If is an (∞,n)-category (including the case of ∞-groupoids, (∞,1)-categories, …) or a model thereof (including Kan complexes or quasi-categories, …) then the spine of is the maximal list of composable edges (morphisms) of .
Let be a simplicial set.
When , an -horn in has the same edges as any of its fillers, so we may speak of the spine of a horn as well.
The above notion generalizes to dendroidal sets
In (Cisinski-Moerdijk) this is called the Segal core of .
For a linear tree this reproduces the above definition of spines of simplices.
More generally, a dendroidal set is the dendroidal nerve of a symmetric operad over Set (a symmetric multicategory), precisely if for all trees the morphisms induced from the spine inclusion are bijections.
For simplicial sets, this is a classical statement (Grothendieck / Segal). Its homotopical weakening leads to the notion of Segal category and complete Segal space. For dendroidal sets this is (Cisinski-Moerdijk, cor. 2.7).
This is (Cisinski-Moerdijk, prop. 2.4).
Dendroidal spines are discussed in section 1 of