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at interval object we have a section that discusses how in a category with interval object for every object there are various incarnations of its "path groupoid".
We had had two such incarnations there: the first one discusses the structure of a Trimble n-category on this "path groupoid", the second one the structure of a simplicial set.
I want one more such incarnation: the structure of a planar dendroidal set.
A proposal for how that should work I have now typed in the new section titled currently Fundamental little 1-cubes space induced from an interval.
(This section title is bad, I need to think of something better...)
Eventually I want to see if this can be pushed to constitute the necessary ingredients for a "May recognition principle" in a general oo-stack oo-topos: over a site C with interval object, I want for each k a dendroidal presheaf that encodes something like the C-parameterized little k-cubes operad, which should act on k-fold loop oo-stacks on C.
That's the motivation, at least.
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