Kan fibration

Definition

For each n ≥ 0, recall that the standard n -simplex, Δ n , is the representable simplicial set

Applying the geometric realization functor to this simplicial set gives a space homeomorphic to the topological standard n -simplex: the convex subspace of ℝⁿ⁺¹ consisting of all points ( t 0 , … , t n ) such that the coordinates are non-negative and sum to 1.

For each k ≤ n, this has a subcomplex Λ k n , the k-th horn inside Δ n , corresponding to the boundary of the n-simplex, with the k-th face removed. This may be formally defined in various ways, as for instance the union of the images of the n maps Δ n − 1 → Δ n corresponding to all the other faces of Δ n . Horns of the form Λ k 2 sitting inside Δ 2 look like the black V at the top of the adjacent image. If X is a simplicial set, then maps

correspond to collections of n + 1 n -simplices satisfying a compatibility condition. Explicitly, this condition can be written as follows. Write the n -simplices as a list ( s 0 , … , s k − 1 , s k + 1 , … , s n + 1 ) and require that

These conditions are satisfied for the ( n − 1 ) -simplices of Λ k n sitting inside Δ n .

A map of simplicial sets f : X → Y is a Kan fibration if, for any n ≥ 1 and 0 ≤ k ≤ n , and for any maps s : Λ k n → X and y : Δ n → Y such that f ∘ s = y ∘ i , there exists a map x : Δ n → X such that s = x ∘ i and y = f ∘ x . Stated this way, the definition is very similar to that of fibrations in topology (see also homotopy lifting property), whence the name "fibration".

Using the correspondence between n -simplices of a simplicial set X and morphisms Δ n → X (a consequence of the Yoneda lemma), this definition can be written in terms of simplices. The image of the map f s : Λ k n → Y can be thought of as a horn as described above. Asking that f s factors through y i corresponds to requiring that there is an n -simplex in Y whose faces make up the horn from f s (together with one other face). Then the required map x : Δ n → X corresponds to a simplex in X whose faces include the horn from s . The diagram to the right is an example in two dimensions. Since the black V in the lower diagram is filled in by the blue 2 -simplex, if the black V above maps down to it then the striped blue 2 -simplex has to exist, along with the dotted blue 1 -simplex, mapping down in the obvious way.

A simplicial set X is called a Kan complex if the map from X to 1, the one-point simplicial set, is a Kan fibration. In the model category for simplicial sets, 1 is the terminal object and so a Kan complex is exactly the same as a fibrant object.

Examples

An important example comes from the singular simplices used to define singular homology. Given a space X , define a singular n -simplex of X to be a continuous map from the standard topological n -simplex (as described above) to X ,

Taking the set of these maps for all non-negative n gives a graded set,

To make this into a simplicial set, define face maps d i : S n ( X ) → S n − 1 ( X ) by

and degeneracy maps s i : S n ( X ) → S n + 1 ( X ) by

Since the union of any n + 1 faces of Δ n + 1 is a strong deformation retract of Δ n + 1 , any continuous function defined on these faces can be extended to Δ n + 1 , which shows that S ( X ) is a Kan complex.

It can be shown that the simplicial set underlying a simplicial group is always fibrant.

Applications

The homotopy groups of a fibrant simplicial set may be defined combinatorially, using horns, in a way that agrees with the homotopy groups of the topological space which realizes it.