Every category $C$ gives rise to an arrow category $Arr(C)$ such that the objects of $Arr(C)$ are the morphisms (or arrows, hence the name) of $C$.
For $C$ any category, its arrow category $Arr(C)$ is the category such that:
in $C$;
This is isomorphic to the functor category
for $I$ the interval category $\{0 \to 1\}$. $Arr(C)$ is also written $[\mathbf{2},C]$, $C^{\mathbf{2}}$, $[\Delta[1],C]$, or $C^{\Delta[1]}$, since $\mathbf{2}$ and $\Delta[1]$ (for the $1$-simplex) are common notations for the interval category.
$Arr(C)$ is equivalently the comma category $(id/id)$ where $id\colon C \to C$ is the identity functor.
$Arr(C)$ plays the role of a directed path object for categories in that functors
are the same as natural transformations between functors between $X$ and $Y$.
Last revised on June 7, 2021 at 05:29:37. See the history of this page for a list of all contributions to it.