nLab arrow category

Idea

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 $C$.

Definition

For $C$ any category, its arrow category $Arr(C)$ is the category such that:

• an object $a$ of $Arr(C)$ is a morphism $a\colon a_0 \to a_1$ of $C$;
• a morphism $f\colon a \to b$ of $Arr(C)$ is a commutative square
  $$\array { a_0 & \overset{f_0}\to & b_0 \\ \llap{a}\downarrow & & \rlap{b}\downarrow \\ a_1 & \underset{f_1}\to & b_1 }$$

  in $C$;

• composition in $Arr(C)$ is given simply by placing commutative squares side by side to get a commutative oblong.

This is isomorphic to the functor category

$$Arr(C) := Funct(I,C) = [I,C] = C^I$$

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.

Properties

Proposition

The arrow category $Arr(C)$ is equivalently the comma category $(id/id)$ for the case that $id\colon C \to C$ is the identity functor.

Remark

$Arr(C)$ plays the role of a directed path object for categories in that functors

$$X \to Arr(Y)$$

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

Example

(arrow category is Grothendieck construction on slice categories)
For $\mathcal{S}$ any category, let

$$\mathcal{S}_{(-)} \,\colon\, \mathcal{S} \longrightarrow Cat$$

be the pseudofunctor which sends

• an object $B \,\in\, \mathcal{S}$ to the slice category $\mathcal{S}_{/B}$,

• a morphism $f \colon B \to B'$ to the left base change functor $f_! \,\colon\, \mathcal{C}_{B} \to \mathcal{C}_{/B'}$ given by post-composition in $\mathcal{C}$.

The Grothendieck construction on this functor is the arrow category $Arr(\mathcal{S})$ of $\mathcal{S}$:

$$Arr(\mathcal{S}) \;\;\; \simeq \;\;\; \int_{B \in \mathcal{S}} \mathcal{S}_{/B} \mathrlap{\,.}$$

This follows readily by unwinding the definitions. In the refinement to the Grothendieck construction for model categories (here: slice model categories and model structures on functors) this equivalence is also considered for instance in Harpaz & Prasma (2015), above Cor. 6.1.2.

The correponding Grothendieck fibration is also known as the codomain fibration.

Related concepts

• path space object

• slice category, undercategory

• arrow (∞,1)-category

• arrow (∞,1)-topos

• twisted arrow category