Maths - Mappings Between Topologies

The most important part of this subject is the relationship between different topologies. The mappings or morphisms between them.

For example, the continuous 'rubber geometry' deformations as between a toroid and a mug.

There are two related mapping types continuous mappings and homeomorphisms, we start with continuous mappings.

Open Set Criterion for Continuity

Continuous Maps

A function is continuous if it doesn't jump, that is, when two inputs of the function get close to each other then the corresponding outputs of the function get close to each other.

Here we discuss the most general form of continuous map which also applies to non-metric topological spaces.

A continuous mapping implies a limited kind of reversibility, at least locally.

Definition 1

Here is one definition of continuity, based on open sets:

Let X and Y be topological spaces. A function f : X->Y is continuous if f-1(V) is open for every open set V in Y. diagram

Definition 2

Another definition of continuity, based on neighbourhoods, is equivalent to the above definition.

Let X and Y be topological spaces. A function f : X->Y is continuous if for every x∈X and every open set U containing f(x), there exists a neighbourhood V of x such that f(V)containsU. continuity diagram

More detail about continuity on page here.

Continuous Surjective Maps

diagram

Here is an extreme case of a Surjective Map which maps to a single point.

This seems to meet the requirements as there is an open set round the whole preimage. We can think of surjections which meet these requirements as 'nice surjections'.

See fibrations

Continuous Injective Maps

diagram

Here is an extreme case of a Injective Map which maps from a single point.

In this case all the open sets in the codomain need to map back to a single open set in the domain. This meets the requirements so it is a 'nice injection'.

See cofibrations

Map from Interval

Start by thinking of the interval as a continuous line segment from 0 to 1.

So a map from I represents all possible values between X and Y. for example, all the intermediate shapes between a toroid and a mug.

A homotopy is a higher level map between two maps.

diagram

Homotopic Maps

diagram

They are homotopic if:

there existsH: X × [0,1] -> Y

where:

  • H(x,0) = F(x)
  • H(x,1) = G(x)

We can think of this as 'filling in' the gap between F and G so that we can take any path through it.

There is a good video about this on a site here.

We can represent a continuous map between shapes as a homotopy by mapping from the identity mapping. diagram
diagram Or we could redraw it like this.

Homotopy

A homotopy is a function between functions with certain properties.

Let X and Y be topological spaces and f,g : X->Y be two continuous maps. Then f is said to be homotopic to g (written f~g) if there exists a map F : X×power set-> Y, called a homotopy such that F(x,0) = f(x) and F(x,1) = g(x) .

where:

  • f~g f is said to be homotopic to g
  • power setrepresents the unit interval [0,1]
  • continuity, in this context, is defined on page here
homotopy

Homotopy Equivalence of Spaces

diagram  

There is a good video about this on a site here.

What does a continuous mapping look like?

A continuous mapping may collapse multiple points to single points.
Here we convert that to neigbourhoods.
 

So this looks like a fibre bundle as discussed on the page here.

Counter Examples

 
 

Examples - Möbius Band to Circle

Here we have a morphism which collapses the width of the Möbius band.
when we reverse this morphism we keep the global structure and loose some local structure. We also loose the twist in the Möbius band.

Examples - Helix to Circle

Here we have a morphism which collapses a helix to a circle. diagram helix map
 

Homeomorphisms

A homeomorphism preserves 'nearness' but allows 'rubber geometry' deformations.

Note: homeomorphism is a different concept from homomorphism.

homeomorphism

Projections

We can project helix onto circle and still keep continuity. projections

Homotopy Extension Property (HEP)

Lets consider mapping f0 from a topological space X to a topological space Y. How can we determine if this mapping is a continuous mapping? We have already seen a possible answer to this above (f0 : X->Y is continuous if f0-1(V) is open for every open set V in Y). There are other ways to determine this if we don't have details of open sets. diagram
diagram Here is a example, its obvious just by looking at it that this map is continuous, we can just deform X to get Y (its got the same number of holes) but some maps may not be continuous and it may not be obvious. So we need to formalise it.
Sometimes we have an additional piece of information. That is that there is a subset of X called A, this has a map into Y that is known to be continuous, this is f0 restricted to A denoted: f0|A . So the question is: when can this continuous map be extended to the whole of f0 ? diagram
diagram

To be sure to map to Y can be extended the map from X to its subset A must be a retraction.

A retraction is a surjective map that is continuous.

This ensures that A has the same number of holes as X.

This diagram is intended to show the homotopy and the mappings along the interval. To do this the topologies X, A and Y are shown as some sort of one dimensional projection.

So the whole mapping is shown as X×I starting at X×{0} and mapping to Y at X×{1}.

diagram
diagram

This is often illustrated in terms of category theory diagrams (for instance wiki here). This diagram is rotated compared to the diagrams above.

Here the map from the interval is curried. So YI is all the maps out of the interval I->Y. Instead of having X×I -> Y we have
X->I->Y which is X->YI.

There is a good video about this on a site here.

Example

Are the two shapes on the right 'homotopy equivalent' ?

Is there a continuous map between them in both directions?

diagram
diagram We can use the homotopy extension property to contract to middle line of X to a point.
All the other points on the boundary of X map to corresponding points on the boundary of A. diagram

HEP and Topos Theory

Relationship to topos theory.

If we look again at the category theory diagram for the HEP. diagram

and then compare it to the diagram for the sub-object classifier (see page here) we can see that there is a similar thing going on:

In the Topos case there may not be a requirement for continuous mappings but S is a subobject of X and the mapping between them is injective.

diagram

HEP and Kan Extensions

Relationship to Kan extensions.

Starting again at the category theory diagram for the HEP. diagram
Compare it with the diagram for the Kan extension (see page here). The HEP appears to be a special case of the Kan extension (most things are!) so, in this diagram, 'F' is a constant mapping and 'K' is injective. diagram
In order to make F constant we factor it through the terminal object. diagram

Next

see pages about:


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see also:

Video about homotopy on a site here.

see pages about:

 

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