# 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.

### Catagorical Products in Directed Graphs

Following on from the concept of catagorical product (discussed on page here) here we have an example where we have the product of two directed graphs.

 In this example the objects to be multiplied together (B and C) both consist of two points and a directed line segment between them. The product will be a plane surface defined by the arrows shown here and their compositions. To see this in more detail see the example of products in a simplicial set on this page.
 Concentrating on the connection between the product and C. There is a map from the points in the product to the points in C (shown as red arrows). We can construct an arrow, in the opposite direction, from the arrows in the product to the arrows in C (shown as blue double-line arrow). So the structure in C 'lifts' up to the same structure between sub-graphs.

This is a similar construct to a fibration (see page here) but here we have not invoked the concepts of a topological space or continuous maps between them.