Kan Fibration¶
A Kan fibration is a simplicial-set map that lifts every compatible horn over a specified simplex in its target.
Core Idea¶
A Kan fibration is a map \(p:X\to S\) of simplicial sets for which every compatible horn in \(X\) can be completed to a simplex that maps to a specified completed simplex in \(S\). The test covers every dimension and every inner and outer horn. It guarantees a lift exists, not that it is unique.[^ref-dac61b20ce20]
Scope of Application¶
For the identity \(\Delta^1\to\Delta^1\), the horn \(\Lambda^1_0=\{0\}\) over the edge \(0\to1\) lifts by that same edge. The vertex inclusion \(\Delta^0\to\Delta^1\) fails the identical square: upstairs only the constant edge at \(0\) exists. A distinct relative example is the constant simplicial abelian quotient \(\mathbb Z\to\mathbb Z/2\). Its horn vertex \(3\) over base \(1\pmod2\) lifts by the constant edge \(3\); Stacks proves all-horn lifting for every degreewise-surjective simplicial abelian homomorphism.[ref-dac61b20ce20][ref-20dc53b46adb][^ref-08ebfbaae361]
Clarity¶
Distinguish Kan complexes, which are objects, from Kan fibrations, which are maps. A filler in the total object must also map to the chosen base simplex; an unrelated filler does not solve the lifting square. Existence does not require a unique or algorithmically chosen filler. These are definition boundaries, not intrinsic design tensions.
Manages Complexity¶
The all-horn lifting predicate packages many extension questions into one map property, and known pullback and composition results let the property be reused.[^ref-08ebfbaae361]
Abstract Reasoning¶
For a proposed map, take a horn and a compatible full simplex downstairs. Ask whether the horn has a full extension upstairs mapping to that exact simplex. One failed square disproves the property; a theorem covering all such squares proves it.
Knowledge Transfer¶
The typed lifting test carries between algebraic and homotopical uses of simplicial sets. Their structure is the prerequisite carrier, but being a simplicial set does not itself entail horn filling. A different structure called a “fibration” does not inherit the Kan condition by name.
[^ref-dac61b20ce20]: Kerodon, “Kan Fibrations,” Definition 3.1.1.1. [^ref-08ebfbaae361]: The Stacks Project, Section 14.31, “Kan fibrations”. [^ref-20dc53b46adb]: Kerodon, “Horns,” Example 1.2.4.3.
Relationships to Other Abstractions¶
Current abstraction Kan Fibration Domain-specific
Parents (1) — more general patterns this builds on
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Kan Fibration presupposes Simplicial set Domain-specific
Kan fibrations are lifting properties of maps between simplicial sets.
Hierarchy path (1) — routes to 1 parentless root
- Kan Fibration → Simplicial set → Compositionality
Neighborhood in Abstraction Space¶
Kan Fibration sits in a sparse region of the domain-specific corpus (93rd percentile for distinctiveness): few abstractions share its structure, so a faithful description tends to retrieve it precisely.
Family — Unclustered & Miscellaneous (2551 abstractions)
Nearest neighbors
- Categorical Lift — 0.81
- Singular Homology — 0.80
- Fiber Product of Schemes — 0.79
- Complete variety — 0.78
- Delta set — 0.78
Computed from structural-signature embeddings · 2026-10-08