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To my shame I have to admit that I have as yet not looked much into opetopes and opetopic sets.

I am in the process of writing nLab entries on dendroidal sets and noticed that some remarks on the relation to opetopic sets should probably be in order.

Now, I know that I should just sit down and read the opetopic literature. But while I am busy doing that, and since the model structure on dendroidal sets wasn't around when most of it was written: does anyone know more about the relation?

edited Nov 17 2009 at 1:20

Greg Kuperberg
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asked Nov 17 2009 at 0:16

Urs Schreiber
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Tom Leinster · Answer 1 · 2009-11-18 06:50:29Z UTC

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Urs, do you have a reason to think that there'll be much to say about this? I can see that opetopic and dendroidal sets are both presheaf categories that arise in higher-dimensional category theory. I can see that in both cases, the small category on which you're taking presheaves has a graphical or geometric interpretation, and there are some "face maps", and there's something tree-like going on. But beyond that, I don't see what there is to say. Do you have something in mind?

I just looked in Ittay Weiss's thesis, Dendroidal Sets. "Opetope" is not in the index, nor is the relevant Baez--Dolan paper (Higher Dimensional Algebra III) cited. So I guess he had no thoughts on the matter.

edited Nov 18 2009 at 6:50

answered Nov 18 2009 at 4:32

Tom Leinster
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	I tried to figure out whether opetopic objects are equivalent to dendroidal objects, or if maybe one is a special case of the other. I gave up just because I did not (yet) find a concise, explicit description of the opetopic base category. – Greg Kuperberg Nov 18 2009 at 5:45
	Yes, defining the category of opetopes (the presheaves on which are the opetopic sets) is no easy task. Defining the objects seems appreciably easier; it's the maps that make it hard. – Tom Leinster Nov 18 2009 at 6:49
	Tom, no, this was just a naive question. You might read it in part as asking "Can anyone give me a better idea of what opetopic sets are like?" I was hoping that the answer would have been that opetopic sets are somehow a vast generalization of dendroidal sets. That would have given me motivation to look into them more closely. But if you say there is no non-superficial relation, I'll take your word for it. Thanks. – Urs Schreiber Nov 18 2009 at 7:45
	Greg, thanks for this comment. Even though a negative "result", that's useful to know. – Urs Schreiber Nov 18 2009 at 7:48
	Well, I don't say there is no non-superficial relation - just that after a few minutes' thought, I can't see one. That doesn't mean much! – Tom Leinster Nov 18 2009 at 9:02

Relation between dendroidal and opetopic sets

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