Kan fibration

Summary

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.

Definitions edit

Definition of the standard n-simplex edit

 
The striped blue simplex in the domain has to exist in order for this map to be a Kan fibration

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

 

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

Definition of a horn edit

For each k ≤ n, this has a subcomplex  , the k-th horn inside  , 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   corresponding to all the other faces of  .[1] Horns of the form   sitting inside   look like the black V at the top of the adjacent image. If   is a simplicial set, then maps

 

correspond to collections of    -simplices satisfying a compatibility condition, one for each  . Explicitly, this condition can be written as follows. Write the  -simplices as a list   and require that

  for all   with  .[2]

These conditions are satisfied for the  -simplices of   sitting inside  .

Definition of a Kan fibration edit

 
Lifting diagram for a Kan fibration

A map of simplicial sets   is a Kan fibration if, for any   and  , and for any maps   and   such that   (where   is the inclusion of   in  ), there exists a map   such that   and  . 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 edit

Using the correspondence between  -simplices of a simplicial set   and morphisms   (a consequence of the Yoneda lemma), this definition can be written in terms of simplices. The image of the map   can be thought of as a horn as described above. Asking that   factors through   corresponds to requiring that there is an  -simplex in   whose faces make up the horn from   (together with one other face). Then the required map   corresponds to a simplex in   whose faces include the horn from  . 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  -simplex, if the black V above maps down to it then the striped blue  -simplex has to exist, along with the dotted blue  -simplex, mapping down in the obvious way.[3]

Kan complexes defined from Kan fibrations edit

A simplicial set   is called a Kan complex if the map from  , the one-point simplicial set, is a Kan fibration. In the model category for simplicial sets,   is the terminal object and so a Kan complex is exactly the same as a fibrant object. Equivalently, this could be stated as: if every map   from a horn has an extension to  , meaning there is a lift   such that

 

for the inclusion map  , then   is a Kan complex. Conversely, every Kan complex has this property, hence it gives a simple technical condition for a Kan complex.

Examples edit

Simplicial sets from singular homology edit

An important example comes from the construction of singular simplices used to define singular homology, called the singular functor[4]pg 7

 .

Given a space  , define a singular  -simplex of X to be a continuous map from the standard topological  -simplex (as described above) to  ,

 

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

 .

To make this into a simplicial set, define face maps   by

 

and degeneracy maps   by

 .

Since the union of any   faces of   is a strong deformation retract of  , any continuous function defined on these faces can be extended to  , which shows that   is a Kan complex.[5]

Relation with geometric realization edit

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

 

giving the isomorphism

 

Simplicial sets underlying simplicial groups edit

It can be shown that the simplicial set underlying a simplicial group is always fibrant[4]pg 12. In particular, for a simplicial abelian group, its geometric realization is homotopy equivalent to a product of Eilenberg-Maclane spaces

 

In particular, this includes classifying spaces. So the spaces  ,  , and the infinite lens 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 edit

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   and is typically denoted  . We could have also replaced   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.

Non-example: standard n-simplex edit

It turns out the standard  -simplex   is not a Kan complex[6]pg 38. The construction of a counter example in general can be found by looking at a low dimensional example, say  . Taking the map   sending

 

gives a counter example since it cannot be extended to a map   because the maps have to be order preserving. If there was a map, it would have to send

 

but this isn't a map of simplicial sets.

Categorical properties edit

Simplicial enrichment and function complexes edit

For simplicial sets   there is an associated simplicial set called the function complex  , where the simplices are defined as

 

and for an ordinal map   there is an induced map

 

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

 

Exponential law edit

This complex has the following exponential law of simplicial sets

 

which sends a map   to the composite map

 

where   for   lifted to the n-simplex  . ^

Kan fibrations and pull-backs edit

Given a (Kan) fibration   and an inclusion of simplicial sets  , there is a fibration[4] pg 21

 

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

 

where   is the pull-back map given by pre-composiiton and   is the pushforward map given by post-composition. In particular, the previous fibration implies   and   are fibrations.

Applications edit

Homotopy groups of Kan complexes edit

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. For a Kan complex   and a vertex  , as a set   is defined as the set of maps   of simplicial sets fitting into a certain commutative diagram:

 

Notice the fact   is mapped to a point is equivalent to the definition of the sphere   as the quotient   for the standard unit ball

 

Defining the group structure requires a little more work. Essentially, given two maps   there is an associated  -simplice   such that   gives their addition. This map is well-defined up to simplicial homotopy classes of maps, giving the group structure. Moreover, the groups   are Abelian for  . For  , it is defined as the homotopy classes   of vertex maps  .

Homotopy groups of simplicial sets edit

Using model categories, any simplicial set   has a fibrant replacement   which is homotopy equivalent to   in the homotopy category of simplicial sets. Then, the homotopy groups of   can be defined as

 

where   is a lift of   to  . 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).

See also edit

References edit

  1. ^ See Goerss and Jardine, page 7
  2. ^ See May, page 2
  3. ^ May uses this simplicial definition; see page 25
  4. ^ a b c Goerss, Paul G.; Jardin, John F. (2009). Simplicial Homotopy Theory. Birkhäuser Basel. ISBN 978-3-0346-0188-7. OCLC 837507571.
  5. ^ See May, page 3
  6. ^ Friedman, Greg (2016-10-03). "An elementary illustrated introduction to simplicial sets". arXiv:0809.4221 [math.AT].
  • An elementary illustrated introduction to simplicial sets

Bibliography edit