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In mathematics, '''Kan complexes''' and '''Kan fibrations''' are part of the theory of [[simplicial set]]s. Kan fibrations are the fibrations of the standard [[model category]] for simplicial sets and are therefore of fundamental importance. Kan complexes are the [[fibrant object]]s in this model category. The name is in honor of [[Daniel Kan]].

==Definition==
[[File:2dKanFibration.svg|thumb|right|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 [[Simplicial_set#The_standard_n-simplex_and_the_simplex_category|standard <math>n</math>-simplex]], <math>\Delta^n</math>, is the representable simplicial set
:<math>\Delta^n(i) = \mathrm{Hom}_{\mathbf{\Delta}} ([i], [n])</math>
Applying the [[Simplicial_set#Geometric_realization|geometric realization]] functor to this simplicial set gives a space homeomorphic to the [[Simplex#The_standard_simplex|topological standard <math>n</math>-simplex]]: the convex subspace of ℝ<sup>n+1</sup> consisting of all points <math>(t_0,\dots,t_n)</math> such that the coordinates are non-negative and sum to 1.

For each ''k'' ≤ ''n'', this has a subcomplex <math>\Lambda^n_k</math>, the ''k''-th horn inside <math>\Delta^n</math>, 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 <math>\Delta^{n-1} \rightarrow \Delta^n</math> corresponding to all the other faces of <math>\Delta^n</math>.<ref>See Goerss and Jardine, page 7</ref> Horns of the form <math>\Lambda_k^2</math> sitting inside <math>\Delta^2</math> look like the black V at the top of the image to the right. If <math>X</math> is a simplicial set, then maps
: <math>s: \Lambda_k^n \to X</math>
correspond to collections of <math>n+1</math> <math>n</math>-simplices satisfying a compatibility condition. Explicitly, this condition can be written as follows. Write the <math>n</math>-simplices as a list <math>(s_0,\dots,s_{k-1},s_{k+1},\dots,s_{n+1})</math> and require that
: <math>d_i s_j = d_{j-1} s_i\,</math> for all <math>i < j</math> with <math>i,j \neq k</math>.<ref>See May, page 2</ref>
These conditions are satisfied for the <math>(n-1)</math>-simplices of <math>\Lambda_k^n</math> sitting inside <math>\Delta^n</math>.

[[File:Kan fibration.png|thumb|right|Lifting diagram for a Kan fibration]]
A map of simplicial sets <math>f: X\rightarrow Y</math> is a '''Kan fibration''' if, for any <math>n\ge 1</math> and <math>0\le k\le n</math>, and for any maps <math>s:\Lambda^n_k\rightarrow X</math> and <math>y:\Delta^n\rightarrow Y\,</math> such that <math>f \circ s=y \circ i</math>, there exists a map <math>x:\Delta^n \rightarrow X</math> such that <math>s=x \circ i</math> and
<math>y=f \circ x</math>. Stated this way, the definition is very similar to that of [[fibration]]s in [[topology]] (see also [[homotopy lifting property]]), whence the name "fibration".

Using the correspondence between <math>n</math>-simplices of a simplicial set <math>X</math> and morphisms <math>\Delta^n \to X</math> (a consequence of the [[Yoneda lemma]]), this definition can be written in terms of simplices. The image of the map <math>fs: \Lambda_k^n \to Y</math> can be thought of as a horn as described above. Asking that <math>fs</math> factors through <math>yi</math> corresponds to requiring that there is an <math>n</math>-simplex in <math>Y</math> whose faces make up the horn from <math>fs</math> (together with one other face). Then the required map <math>x: \Delta^n\to X</math> corresponds to a simplex in <math>X</math> whose faces include the horn from <math>s</math>. 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 <math>2</math>-simplex, if the black V above maps down to it then the striped blue <math>2</math>-simplex has to exist, along with the dotted blue <math>1</math>-simplex, mapping down in the obvious way.<ref>May uses this simplicial definition; see page 25</ref>

A simplicial set ''X'' is called a '''Kan complex''' if the map from ''X'' to 1, the one-point simplicial set, is a Kan fibration. In the [[model category]] for simplicial sets, <math>1</math> is the terminal object and so a Kan complex is exactly the same as a [[fibrant object]].

==Examples==
An important example comes from the [[Singular_homology#Singular_simplices|singular simplices]] used to define [[singular homology]]. Given a space <math>X</math>, define a singular <math>n</math>-simplex of X to be a continuous map from the standard topological <math>n</math>-simplex (as described above) to <math>X</math>,
: <math>f: \Delta_n \to X</math>
Taking the set of these maps for all non-negative <math>n</math> gives a graded set,
: <math>S(X) = \coprod_n S_n(X)</math>.
To make this into a simplicial set, define face maps <math>d_i: S_n(X)\to S_{n-1}(X)</math> by
: <math>(d_i f)(t_0,\dots,t_{n-1}) = f(t_0,\dots,t_{i-1},0,t_i,\dots,t_{n-1})\,</math>
and degeneracy maps <math>s_i: S_n(X)\to S_{n+1}(X)</math> by
: <math>(s_i f)(t_0,\dots,t_{n+1}) = f(t_0,\dots,t_{i-1},t_i + t_{i+1},t_{i+2},\dots,t_{n+1})\,</math>.
Since the union of any <math>n+1</math> faces of <math>\Delta_{n+1}</math> is a strong [[deformation retract]] of <math>\Delta_{n+1}</math>, any continuous function defined on these faces can be extended to <math>\Delta_{n+1}</math>, which shows that <math>S(X)</math> is a Kan complex.<ref>See May, page 3</ref>

It can be shown that the simplicial set underlying a simplicial [[group (mathematics)|group]] is always fibrant.

==Applications==
The [[homotopy group]]s 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.

==See also==
*[[Weak Kan complex]] (also called quasi-category, ∞-category)
*[[∞-groupoid]]

==References==
<references/>

==Bibliography==
*{{cite book
| last = Goerss
| first = Paul
| coauthors = Jardine, John
| title = Simplicial homotopy theory
| publisher = Birkhäuser
| year = 1999
| isbn = 3-7643-6064-X}}
*{{cite book
| last = May | first = Peter
| authorlink =J. Peter May
| title = Simplicial objects in algebraic topology | publisher = The university of Chicago press | year = 1967
| isbn = 0-226-51180-4}}

[[Category:Simplicial sets]]
[[Category:Homotopy theory]]

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en>Lychee512 at 17:51, 7 January 2015

en>Mogism: Typo fixing and cleanup, typos fixed: , → , using AWB