Let the category of Abelian Groups. I know that product and coproduct of a finite number of objects are the same in this category. Then, it follows that the projective and injective limits of finite diagrams are the same? Thanks.
2026-03-25 09:48:33.1774432113
Category of Abelian Groups: Limits
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No. For instance, for a diagram of the form $A\to B$ with two objects and an arrow between them, the limit will be $A$ and the colimit will be $B$ (this is true in any category).