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# exact sequence theorem in $C_{3}$–category

###### Theorem 0.1.

(Proposition 1.6. in ref. [1]) A cocomplete Abelian category $\mathcal{A}$ is $C_{3}$ if and only if for every direct family of subobjects $\left\{A_{i}\right\}$ of an object $A$ , and every morphism $g:B\to A$, one has the following equation:

$g^{{-1}}(\bigcup A_{i})=\bigcup g^{{-1}}(A_{i}).$ |

Remark: The proof involves the exact sequence:

$0\to g^{{-1}}(A_{i})\to B\to Im/A_{i}\bigcap Im\to 0,$ |

# References

- 1 See p.83 and eq. (3) in ref. $[266]$ in the Bibliography for categories and algebraic topology

Defines:

$Im$,cocomplete Abelian category, exact sequence

Keywords:

$C_3$--category for direct family and exact sequence

Related:

C_3Category,ExactSequence2,CategoricalSequence , CategoricalSequence

Synonym:

$C_3$ category theorem

Type of Math Object:

Theorem

Major Section:

Reference

Parent:

Groups audience:

## Mathematics Subject Classification

18A99*no label found*18-00

*no label found*18E10

*no label found*18E15

*no label found*

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