# Maths - Fibration

## Introduction

This page continues from the model category page which is introduced on the page here.

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

## 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.

### Lifting Property in Topology

 This fibration is like a fibre bundle in sets, the differences are: The arrows are continuous. This gives extra structure which allows arrows in B (shown in red) to be 'lifted' to arrows between bundles. That is: the fibration goes from E to B (the vertices map in that direction) but the open sets map in the other direction so if we first map each point to an open set containing only that point we can then map to E. So composing these arrows gives 'lift'.

For more about the lifting property see the page here.

### Fibration - Line to Point Map Example

 To keep things simple, take the example of a line continuously retracted down to a point.
 To see this, in its most general form, as a continuous map between topological spaces I have drawn the line consisting of representative points and open sets.
 For it to be a continuous map the preimage of an open set must be an open set. We don't want to have to make any arbitrary decisions about which open set to map back to, so the preimage is the open set around the whole line. The red arrow is a map between open sets.

This is the homotopy lifting property. When we look at more complicated examples (see fibration page) we will see structure between points being lifted to structure between open sets.

### Fibration and Simplicial Sets

 There are multiple surjective maps, from the triangle to the line, depending on which points are merged together. Its a bit messy to show all the mappings in this diagram so, for each mapping, I have only shown the points that are merged together. In each case the other point goes to the other end of the line.
 For it to be a continuous map the preimage of an open set must be an open set. Since we are mapping between simplicies can we treat them as topologies and lines as open sets? So a line must have a preimage.

This is the homotopy lifting property.

## Fibrations and Indexing

In sets we can look at fibrations as an 'I' indexed family of sets. To denote this I have replaced B with the indexing set 'I' and replaced E with the family of sets X, like this: In the slice category of sets:

 These are the objects These are the arrows

That is, in the slice category the indexing set remains the same.

Here are some examples of comma categories. In these cases all the objects are in the same category (including the fixed object). The comma category generalises this by allowing the objects to come from other categories.

#### Arrow category

see page here

• Objects f:X->I
• Morphisms <s,t>

fibration

#### Slice category

see page here

• Objects f:X->I
• Morphisms s

#### Co-slice category

see page here

• Objects f:X->I
• Morphisms t

substitution

## Fibrations and 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.

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

 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.

#### 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.

 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.

 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.

 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.

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

 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.

 But E only needs to be like a product locally.

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

## 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.

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 MÃ¶bius band.

### 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.

## 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.

## 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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