spine

If $S$ 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 $S$ is the maximal list of *composable* edges (morphisms) of $S$.

Similarly, if $S$ is an (∞,1)-operad or a model thereof (given by a dendroidal set or Segal operad, etc. ), the spine $Sp(T) \hookrightarrow T \to S$ of a tree $T$ in $S$ is a collection of composable operations in the $(\infty,1)$-operad.

Let $S : \Delta^{op} \to Set$ be a simplicial set.

The **spine** of an $n$-simplex $\sigma \in S_n$, also called its **backbone**, is the union of the edges (1-cells) $0 \to 1$, $1 \to 2$, $\cdots$ , $(n-1) \to n$ between the successive vertices of $\sigma$.

When $n \gt 2$, an $(n,i)$-horn in $S$ 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

The **spine** of a tree $T$ is the union over its corollas?

$Sp(T) = \bigcup_{C_{k} \to T} C_k
\,.$

In (Cisinski-Moerdijk) this is called the *Segal core* of $T$.

For a linear tree this reproduces the above definition of spines of simplices.

A simplicial set $X$ is the nerve of a category precisely if for all $n \in \mathbb{N}$ all the morphisms induced from the spin inclusion

$X^{Sp[n] \hookrightarrow \Delta[n]} : X_n \to X_1 \times_{X_0} \cdots \times_{X_0} X_1$

are bijections.

More generally, a dendroidal set $X$ is the dendroidal nerve of a symmetric operad over Set (a symmetric multicategory), precisely if for all trees $T$ the morphisms induced from the spine inclusion $X^{Sp[T] \hookrightarrow \Delta[n]}$ 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 $T$, the spine inclusion $Sp[T] \hookrightarrow \Omega[T]$ is an inner anodyne morphism.

This is (Cisinski-Moerdijk, prop. 2.4).

Dendroidal spines are discussed in section 1 of

- Denis-Charles Cisinski, Ieke Moerdijk,
*Dendroidal Segal spaces and infinity-operads*(arXiv:1010.4956)

Revised on April 18, 2012 01:01:56
by Tim Campion?
(140.180.17.71)