PlanetMath: $C_3$-category theorem

(more info)

Math for the people, by the people.

donor list - find out how

Main Menu

sections Encyclopædia
Papers
Books
Expositions

meta Requests (236)
Orphanage
Unclass'd (1)
Unproven (540)
Corrections (51)
Classification

talkback Polls
Forums
Feedback
Bug Reports

downloads Snapshots
PM Book

information News
Docs
Wiki
ChangeLog
TODO List
Copyright
About

Entry average rating: No information on entry rating

-category theorem

(Theorem)

Theorem 0.1 (Proposition 1.2. in ref. [1].)

A cocomplete Abelian category is $C_3$ if and only if the direct limit of every direct family of subobjects $\left\{A_i\right\}$ of an object $A$ is equal to $\bigcup A_i$ .

Bibliography

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

-category theorem" is owned by bci1.

(view preamble | get metadata)

See Also:

-category, alternative definition of an Abelian category, $C_3$

-category corollary, $C_3$

-category generators corollary

Other names:

Ab5 category, cocomplete Abelian category

Keywords:

cocomplete Abelian category, $C_3$

-category theorem

Attachments:: -category corollary (Corollary) by bci1
-category generators corollary (Corollary) by bci1

Log in to rate this entry.
(view current ratings)

Cross-references: object, subobjects, direct family, direct limit, proposition
There is 1 reference to this entry.

This is version 5 of $C_3$

-category theorem, born on 2008-09-27, modified 2009-02-03.
Object id is 11098, canonical name is C_3CategoryTheorem.
Accessed 870 times total.

Classification:

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

Pending Errata and Addenda

None.

Discussion

forum policy

No messages.

Interact