Axiom of Abelian Category: AB5 & AB5*

[jose80]

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).

 

[micromass]

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.