Kan fibration

Kan fibrations are the fibrations of the standard model category structure on simplicial sets and are therefore of fundamental importance.

Kan complexes are the fibrant objects in this model category.

The name is in honor of Daniel Kan. For each n ≥ 0, recall that the standard

, is the representable simplicial set Applying the geometric realization functor to this simplicial set gives a space homeomorphic to the topological standard

, 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

is a simplicial set, then maps correspond to collections of

Stated this way, the definition is very similar to that of fibrations in topology (see also homotopy lifting property), whence the name "fibration".

(a consequence of the Yoneda lemma), this definition can be written in terms of simplices.

Since the black V in the lower diagram is filled in by the blue

, the one-point simplicial set, is a Kan fibration.

In the model category for simplicial sets,

Conversely, every Kan complex has this property, hence it gives a simple technical condition for a Kan complex.

, any continuous function defined on these faces can be extended to

In particular, for a simplicial abelian group, its geometric realization is homotopy equivalent to a product of Eilenberg-Maclane spaces

are correspond to Kan complexes of some simplicial set.

In fact, this set can be constructed explicitly using the Dold–Kan correspondence of a chain complex and taking the underlying simplicial set of the simplicial abelian group.

Another important source of examples are the simplicial sets associated to a small groupoid

This is defined as the geometric realization of the simplicial set

gives a counter example since it cannot be extended to a map

but this isn't a map of simplicial sets.

there is an associated simplicial set called the function complex

(since the first factor of Hom is contravariant) defined by sending a map

This complex has the following exponential law of simplicial sets

is in the function complex in the category of simplicial sets) induced from the commutative diagram

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.

of simplicial sets fitting into a certain commutative diagram:

is mapped to a point is equivalent to the definition of the sphere

Defining the group structure requires a little more work.

in the homotopy category of simplicial sets.