nLab spine

Idea

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.

Definition

In simplicial sets

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

Definition

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.

In dendroidal sets

The above notion generalizes to dendroidal sets

Definition

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 $T$.

Remark

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

Properties

Characterizations of compositions

Proposition

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

Closure and lifting properties

In the following, tree means “finite non-planar rooted tree” as used in the definition of dendroidal set .

Proposition

For any tree $T$, the spine inclusion $Sp[T] \hookrightarrow \Omega[T]$ is an inner anodyne morphism.

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

Related concepts

• horn, boundary of a simplex

References

Dendroidal spines are discussed in section 1 of

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