Maths - Fibration

Introduction

This page continues from the page about fiber bundles here. Here this is generalised to the notion of a fibration.

Fibrations have additional structure to fiber bundles which allows some structure in B to be 'lifted' to E. (see Wikipedia page)

Lifting

Given a path [0,1] in the base space B and a point e0 in the total space E.

We can 'lift' this path into the total space E as a path starting at e0.

diagram

Here I am mostly concentrating on discrete structures so we need a looser concept of continuous mappings.

diagram

An inclusion from [0,1] into B can be lifted into E:

If we are using CW-complexes to model our topological spaces then we can use a slightly weaker form of fibration known as a Serre fibration.

diagram

Simple Example

Fibration and co-fibration is related to fibre bundles as discussed on the page here. To recap from this page we can start with a surjective mapping between sets and then add extra structure to these sets.

In this example we have two sets B the base set and E the total set and a surjective mapping p between them.

diagram

We get a fibre bundle by trying to get as close as we can to the inverse of the function p without making arbitrary choices. diagram

On this page want to expand the idea of fibre bundles on sets to add more structure.

As an example lets go from sets to directed graphs. The function p must now map the arrows as well as the vertices.

diagram

What is the closest we can get to an inverse function?

Here is a possibility. This 'lifts' the vertices to sets which may contain multiple vertices.

diagram

Is this valid because two paths join and this is like a tear which is not allowed in topology/homotopy transforms?

diagram

So E is like a product.

Here we lift the interval to the red arrow in E.

note: we think of this as being continuous even though this example has discrete vertices.

diagram

But E only needs to be like a product locally.

diagram

Can we think of this lifted mapping as being between the sets?

diagram

Fibration

We can generalise a fibre bundle to have more structure than a type indexed by a set:

  • A fibration allows us to use something more complicated than a set to index types.
  • A sheaf allows us to relate local and global structure.

Fibration

In this diagram B represents the base space and E represents the total space. B is now more than the index set because it now has structure. p is a projection of the total space into the base space.

diagram
diagram

The diagram needs to have the lifting property which says that there must be an arrow h~ from the path to E.

So given the structure in E0 we can get the structure at other points.

Fibration Example

A cylinder (S1*I) mapped onto a circle (S1).

This map is a projection locally (upto homeomorphism).

Since we only require a projection locally we could also project from a helix or a Möbius band.

diagram

Fibrations of Graphs

theory: topological graph theory

undirected graph: covering projection
directed graph: fibration (weaker form of covering projection)

Here is an example for directed multigraphs.

Each node in the top graph maps to the bottom graph (fibration). The corresponding node always has the same number and colour of incoming arcs (but not necessarily outgoing arcs).

This gives some sort of local invariance.

graph fibration

Fibration and Co-fibration

Homotopy has the concept of:

 

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.

Next

If we reverse the arrows in the diagram for fibrations we get the diagram for cofobrations:

Co-fibration involves the concept of extension

Extension is dual to lift.


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see also:

 

Correspondence about this page

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