A version of Alperinʼs weight conjecture for finite category algebras

Let p be a prime and k an algebraically closed field of characteristic p. We construct a functor C→OC on the category of finite categories with the property that if G=C is a finite group, then OC is the orbit category of p-subgroups of G. This leads to an extension of Alperinʼs weight conjecture to any finite category C, stating that the number of isomorphism classes of simple kC-modules should be equal to that of the weight algebra W(kOC) of OC. We show that the versions of Alperinʼs weight conjecture for finite groups and for finite categories are in fact equivalent.