category theory

# Contents

## 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 $C$ is given simply by placing commutative squares side by side to get a commutative oblong.

Up to equivalence, this is the same as 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

• $Arr(C)$ is the 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

$X \to Arr(Y)$

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

Revised on August 22, 2014 19:53:08 by Toby Bartels (98.19.44.147)