# PlanetPhysics/C 3CategoryTheorem

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\begin{theorem} (proposition 1.2. in ref. [1].)


A cocomplete Abelian category is ${\displaystyle C_{3}}$ if and only if the direct limit of every direct family of subobjects ${\displaystyle \left\{A_{i}\right\}}$ of an object ${\displaystyle A}$ is equal to ${\displaystyle \bigcup A_{i}}$. \end{theorem}

[1]

## References

1. See p.82 and eq. (1) in ref. ${\displaystyle [266]}$  in the Bibliography for categories and algebraic topology