homotopy theory, (∞,1)-category theory, homotopy type theory
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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 .
Similarly, if is an (∞,1)-operad or a model thereof (given by a dendroidal set or Segal operad, etc. ), the spine of a tree in is a collection of composable operations in the -operad.
Let be a simplicial set.
The spine of an -simplex , also called its backbone, is the union of the edges (1-cells) , , , between the successive vertices of .
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.
A simplicial set is the nerve of a category precisely if for all all the morphisms induced from the spine inclusion
are bijections.
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).
In the following, tree means “finite non-planar rooted tree” as used in the definition of dendroidal set .
For any tree , the spine inclusion is an inner anodyne morphism.
This is (Cisinski-Moerdijk, prop. 2.4).
Dendroidal spines are discussed in section 1 of
Last revised on June 1, 2017 at 10:07:26. See the history of this page for a list of all contributions to it.