Maths - Kan Fibration

Introduction

Kan fibrations (AKA Kan complexes) are part of the theory of simplicial sets. Kan fibrations are the fibrations of the standard model category structure on simplicial sets.

For a general discussion about fibrations see the page here.

Fibration and Co-fibration

Homotopy has the concept of:

• a fibration which has the lifting property.
• a co-fibration which has the extension property -extension is dual to lift.

	Fibration (lifting property)	Co-fibration (Extension Property)
Homotopy	Fibration (see page here)	Co-fibration (see page here)
Combinatorics (simplicial sets)	Kan fibration (see page here)	Kan extension (see page here)

Kan fibrations are combinatorial analogs of Serre fibrations of topological spaces.

(ncatlab).

Extension is dual to lift.

Fibrations and Cofibrations are used in model theory.

 

 

 

 

 

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Where Are You

This is one of a series of pages about topology, these pages start here.

For other pages, on the site, about topology see:

• Open Sets
• Fibre bundles
• Simplectical Complexes
• Delta Complexes
• Homotopy
• Homology
• Cubical Type Theory

Other pages

Other pages on this site which discuss fibres:

• In type theory terms.
• In category theory terms.

Computer Code Implementation

I have written some code to implement algebraic topology structures. For an explanation of this code see the page here.

Prerequisites

• Topology explained on this page.

Related Subjects

• Representables and Yoneda