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For a category, a class of morphisms in is said to satisfy 2-out-of-3 if for all composable we have that if two of the three morphisms , and the composite is in , then so is the third.
So in particular this means that 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 , and are in , then so is the third.
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.
Last revised on June 10, 2017 at 18:17:44. See the history of this page for a list of all contributions to it.