It is known that finitely generated FI-modules over a field of characteristic 0 are Noetherian. We generalize this result to the abstract setting of an infinite EI category satisfying certain combinatorial conditions.

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In the first part of this paper, we study Koszul property of directed graded categories. In the second part of this paper, we prove a general criterion for an infinite directed category to be Koszul. We show that infinite directed categories in the theory of representation stability are Koszul over a field of characteristic zero

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