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Apr 19, 2021 at 16:28 comment added Tyrone It might not be the exact direction you are looking for, but Hemmi-Kawamoto made use of the permuto-associahedra to define what they call $AC_n$-spaces. Their paper is very readable and the pair authored several similar papers.
Apr 17, 2021 at 22:20 comment added user1092847 Intuitively, I would expect the $1$-skeleton of Kapranov's permutoassociahedron to be a model: doi.org/10.1016/0022-4049(93)90049-Y
Apr 17, 2021 at 21:13 history asked Tim Campion CC BY-SA 4.0