PlanetMath: exact sequence theorem in $C

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exact sequence theorem in $C_3$

--category

(Theorem)

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: $$ 0 \to g^{-1}(A_i) \to B \to Im / A_i \bigcap Im \to 0 ,$$

where $Im$ is the image of the morphism $g$ .

Bibliography

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

"exact sequence theorem in $C_3$

--category" is owned by bci1.

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See Also:

-category, exact sequence, categorical sequence, categorical sequence

Other names:

category theorem

Also defines:

Keywords:

--category for direct family and exact sequence

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Cross-references: image, exact sequence, proof, equation, morphism, object, subobjects, direct family, cocomplete Abelian category, proposition
There are 2 references to this entry.

This is version 6 of exact sequence theorem in $C_3$

--category, born on 2008-09-27, modified 2009-06-05.
Object id is 11101, canonical name is ExactSequenceTheoremInC_3Category.
Accessed 929 times total.

Classification:

AMS MSC:	18E15 (Category theory; homological algebra :: Abelian categories :: Grothendieck categories)
	18E10 (Category theory; homological algebra :: Abelian categories :: Exact categories, abelian categories)
	18-00 (Category theory; homological algebra :: General reference works )
	18A99 (Category theory; homological algebra :: General theory of categories and functors :: Miscellaneous)

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