Polarised noncrossing partititions and the coherent self-dual $\omega$-equivalence

We construct an acyclic augmented chain complex of abelian groups whose entry in degree $n > 0$ is free on the set of noncrossing partitions of degree $n-1$ equipped with a $\{0, 1\}$-labelling of their gaps.

The definition of the differential in this complex is related, via a restricted Leibniz rule, to the gap-insertion operad of Ebrahimi-Fard, Foissy, Kock, and Patras.

We conjecture that this augmented chain complex is the linearisation of a polygraph presenting a self-dual model of the coherent walking $\omega$-equivalence constructed by the author, Loubaton, Ozornova, and Rovelli, and provide evidence for this conjecture.