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Concept
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 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 each n ≥ 0, recall that the , , 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.
Horn of a simplex
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 . 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 .
These conditions are satisfied for the -simplices of sitting inside .
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".
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).
Official source
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