nLab
arrow category

Definition

For $C$ any category, its arrow category is the functor category

$$\mathrm{Arr}(C):=\mathrm{Funct}(I,C)$$Arr(C) := Funct(I,C)

for $I$ the interval category $\{0\to 1\}$. $\mathrm{Arr}(C)$ is also written $[2,C]$ or $C^2$ or $C^{\Delta [1]}$, since 2 and $\Delta [1]$ (the 1-simplex) etc. are common notation for the interval category.

This means that the objects of $\mathrm{Arr}(C)$ are the morphisms (the “arrows”, therefore the name) of $C$, while the morphisms of $\mathrm{Arr}(C)$ are pairs of $C$ morphisms constituting commuting square diagrams in $C$.

Properties

• $\mathrm{Arr}(C)$ is the equivalently the comma category $(\mathrm{id}/\mathrm{id})$ where $\mathrm{id}:C\to C$ is the identity functor.

• $\mathrm{Arr}(C)$ plays the role of a directed path object for categories in that functors

  $$X\to \mathrm{Arr}(Y)$$X \to Arr(Y)

  are the same as natural transformations between functors between $X$ and $Y$.

Related concepts

• path space object

• slice category, undercategory

• arrow (∞,1)-category

• arrow (∞,1)-topos