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The meta picture is: (non-planar) dendroidal sets are to symmetric colored operads as simplicial sets are to categories.

This suggests that one should have the notion of planar dendroidal sets (with a straightforwards definition?) and they should be in the same relation as above with non-symmetric colored operad. Is the theory of planar dendroidal sets developed anywhere? I'm somehow unable to google a link.

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Planar dendroidal sets (aka non-symmetric dendroidal sets) are discussed in these lectures of Moerdijk (Lecture 3) with the graphical category denoted $\Omega_p$. For more details, see the textbook by Heuts and Moerdijk.

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  • $\begingroup$ This answer (and other recent ones) should have more information, instead of only the link to the paper of Heuts and Moerdijk. Try to understand the background of each OP and then explain it to them. $\endgroup$
    – David White
    Commented Apr 3, 2023 at 11:33

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