Kan Fibration

Definition

A Kan fibration is a map of simplicial sets $p:X\to Y$ with the right lifting property against all horn inclusions ${\Lambda }_k^n\hookrightarrow {\Delta }^n$ for $n\ge 1$ and $0\le k\le n$. Equivalently, every commutative square

$$\begin{array}{rlr}{\Lambda }_k^n& ⟶X& \\ \downarrow & & \downarrow p\\ {\Delta }^n& ⟶Y& \end{array}$$

admits a diagonal filler ${\Delta }^n\to X$ making both triangles commute.

Properties

• Stable under pullback, composition, and retracts in simplicial sets.
• In the standard Quillen model structure on simplicial sets, fibrations are precisely the Kan fibrations; trivial fibrations are the maps with RLP against all boundary inclusions $\partial {\Delta }^n\hookrightarrow {\Delta }^n$.
• The terminal map $X\to 1$ is a Kan fibration iff $X$ is a Kan complex (all horn fillers exist in $X$).

Related Concepts

• Fibration
• Topological Fibration
• Hurewicz Fibration
• Serre Fibration
• Grothendieck Fibration

References

¹²

Footnotes

1. https://ncatlab.org/nlab/show/Kan+fibration

2. https://people.math.rochester.edu/faculty/doug/otherpapers/gj.pdf