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.
Definitions.
Definition of the standard n-simplex.
For each "n" ≥ 0, recall that the standard formula_1-simplex, formula_2, is the representable simplicial set
Applying the geometric realization functor to this simplicial set gives a space homeomorphic to the topological standard formula_1-simplex: the convex subspace of ℝn+1 consisting of all points formula_5 such that the coordinates are non-negative and sum to 1.
Definition of a horn.
For each "k" ≤ "n", this has a subcomplex formula_6, the "k"-th horn inside formula_2, 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 formula_8 corresponding to all the other faces of formula_2. Horns of the form formula_10 sitting inside formula_11 look like the black V at the top of the adjacent image. If formula_12 is a simplicial set, then maps
correspond to collections of formula_1 formula_15-simplices satisfying a compatibility condition, one for each formula_16. Explicitly, this condition can be written as follows. Write the formula_15-simplices as a list formula_18 and require that
These conditions are satisfied for the formula_15-simplices of formula_23 sitting inside formula_2.
Definition of a Kan fibration.
A map of simplicial sets formula_25 is a Kan fibration if, for any formula_26 and formula_27, and for any maps formula_28 and formula_29 such that formula_30 (where formula_31 is the inclusion of formula_6 in formula_2), there exists a map formula_34 such that formula_35 and 
formula_36. 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.
Using the correspondence between formula_1-simplices of a simplicial set formula_12 and morphisms formula_39 (a consequence of the Yoneda lemma), this definition can be written in terms of simplices. The image of the map formula_40 can be thought of as a horn as described above. Asking that formula_41 factors through formula_42 corresponds to requiring that there is an formula_1-simplex in formula_44 whose faces make up the horn from formula_41 (together with one other face). Then the required map formula_46 corresponds to a simplex in formula_12 whose faces include the horn from formula_48. 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 formula_49-simplex, if the black V above maps down to it then the striped blue formula_49-simplex has to exist, along with the dotted blue formula_51-simplex, mapping down in the obvious way.
Kan complexes defined from Kan fibrations.
A simplicial set formula_12 is called a Kan complex if the map from formula_53, the one-point simplicial set, is a Kan fibration. In the model category for simplicial sets, formula_54 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 formula_55 from a horn has an extension to formula_2, meaning there is a lift formula_57 such thatformula_58for the inclusion map formula_59, then formula_12 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 7formula_61.Given a space formula_12, define a singular formula_1-simplex of X to be a continuous map from the standard topological formula_1-simplex (as described above) to formula_12,
Taking the set of these maps for all non-negative formula_1 gives a graded set,
To make this into a simplicial set, define face maps formula_69 by
and degeneracy maps formula_71 by
Since the union of any formula_73 faces of formula_74 is a strong deformation retract of formula_74, any continuous function defined on these faces can be extended to formula_74, which shows that formula_77 is a Kan complex.
Relation with geometric realization.
It is worth noting the singular functor is right adjoint to the geometric realization functorformula_78giving the isomorphismformula_79
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 spacesformula_80In particular, this includes classifying spaces. So the spaces formula_81, formula_82, and the infinite lens spaces formula_83 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.
Another important source of examples are the simplicial sets associated to a small groupoid formula_84. This is defined as the geometric realization of the simplicial set formula_85 and is typically denoted formula_86. We could have also replaced formula_84 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.
It turns out the standard formula_1-simplex formula_2 is not a Kan complexpg 38. The construction of a counter example in general can be found by looking at a low dimensional example, say formula_90. Taking the map formula_91 sendingformula_92gives a counter example since it cannot be extended to a map formula_93 because the maps have to be order preserving. If there was a map, it would have to sendformula_94but this isn't a map of simplicial sets.
Categorical properties.
Simplicial enrichment and function complexes.
For simplicial sets formula_95 there is an associated simplicial set called the function complex formula_96, where the simplices are defined asformula_97and for an ordinal map formula_98 there is an induced mapformula_99(since the first factor of Hom is contravariant) defined by sending a map formula_100 to the compositionformula_101
Exponential law.
This complex has the following exponential law of simplicial sets
formula_102
which sends a map formula_103 to the composite map
formula_104
where formula_105 for formula_106 lifted to the n-simplex formula_2.
Kan fibrations and pull-backs.
Given a (Kan) fibration formula_108 and an inclusion of simplicial sets formula_109, there is a fibration pg 21formula_110(where formula_111 is in the function complex in the category of simplicial sets) induced from the commutative diagramformula_112where formula_113 is the pull-back map given by pre-composiiton and formula_114 is the pushforward map given by post-composition. In particular, the previous fibration implies formula_115 and formula_116 are fibrations.
Applications.
Homotopy groups of Kan complexes.
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 formula_12 and a vertex formula_118, as a set formula_119 is defined as the set of maps formula_120 of simplicial sets fitting into a certain commutative diagram: formula_121Notice the fact formula_122 is mapped to a point is equivalent to the definition of the sphere formula_123 as the quotient formula_124 for the standard unit ballformula_125Defining the group structure requires a little more work. Essentially, given two maps formula_126 there is an associated formula_127-simplice formula_128 such that formula_129 gives their addition. This map is well-defined up to simplicial homotopy classes of maps, giving the group structure. Moreover, the groups formula_119 are Abelian for formula_131. For formula_132, it is defined as the homotopy classes formula_133 of vertex maps formula_118.
Homotopy groups of simplicial sets.
Using model categories, any simplicial set formula_12 has a fibrant replacement formula_136 which is homotopy equivalent to formula_12 in the homotopy category of simplicial sets. Then, the homotopy groups of formula_12 can be defined asformula_139where formula_140 is a lift of formula_118 to formula_136. 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).