Operads and a Broken Pentagon
It is a broken pentagon, but the top side is an identity if we want the sequence to form a 1-operad in the usual sense. Otherwise, the sequence of permutohedra form a kind of non-commutative operad. Moreover, there is a map of operads from these permutohedra to the diagrams with collapsing identities. This was outlined concretely by Loday.
This is an example of a low dimensional operadic map which we might otherwise have viewed as a broken parity cube in a tetracategorical context.