  Reading's "shard intersection order" on the symmetric group can be realized
as the "lattice of wide subcategories" of the corresponding preprojective
algebra. In this paper, we first use Bancroft's combinatorial model for the
shard intersection order to associate a unique shard to each downward cover
relation. We then show that, under the correspondence with wide subcategories,
this process coincides with Jasso's "$\tau$-tilting reduction". In particular,
this yields a combinatorial model for this algebras's $\tau$-exceptional
sequences" (defined by Buan and Marsh). We formulate this model using the
combinatorics of arc diagrams. Finally, we use our model to give a new
representation-theoretic proof that the shard intersection order is
EL-shellable in type A.
