Koszulity of directed categories in representation stability theory
  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. 
Wee Liang Gan, Liping Li
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