category theory

# Contents

## Definition

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

$Arr(C) := Funct(I,C)$

for $I$ the interval category $\{0 \to 1\}$. $Arr(C)$ is also written $[\mathbf{2},C]$ or $C^{\mathbf{2}}$ 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 $Arr(C)$ are the morphisms (the “arrows”, therefore the name) of $C$, while the morphisms of $Arr(C)$ are pairs of $C$ morphisms constituting commuting square diagrams in $C$.

## 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 March 21, 2013 09:13:06 by Chris Waggoner? (107.205.138.70)