# Commutative triangles

## Definition

Let $C$ be a category. A triangle of morphisms of $C$ consists of objects $X,Y,Z$ of $C$ and morphisms $f\colon X \to Y$, $g\colon Y \to Z$, and $h\colon X \to Z$. This is often pictured as a triangle

$\array { X & \overset{f}\rightarrow & Y \\ & \searrow^{h} & \downarrow^{g} \\ & & Z }$

The triangle is commutative if $h = g \circ f$.

## Characterisation

A commutative triangle is determined entirely by $f$ and $g$; therefore, a commutative triangle is equivalent to a composable pair of morphisms.

Accordingly, one rarely hears of commutative triangles on their own; instead, the concept only comes up when one already has a triangle and asks whether it commutes. (This is different from the situation with commutative squares.)

Created on September 3, 2010 20:01:23 by Toby Bartels (173.190.153.41)