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




In simplicial homotopy theory a horn of a simplex is a simplicial incarnation of the notion of a boundary segment (as opposed to the full boundary) of a cell.

Specifically, the kkth horn Λ k[n]Δ[n]\Lambda_k[n] \hookrightarrow \Delta[n] of the nn-dimensional simplex (defined for n1n \geq 1, see Rem. ) is the simplicial set obtained from the boundary of the n-simplex Δ n\partial \Delta^n of the standard simplicial n-simplex Δ n\Delta^n by discarding its kkth face.

The property of a morphism of simplicial set to have the right lifting property against the inclusions of horns into their simplices (have “horn fillers”) characterizes the basic types of fibrations and fibrant objects in simplicial homotopy theory:

Specifically, Kan fibrations are the simplicial analog of Serre fibrations (in a precise sense so under the Quillen equivalence between the classical model structure on simplicial sets and on topological spaces) and under this correspondence the horn inclusions into the nnth simplex correspond to the inclusion of the (n1)(n-1)-dimensional topological disk into its cylinder:

Λ k[n]Δ[n]||D n1(id,0)D n1×[0,1], \Lambda_k[n] \hookrightarrow \Delta[n] \;\;\;\; \overset{\left\vert-\right\vert}{\mapsto} \;\;\;\; D^{n-1} \overset{(id,0)}{\hookrightarrow} D^{n-1} \times [0,1] \,,

as both are the generating cofibrations in their respective model categories (on sSet, on Top, respectively).



Δ[n]=Δ(,[n])SimpSet \Delta[n] = \mathbf{\Delta}( -, [n]) \in Simp Set

denote the standard simplicial nn-simplex in SimpSet.



For all n,in,i with n1n\geq 1 and 0in0\leq i \leq n, the (n,i)(n,i)-horn or (n,i)(n,i)-box is the sub-simplicial set

Λ i[n]Δ[n] \Lambda^i[n] \hookrightarrow \Delta[n]

which is the union of all faces except the i thi^{th} one.

This is called an outer horn if k=0k = 0 or k=nk = n, otherwise it is an inner horn.


Since sSet is a presheaf topos, unions of subobjects make sense and they are calculated objectwise, thus in this case dimensionwise. This way it becomes clear what the structure of a horn as a functor Λ k[n]:Δ opSet\Lambda^k[n]: \Delta^{op} \to Set must therefore be: it takes [m][m] to the collection of ordinal maps f:[m][n]f: [m] \to [n] which factor through some coface map [n1][n][n-1] \to [n] which is not the i thi^{th} one.


(the 0-simplex has no horns)
Beware that any would-be horn of the 0-simplex (whose boundary is the empty simplicial set) is ex-cluded in Def. : The 0-simplex has no horn.

This is not a matter of convention if one sticks to the usual definition of Kan fibration as having right lifting against all horns: In particular one must not declare a would-be horn of the 0-simplex to be itself the empty simplicial set, as that would make the nerve of the empty groupoid fail to be a Kan complex (Def. ).

Conversely, the fact that every horn is inhabited means that any morphism out of the empty simplicial set (an empty bundle X\varnothing \to X) is a Kan fibration (Def. ).



The inner horn of the 2-simplex

Δ 2={ 1 0 2}\Delta^2 = \left\{ \array{ && 1 \\ & \nearrow &\Downarrow& \searrow \\ 0 &&\to&& 2 } \right\}

with boundary

Δ 2={ 1 0 2}\partial \Delta^2 = \left\{ \array{ && 1 \\ & \nearrow && \searrow \\ 0 &&\to&& 2 } \right\}

looks like

Λ 1 2={ 1 0 2}. \Lambda^2_1 = \left\{ \array{ && 1 \\ & \nearrow && \searrow \\ 0 &&&& 2 } \right\} \,.

The two outer horns look like

Λ 0 2={ 1 0 2}\Lambda^2_0 = \left\{ \array{ && 1 \\ & \nearrow && \\ 0 &&\to&& 2 } \right\}


Λ 2 2={ 1 0 2}\Lambda^2_2 = \left\{ \array{ && 1 \\ & && \searrow \\ 0 &&\to&& 2 } \right\}


Lifting against horn inclusion


(Kan fibration)
A Kan fibration is a morphism of simplicial sets which has the right lifting property with respect to all horn inclusions Λ k[n]Δ n\Lambda^k[n] \hookrightarrow \Delta^n.


(Kan complex)
A Kan complex is a simplicial set in which “all horns have fillers”: a simplicial set for which the morphism to the point is a Kan fibration (Def. ).


A quasi-category is a simplicial set in which all inner horns have fillers, hence such that the morphism to the point is an inner fibration.


See the references at simplicial homotopy theory.

Last revised on April 16, 2021 at 02:18:06. See the history of this page for a list of all contributions to it.