Maths - Cofibration

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.

If we reverse the arrows in the diagram for fibrations (on the previous page) we get the diagram for cofibrations:

Co-fibration involves the concept of extension

Extension is dual to lift.

diagram
If we have a path and part of that path is specified by an interval, how do we extend that interval? diagram
When we inject A×I U X×0 into X×I we seem to be able to fill in the missing corner. diagram
Here the diagram has been flipped, to go from left to right, to correspond to the diagrams in Wiki and nCatLab: diagram
diagram This looks complicated so lets approach it differently
  diagram
  diagram

As a motivating example lets look at a simplicial complex as discussed on the page here.

Each face (above dimension 0) will contain multiple sub-faces so this starts the look like a many:one relationship.

diagram
face maps
diagram However, in a complex, faces can share the same boundary so there is a many:one relationship in the opposite direction. For instance, 'ab' is contained in both 'abc' and 'abd'.
So the relationship between say, a triangle and a line is a many:many relationship. This can be modeled like this:
diagram
co fibre sequence

So each of these maps is like a subset of a product.

This can be modeled using linear algebra (vector and matrix) although not quite in the usual way. Where the shapes are the vectors (just a list of subshapes) and the relationships between them are matrices. Say:

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