nLab two-out-of-three

Definition

For $C$ a category, a class $K \subset Mor(C)$ of morphisms in $C$ is said to satisfy 2-out-of-3 if for all composable $f,g \in Mor(C)$ we have that if two of the three morphisms $f$, $g$ and the composite $g \circ f$ is in $K$, then so is the third.

$$\array{ \\ {}^{\mathllap{f}}\nearrow \searrow^{\mathrlap{g}} \\ \stackrel{g \circ f}{\to} } \,.$$

So in particular this means that $K$ is closed under composition of morphisms.

This definition has immediate generalization also to higher category theory. For instance in (∞,1)-category theory its says that:

a class of 1-morphisms in an (∞,1)-category satisfies two out of 3, if for every 2-morphism of the form

$$\array{ & {}^{\mathllap{}f}\nearrow &\Downarrow^{\simeq}& \searrow^{\mathrlap{g}} \\ &&\stackrel{h}{\to}&& }$$

we have that if two of $f$, $g$ and $h$ are in $C$, then so is the third.

Examples

• The class of isomorphisms in any category satisfies 2-out-of-3. This case is the archetype of most of the cases in which the property is invoked: 2-out-of-3 is characteristic of morphisms that have a notion of inverse.

• A category with weak equivalences is defined as a category with a subcategory that contains all isomorphisms and satisfies 2-out-of-3.

Related concepts

• category with weak equivalences
• weak factorization system
• 2-out-of-6 property