nLab
commutative triangle

Definition

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

$$\begin{array}{ccc}X& \stackrel{f}{\to }& Y\\ & {\searrow }^h& {\downarrow }^g\\ & & Z\end{array}$$\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.)