Homotopy theory

homotopy theory, (∞,1)-category theory, homotopy type theory

flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed

models: topological, simplicial, localic, …

see also algebraic topology



Paths and cylinders

Homotopy groups

Basic facts




If SS 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 SS is the maximal list of composable edges (morphisms) of SS.

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


In simplicial sets

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


The spine of an nn-simplex σS n\sigma \in S_n, also called its backbone, is the union of the edges (1-cells) 010 \to 1, 121 \to 2, \cdots , (n1)n(n-1) \to n between the successive vertices of σ\sigma.

When n>2n \gt 2, an (n,i)(n,i)-horn in SS 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


The spine of a tree TT is the union over its corollas?

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

In (Cisinski-Moerdijk) this is called the Segal core of TT.


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


Characterizations of compositions


A simplicial set XX is the nerve of a category precisely if for all nn \in \mathbb{N} all the morphisms induced from the spine inclusion

X Sp[n]Δ[n]:X nX 1× X 0× X 0X 1 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 XX is the dendroidal nerve of a symmetric operad over Set (a symmetric multicategory), precisely if for all trees TT the morphisms induced from the spine inclusion X Sp[T]Δ[n]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 .


For any tree TT, the spine inclusion Sp[T]Ω[T]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

Revised on June 1, 2017 06:07:26 by Igor Khavkine (