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




The horn Λ k[n]=Λ k nΔ n\Lambda_k[n] = \Lambda^n_k \hookrightarrow \Delta^n is the simplicial set obtained from the boundary of the n-simplex Δ n\partial \Delta^n of the standard simplicial nn-simplex Δ n\Delta^n by discarding the kkth face.



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

be 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 subsimplicial 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 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\}


Relation to other concepts

  • 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.

  • 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.

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

  • The boundary of a simplex is the union of its faces.

  • The spine of a simplex is the union of all its generating 1-cells.

Last revised on April 17, 2018 at 15:59:11. See the history of this page for a list of all contributions to it.