Kan fibration

The notion of a Kan fibration is a part of the theory of simplicial sets in mathematics. Kan fibrations are the fibrations in the model category theoretic sense on the category of simplicial sets and are therefore of fundamental importance. The name is in honor of Daniel Kan.

Definition

For each "n ≥ 0", recall that the "standard n-simplex" $Delta^n$, is the representable simplicial set:$Delta^n(i)\; =\; Hom\_\{mathbf\{Delta\; (\; [i]\; ,\; [n]\; )$Geometrically, this corresponds to a solid "n"-simplex, such as the convex subspace of ℝⁿ⁺¹ consisting of all points ("t₀, ..., t_n") such that each "t_i" is positive and their sum equals "1".

For each "k ≤ n", this has a subcomplex $Lambda^n\_k$, the "k"-th horn inside $Delta^n$, corresponding to the boundary of the "n"-simplex, with the "k"-th face removed. (It may be formally defined in various ways, as for instance the union of the images of the "n" maps $Delta^\{n-1\}\; ightarrow\; Delta^n$ corresponding to all the other faces of $Delta^n$.)

A map of simplicial sets $f:\; X\; ightarrow\; Y$ is a "Kan fibration" if for any $kle\; n$and for any maps $s:Lambda^n\_k\; ightarrow\; X$ and $y:Delta^n\; ightarrow\; Y$ such that $f\; circ\; s=y\; circ\; i$, there exists a map $x:Delta^n\; ightarrow\; X$ such that $s=x\; circ\; i$ and $y=f\; circ\; x$. Therefore, this definition very much resembles the one of fibrations in topology, see also homotopy lifting property, whence the name "fibration".

Using the correspondence of "n"-simplices of a simplicial set "X" and morphisms$Delta\_n\; ightarrow\; X$ (this is an instance of the Yoneda lemma), the Kan-fibration property means that if there are given "n+1" "n"-simplices :"s₀, ..., s_k-1, s_k+1, ..., s_n+1 ∈ X_n", "fitting together", i.e. :"d_u s_t = d_t s_u-1"for all suitable "u, t" ("d_u" denotes the "u"-th boundary operator of the simplicial set "X"), together with an "n+1"-simplex "y ∈ Y_n+1", such that "f(s_u) = d_u(y)",there is a (not necessarily unique) "n+1"-simplex "x ∈ X_n+1" mapping to "y" under "f" and, applying the boundary operators, restricting to the given "s_u": "f(x) = y", "d_u(x)=s_u".

Morally, one may say that it means that a horn in $X$ extendsto a simplex in $Y$ then it extends to a simplex in $X$ in a way thatlifts the simplex in $X$.

A simplicial set "X" is called fibrant, if the unique morphism to the one-point simplicial set is a Kan fibration. This amounts to saying that for a collection of "n"-simplices fitting together as above, there is a "n+1"-simplex restricting to the given simplices, or in other words that every horn in "X" has a filler.

Examples

* An important example is the simplicial set of simplices, i.e. continuous maps from the standard (topological) simplex to a topological space "X"::$S(X)\; :=\; \{\; Delta\_n^\{top\}\; ightarrow\; X\; \}$

Using the fact that the topological horn is a strong deformation retract of the full simplex, one shows that this simplicial set is fibrant.
* It can be shown that the simplicial set underlying a simplicial group is always fibrant.

Applications

The homotopy groups of a fibrant simplicial set may be defined combinatorially, using horns, in a way that agrees withthe homotopy groups of the topological space which realizes it.

References

cite book
last = Goerss
first = Paul
coauthors = Jardine, John
title = Simplicial homotopy theory
publisher = Birkhäuser
year = 1999
isbn = 376436064X

cite book
last = May | first = Peter
authorlink = Peter May
title = Simplicial objects in algebraic topology | publisher = The university of Chicago press | year = 1967
isbn = 0226511804