with simplicial/cubical/globular categories

Working directly with simplicial/cubical/globular categories has the following advantages:

1. it does allow to work directly with higher categories without going through an inductive process.
2. it allows to deal with weak $(\infty,1)$-categories, as simplicial/cubical/globular are models for weak $\infty$-groupoids (here $(\infty,1)$ stands for "$\infty$-categories such that $n$-arrows for $n\geq2$ are weakly invertible").

Disclaimer: this is not an answer to the question as I have no explanation for why people don't introduce natural transformations in the way explained in the question, but I am posting this in order to expand a comment I made. The comment was

this is the starting observation to make for introducing simplicial categories as a model for $\infty$--categories

Moreover, I am not a specialist neither of category theory nor of homotopy theory (and a posteriori of higher categories).

## The $2$-category of categories

The starting point is that the category $Cat$ of categories is actually a $2$-category. For any to objects (i.e. categories) $\mathcal C$ and $\mathcal D$ we have that $Hom_{Cat}(\mathcal C,\mathcal D)$ is itself a category.

This is very transparent when using the definition $$Hom_{Cat}(\mathcal C,\mathcal D):=Hom_{t_{\geq1}(Cat)}(\mathcal D):=Hom_{t_{\leq0}(Cat)}(\mathcal C\times\Delta^1,\mathcal D)\,,$$ where $\Delta^1=\Box^1=G^1$ is the arrow category $0\to 1$ and $t_{\geq 1}(Cat)$ t_{\leq0}(Cat)$is the underlying$1$-category of$Cat$. Remark: In general one can see a$2$-category$\mathcal C$as a simplicial category by replacing the$Hom$-categories by their nerves. In the case of$Cat$, we see that the$Hom$-category Hom$-categories naturally appear as $1$-truncations of simplicial sets (one can replace here "simplicial" by "cubical" of "globular").

## The $3$-category of $2$-categories

Le us now go to natural transformations of (strict) $2$-functors between (strict) $2$-categories. Given two such $2$-functors $F,G:\mathcal C\to\mathcal D$ one can see that a natural transformation $F\Rightarrow G$ is the same as a $2$-functors $$\phi:\mathcal C\times G^2\to\mathcal D$$ such that $\phi(-,0)=F$ and $\phi(-,0)=G$, where $G^2$ is the $2$-category with two objects $0$ and $1$ and such that $Hom_{G^2}(0,1)$ is the arrow category $G^1=(0\to 1)$.

Therefore the "set" of $2$-functors is a naturally a $2$-category.

Remark: as before we can then see any $3$-category as a simplicial/cubical/globular category by replacing the $Hom$-$2$-categories by their (simplicial/cubical/globular) nerves.

In the case of $2-Cat$, we see that the $Hom$-$2$-category Hom$-$2$-categories naturally appear as$2$-truncations of globular sets. ## Simplices, Cubes, and globes The globe category$G$, the cubical category$\Box$and the simplicial category$\Delta$are known to be suitable geometric shape to model higher structures. Simplicial sets are good models for (weak)$\infty$-groupoids. It was proved (by Jardine ... with some improvement by Cisinski if I remember well) that cubical sets also provide a model for (weak)$\infty$-groupoids. I don't know any reference but I guess that the same holds for globular sets (which are quite more used by people working with automata). ## The$(n+1)$-category of$n$-categories Let me consider the category$n-Cat$of (strict)$n$-categories. A (strict)$n$-functor can be written as a functor $$\phi:\mathcal C\times G^n\to\mathcal D$$ such that$\phi(-,0)=F$and$\phi(-,0)=G$, where$G^n$is the$n$-category with two objects$0$and$1$and such that$Hom_{G^n}(0,1)$is the$(n-1)$-category G^{n-1}$.

Therefore the "set" of $n$-functors is a naturally a (strict) $n$-category, and thus $n-Cat$ is a (strict) $n+1$-category. It also naturally appears as a $n$-truncation of a globular category.

## The advantage of working with simplicial/cubical/globular categories

Working directly with simplicial/cubical/globular categories has the following advantages:

1. it does allow to work directly with higher categories without going through an inductive process.
2. it allows to deal with weak $(\infty,1)$-categories, as simplicial/cubical/globular are models for weak $\infty$-groupoids (here $(\infty,1)$ stands for "$\infty$-categories such that $n$-arrows for $n\geq2$ are weakly invertible").
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