category theory

# Contents

## Definition

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

$\begin{array}{c}\\ {}^{f}↗{↘}^{g}\\ \stackrel{g\circ f}{\to }\end{array}\phantom{\rule{thinmathspace}{0ex}}.$\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

$\begin{array}{cccc}& {}^{f}↗& {⇓}^{\simeq }& {↘}^{g}\\ & & \stackrel{h}{\to }& & \end{array}$\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

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