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Wikipedia / Kan fibration

In mathematics, Kan complexes and Kan fibrations are part of the theory of simplicial sets. 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 various kinds of fibrations for simplicial sets, see Fibration of simplicial sets.

Definitions

Definition of the standard n-simplex

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

Applying the geometric realization functor to this simplicial set gives a space homeomorphic to the n-simplex such that the coordinates are non-negative and sum to 1.

Definition of a horn

For each k≤n, the k, 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. is a simplicial set, then maps

and require that

  1. .

.

Definition of a Kan fibration

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

Technical remarks

(a consequence of the Yoneda lemma-simplex, mapping down in the obvious way.

Kan complexes defined from Kan fibrations

is called a Kan complex, the one-point simplicial set, is a Kan fibration. In the model category is the terminal object and so a Kan complex is exactly the same as a fibrant object such thatfor the inclusion map is a Kan complex. Conversely, every Kan complex has this property, hence it gives a simple technical condition for a Kan complex.

Examples

Simplicial sets from singular homology

An important example comes from the construction of singular simplices used to define singular homology, called the singular functorpg 7.,

gives a graded set,

  1. .

by

by

  1. .

is a strong deformation retract, any continuous function is a Kan complex.

Relation with geometric realization

It is worth noting the singular functor is right adjoint to the geometric realization functorgiving the isomorphism

See also: Milnor's theorem on Kan complexes.

Simplicial sets underlying simplicial groups

It can be shown that the simplicial set underlying a simplicial group is always fibrantpg 12. In particular, for a simplicial abelian group, its geometric realization is homotopy equivalent to a product of Eilenberg-Maclane spacesIn particular, this includes classifying 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.

Geometric realizations of small groupoids

with an infinity groupoid. It is conjectured that the homotopy category of geometric realizations of infinity groupoids is equivalent to the homotopy category of homotopy types. This is called the homotopy hypothesis.

Mapping space

is a Kan complex.

Postnikov section

Let X be a Kan complex. Then the n-th Postnikov section is the coequalizer. Then the following can be verified directly:

  • is a Kan fibration.

  • and choice of a base point on X.

  • and choice of a base point on X.

For a simplicial set X is an Ex∞ functor.

Non-example: standard n-simplex

is not a Kan complexpg 38 sending because the maps have to be order preserving. If there was a map, it would have to sendbut this isn't a map of simplicial sets.

Categorical properties

Simplicial enrichment and function complexes

there is an associated simplicial set called the function complex, where the simplices are defined as there is an induced map to the composition

Exponential law

This complex has the following exponential law of simplicial sets to the composite map.

Kan fibrations and pull-backs

, there is a fibration pg 21 is in the function complex in the category of simplicial sets) induced from the commutative diagram are fibrations.

The above is a consequence of a theorem of Gabriel and Zisman.

Homotopy groups of Kan complexes

The homotopy groups of simplicial sets fitting into a certain commutative diagram: for the standard unit ball.

Homotopy groups of simplicial sets

can be defined as. These fibrant replacements can be thought of a topological analogue of resolutions of a chain complex (such as a projective resolution or a flat resolution).

Kan

Kan complexes themselves form a weak Kan complex called Kan. Namely, first consider the category K where objects are Kan complexes and morphisms maps of simplicial sets. As a category of presheaves has internal Hom, each hom-set in K has a structure of a simplicial set; in short, K is a simplicial category. The homotopy coherent nerve of it

is then a weak Kan complex (∞-category). In view of homotopy hypothesis or some other variants.

See also: universal left fibration.

See also

Footnotes

References