ct.category theory - When is the Yoneda product graded commutative?

Another starting point is to think of ${rm Ext}(A,A)$ as the derived endomorphism ring of the object $A$ and recall Schur's lemma. If $A$ is a finitely-generated simple module over a ring $R$, then ${rm Hom}_R(A,A)$ is a division algebra. For example, if $R$ is a $k$-algebra over an algebraically closed field $k$, then ${rm Hom}_R(A,A)$ is isomorphic to $k$ (so, in particular, it is commutative.) Via Freyd-Mitchell embedding, this should give some idea what to expect in degree $0$.

Going back the question, then, the examples one might have in mind are categories of modules over a group ring or enveloping algebra of a graded Lie algebra: in these cases, ${rm Ext}(k,k)$ is group- or Lie algebra cohomology, respectively, and has a graded-commutative cup product, where $k$ is the trivial module.

Perhaps there is a suitable "semisimplicity" hypothesis one could impose on the category so that ${rm Ext}(A,A)$ is graded-commutative for all simple objects $A$?