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.
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Definitions
Definition of the standard nsimplex
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 consisting of all points such that the coordinates are nonnegative and sum to 1.
Definition of a horn
For each k ≤ n, this has a subcomplex , the kth horn inside , corresponding to the boundary of the nsimplex, with the kth 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
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
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
A simplicial set is called a Kan complex if the map from , the onepoint 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
Simplicial sets from singular homology
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 nonnegative 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
It is worth noting the singular functor is right adjoint to the geometric realization functor
giving the isomorphism
Simplicial sets underlying simplicial groups
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 EilenbergMaclane 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
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.
Nonexample: standard nsimplex
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
Simplicial enrichment and function complexes
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
This complex has the following exponential law of simplicial sets
which sends a map to the composite map
where for lifted to the nsimplex . ^
Kan fibrations and pullbacks
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 pullback map given by precomposiiton and is the pushforward map given by postcomposition. In particular, the previous fibration implies and 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 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 welldefined 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
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
 Model category
 Simplicial homotopy theory
 Simplicially enriched category
 Weak Kan complex (also called quasicategory, ∞category)
 ∞groupoid
 Fibration of simplicial sets
References
 ^ See Goerss and Jardine, page 7
 ^ See May, page 2
 ^ May uses this simplicial definition; see page 25
 ^ ^{a} ^{b} ^{c} Goerss, Paul G.; Jardin, John F. (2009). Simplicial Homotopy Theory. Birkhäuser Basel. ISBN 9783034601887. OCLC 837507571.
 ^ See May, page 3
 ^ Friedman, Greg (20161003). "An elementary illustrated introduction to simplicial sets". arXiv:0809.4221 [math.AT].
Bibliography
 Goerss, Paul G.; Jardine, John F. (1999). Simplicial Homotopy Theory. Basel: Birkhäuser Basel. doi:10.1007/9783034887076. ISBN 9783034897372. MR 1711612.
 May, J. Peter (1992) [1967]. Simplicial objects in algebraic topology. Chicago Lectures in Mathematics. Chicago, IL: University of Chicago Press. ISBN 0226511804. MR 1206474.