Axiom of Abelian Category: AB5 & AB5*

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SUMMARY

The discussion centers on the Axiom of Abelian Categories, specifically addressing why only the zero category satisfies the axioms AB5 and AB5*. It establishes that a category classified as AB5 and C2* must contain only zero objects. The proof supporting this conclusion is referenced from the book "Abelian Categories with Applications to Rings and Modules" by Popescu, highlighting the relationship between direct limit preservation of exact sequences and projective limit preservation.

PREREQUISITES
  • Understanding of Abelian categories
  • Familiarity with exact sequences in category theory
  • Knowledge of the concepts AB5 and AB5*
  • Basic comprehension of monomorphisms in category theory
NEXT STEPS
  • Study the definitions and implications of AB5 and AB5* in depth
  • Explore the concept of C2* and its relationship with AB3*
  • Read "Abelian Categories with Applications to Rings and Modules" by Popescu for detailed proofs
  • Investigate examples of categories that do not satisfy AB5 and AB5*
USEFUL FOR

This discussion is beneficial for mathematicians, particularly those specializing in category theory, as well as graduate students and researchers exploring the properties of Abelian categories and their applications in algebra.

jose80
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In the axoim of Abelian category I am trying to see why does only the zero category satisfy axioms AB5 (direct limit preseve exact sequences) and AB5* (projective limit preserve exact sequences).
 
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In fact, you can go with something weaker then AB5*. A category is called C2* if it is AB3* and

\coprod A_i\rightarrow \prod A_i

is a monomorphism. Then, every category which is AB5 and C2* has only zero objects.

A nice proof of this fact can be found in the book "Abelian categories with applications to rings and modules" by Popescu.
 

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