### Operads and a Broken Pentagon

In The Combinatorics of Iterated Loop Spaces, Batanin describes an operad based on the poset of faces of the nth Stasheff associahedron. The case of the pentagon looks like this:

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.

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.

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