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Let us say that an abelian category admits a generator $g$, if for every object $x$ in the category there is an epimorphism $g^{\oplus I} \to x$ for some index set $I$. I am interested in weakening this (in the context of categories enriched over a "finite" category such as finite-dimensional vector-spaces or finitely generated modules) to the condition that the index set $I$ has to be finite. That is, we say $g$ to be a finite generator, if for every object $x$ in the category there is an epimorphism $g^n \to x$ for some $n \in \mathbb{N}$.

Is there any study or appearance of such generators in the literature?

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    $\begingroup$ Note that the first notion only makes sense if you know that $\mathscr A$ has arbitrary direct sums (at least of the same object), whereas the second notion implies that $\mathscr A$ is small (every object is the cokernel of some map $g^m \to g^n$, and presumably categories are always locally small). So the two criteria live in different worlds, so to speak. $\endgroup$
    – R. van Dobben de Bruyn
    Commented Sep 16 at 14:12
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    $\begingroup$ @R.vanDobbendeBruyn Yes, this is the reason I am interested in this: Generators in the context of a category not admitting arbitrary direct sums. $\endgroup$
    – Jannik Pitt
    Commented Sep 16 at 14:21
  • $\begingroup$ Probably just means that the ind-category has a generator. $\endgroup$
    – Martin Brandenburg
    Commented Sep 27 at 0:28

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