nLab
two-out-of-three

Context

Category theory

Homotopy theory

Contents

Definition

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.

Examples

Revised on May 22, 2012 08:54:22 by Anonymous Coward (98.223.186.49)