A Kan fibration is one of the notions of fibrations of simplicial sets.
A Kan fibration is a morphism $\pi : Y \to X$ of simplicial sets with the lifting property for all horn inclusions.
This means that for
a commuting square with $n\ge 1$ and $0\le k\le n$, there always exists a lift
In terms of the canonical powering of simplicial sets over sets, this is equivalent to the morphisms
all being epimorphisms. (Here, for instance, $Y^{\Lambda^k[n]}$ is the set of tuples of $(n-1)$-cells in $Y$ that glue along their boundaries to an image of the $k$th $n$-horn.)
Kan fibrations are combinatorial analogs of Serre fibrations of topological spaces. In fact, under the Quillen equivalence of the standard model structure on topological spaces and the standard model structure on simplicial sets, Kan fibrations map to Serre fibrations.
Recall the shape of the horns in low dimension.
-$n=1$- The horns $\Lambda^1_0$ and $\Lambda^1_1$ of the 1-simplex are just copies of the 0-simplex $\Delta^0$ regarded as the left and right endpoint of $\Delta^1$. For $n= 1$ the above condition says that for $\pi : Y \to X$ a Kan fibration we have
corresponding to the lifting diagram
-$n=2$- the horn $\Lambda^2_1$ consists of the two top sides of a triangle. For this the Kan condition says that for any two composable 1-cells in $Y$ that have a “composite up to a 2-cell” in $X$, there exists a corresponding “composite up to a 2-cell” in $Y$ that projects down to the one in $X$:
This corresponds to the lifting diagram
This corresponds to the lifting diagram
A quasi-fibration or weak Kan fibration or inner Kan fibration of simplicial sets is defined as above, but with the lifting property only imposed in inner horns: $\Lambda^n_k$ with $0 \lt k \lt (n-1)$, not the outer horns $\Lambda^n_0$ and $\Lambda^n_n$.
This weakened condition then says that composition of cells may be lifted through the quasi-fibration, but not necessarily inversion of 1-cells. See fibrations of quasi-categories for more details.
Similarly, a left Kan fibration is one that has the lifting property for all horns except possibly the last one. and a right Kan fibration is one that has the lifting property for all horns except possibly the first one. See fibrations of quasi-categories for more details.
A Kan fibration $p : E \to B$ is called a minimal Kan fibration if for all cells $x,y : \Delta[n] \to E$ the condition $p(x) = p(y)$ and $\partial_i x = \partial_i y$ implies for all $k$ that $\partial_k x = \partial_k y$.
The acyclic Kan fibrations morphisms $f : X \to Y$ of Kan complexes that are both Kan fibrations as well as weak equivalences in that they induce isomorphisms on all simplicial homotopy groups (i.e. the acyclic fibrations of Kan complexes) are precisely the morphisms that have the right lifting property with respect to all boundary inclusions $\partial \Delta^n \hookrightarrow \Delta^n$:
e.g. (Goerss-Jardine, chapter I)
Kan fibrations and acyclic Kan fibrations are both stable under pullback.
Because every class of morphisms defined by a right lifting property is stable under pullback.
From this it follows readily that Kan complexes form a Brownian category of fibrant objects.
A Kan fibration $f\colon X\to Y$ is acyclic precisely if the fiber $f^{-1}(y)$ over each vertex $y$ is contractible.
Purely combinatorial proofs of this statement include (Joyal, prop. 8.23, Riehl-Verity 13, lemma 5.4.16)
A functor $F \colon C \to D$ between groupoids is k-surjective for all $k$ and hence a surjective equivalence of categories precisely if under the nerve $N(F) : N(C) \to N(D)$ it induces an acyclic fibration of Kan complexes;
We know that both $N(C)$ and $N(D)$ are Kan complexes. By the above theorem it suffices to show that $N(f)$ being a surjective equivalence is the same as having all lifts
We check successively what this means for increasing $n$:
$n= 0$. In degree 0 the boundary inclusion is that of the empty set into the point $\emptyset \hookrightarrow {*}$. The lifting property in this case amounts to saying that every point in $N(D)$ lifts through $N(F)$.
This precisely says that $N(F)$ is surjective on 0-cells and hence that $F$ is surjective on objects.
$n=1$. In degree 1 the boundary inclusion is that of a pair of points as the endpoints of the interval $\{\circ, \bullet\} \hookrightarrow \{\circ \to \bullet\}$. The lifting property here evidently is equivalent to saying that for all objects $a,b \in Obj(C)$ all elements in $Hom(F(a),F(b))$ are hit. Hence that $F$ is a full functor.
$n=2$. In degree 2 the boundary inclusion is that of the triangle as the boundary of a filled triangle. It is sufficient to restrict attention to the case that the map $\partial \Delta[2] \to N(C)$ sends the top left edge of the triangle to an identity. Then the lifting property here evidently is equivalent to saying that for all objects $a,b \in Obj(C)$ the map $F_{a,b} : Hom(a,b) \to Hom(F(a),F(b))$ is injective. Hence that $F$ is a faithful functor.
Kan fibrations and quasi-fibrations are fibrations in two common model structures on simplicial sets.
Recall that the horn $\Lambda^k[n]$ is the boundary of the $n$-simplex $\Delta^n$ with one face removed. If in the above definition one replaces horns with the full boundaries of simplices, one obtaines the definition of a hypercover, the acyclic fibrations in the classical model structure on simplicial sets.
A simplicial set $X$ for which the unique morphism $X \to pt$ to the terminal simplicial set is a Kan fibration is called a Kan complex.
A simplicial set $X$ for which the unique morphism $X \to pt$ to the terminal simplicial set is a quasi-fibration/weak Kan fibration is called a quasi-category.
Just as the underlying simplicial set of a simplicial group is a Kan complex (see algorithm at simplicial group), so also given any simplicial morphism $f : G\to H$ of simplicial groups for which in each dimension, $n$, the homomorphism $f_n : G_n \to H_n$ is an epimorphism, then the underlying simplicial map of simplicial sets is a Kan fibration. (Apart from a careful choice of section in each dimension, the proof can be constructed from the algorithm given in simplicial group.)
A morphism of simplicial sets that has the left lifting property with respect to all Kan fibrations is called an anodyne morphism.
Kan fibration, anodyne morphism
A standard textbook account is
See also
Andre Joyal Theory of Quasi-Categories and its applications pdf
Emily Riehl, Dominic Verity, Homotopy coherent adjunctions and the formal theory of monads, (arXiv:1310.8279)