nLab
spine

Definition

In simplicial sets

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

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

When $n>2$, an $(n,i)$-horn in $S$ has the same edges as any of its fillers, so we may speak of the spine of a $\mathrm{horn}$ as well.

In dendroidal sets

The above notion generalizes to dendroidal sets: the spine of a tree $T$ is the union over its corollas

$$\mathrm{Sp}(T)=\bigcup _{C_k\to T}{C}_k.$$Sp(T) = \bigcup_{C_{k} \to T} C_k \,.

For a linear tree this reproduces the above definition.

Interpretation in terms of higher categories

If $S$ is a Kan complex or quasi-category, then the spine of $\sigma$ is the maximal list of composable edges (1-morphisms) of $\sigma$.

Similarly, if $S$ is a fibrant dendroidal set or (∞,1)-operad, the spine $\mathrm{Sp}(T)\hookrightarrow T\to S$ of a tree $T$ in $S$ is a collection of composable operations in the $(\mathrm{\infty },1)$-operad.