nLab
commutative triangle

Commutative triangles

Definition

Let CC be a category. A triangle of morphisms of CC consists of objects X,Y,ZX,Y,Z of CC and morphisms f:XYf\colon X \to Y, g:YZg\colon Y \to Z, and h:XZh\colon X \to Z. This is often pictured as a triangle

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

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

Characterisation

A commutative triangle is determined entirely by ff and gg; 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)