Commuting categories for blocks and fusion systems

We extend the notion of a commuting poset for a finite group to p-blocks and fusion systems, and we generalize a result, due originally to Alperin and proved independently by Aschbacher and Segev, to commuting graphs of blocks, with a very short proof based on the G-equivariant version, due to Thevenaz and Webb, of a result of Quillen.