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 standard n {\displaystyle n} -simplex, Δ n {\displaystyle \Delta ^{n}} , is the representable simplicial set
Δ n ( i ) = H o m Δ ( [ i ] , [ n ] ) {\displaystyle \Delta ^{n}(i)=\mathrm {Hom} _{\mathbf {\Delta } }([i],[n])}
Applying the geometric realization functor to this simplicial set gives a space homeomorphic to the topological standard n {\displaystyle n} -simplex: the convex subspace of R n + 1 {\displaystyle \mathbb {R} ^{n+1}} consisting of all points ( t 0 , … , t n ) {\displaystyle (t_{0},\dots ,t_{n})} such that the coordinates are non-negative and sum to 1.
Definition of a horn For each k ≤ n, this has a subcomplex Λ k n {\displaystyle \Lambda _{k}^{n}} , the k-th horn inside Δ n {\displaystyle \Delta ^{n}} , 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 Δ n − 1 → Δ n {\displaystyle \Delta ^{n-1}\rightarrow \Delta ^{n}} corresponding to all the other faces of Δ n {\displaystyle \Delta ^{n}} . Horns of the form Λ k 2 {\displaystyle \Lambda _{k}^{2}} sitting inside Δ 2 {\displaystyle \Delta ^{2}} look like the black V at the top of the adjacent image. If X {\displaystyle X} is a simplicial set, then maps
s : Λ k n → X {\displaystyle s:\Lambda _{k}^{n}\to X}
correspond to collections of n {\displaystyle n} ( n − 1 ) {\displaystyle (n-1)} -simplices satisfying a compatibility condition, one for each 0 ≤ k ≤ n {\displaystyle 0\leq k\leq n} . Explicitly, this condition can be written as follows. Write the ( n − 1 ) {\displaystyle (n-1)} -simplices as a list ( s 0 , … , s k − 1 , s k + 1 , … , s n ) {\displaystyle (s_{0},\dots ,s_{k-1},s_{k+1},\dots ,s_{n})} and require that
d i s j = d j − 1 s i {\displaystyle d_{i}s_{j}=d_{j-1}s_{i}\,} for all i < j {\displaystyle i<j} with i , j ≠ k {\displaystyle i,j\neq k} . These conditions are satisfied for the ( n − 1 ) {\displaystyle (n-1)} -simplices of Λ k n {\displaystyle \Lambda _{k}^{n}} sitting inside Δ n {\displaystyle \Delta ^{n}} .
Definition of a Kan fibration
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