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Definition

For $C$ a category, a class $K\subset \mathrm{Mor}(C)$ of morphisms in $C$ is said to satisfy 2-out-of-3 if for all composable $f,g\in \mathrm{Mor}(C)$ 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.

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

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.

• The collection of “admissible” (“open”) morphisms in a geometry (for structured (∞,1)-toposes).

Related concepts

• category with weak equivalences

• 2-out-of-6 property