Category theory

Homotopy theory



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

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

So in particular this means that KK 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

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

we have that if two of ff, gg and hh are in CC, 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.

Revised on June 10, 2017 14:17:44 by Peter Heinig (2003:58:aa10:db00:a04:639a:b7ef:8b78)