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 $f\colon X \to Y$, $g\colon Y \to Z$, 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 $h = g \circ f$.

$f$ and $g$ are the composable pair of morphisms of a commutative triangle, and $h$ is the composite or unique composite of the morphisms $f$ and $g$ in the commutative triangle.

Equivalently, every commutative triangle in a category $C$ is represented by a functor $F:\Delta^2 \to C$ from the triangle category $\Delta^2$ to $C$.

Characterization

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.)

Related concepts

• triangle category

• triangle, 2-simplex

• hom set, hom type

• composite

• commutative triangle

• unique composite

• Segal type

References

The phrase “commutative triangle” appears in:

• Denis-Charles Cisinski, Bastiaan Cnossen, Hoang Kim Nguyen, Tashi Walde, section 1.5.1 of: Formalization of Higher Categories, work in progress (pdf)