Introduction
This page continues from the model category page here and fibration page here.
Fibration and Cofibration
Homotopy has the concept of:
 a fibration which has the lifting property.
 a cofibration which has the extension property extension is dual to lift.
Fibration 
Cofibration (Extension Property) 


Homotopy  Fibration 
Cofibration (see page here) 
Combinatorics 
Kan fibration (see page here) 
Kan extension (see page here) 
Kan fibrations are combinatorial analogs of Serre fibrations of topological spaces.
Following on from the example lets try to show the HEP in a more general form.
There are conditions to allow this. the arrows from A must play well with the map F. 
Mapping Cylinder and Mapping Cone
A sort of generalised test diagram.
Extension Property in Topology
This is the inverse of the lift property. There is an injection and in this example the open sets in the other direction collapse to an open set with a single vertex. In the more general case maps from an open subset can be expanded to the whole (open) set. 
For more about the extension property see the page here.
Extension as Dual to Lift.How can we get HEP by reversing the arrows in the diagram for fibrations (on the previous page)? Cofibration involves the concept of extension

If we have a path and part of that path is specified by an interval, how do we extend that interval? 
When we inject A×I U X×0 into X×I we seem to be able to fill in the missing corner. 
HEP in Simplicial Complex
A simplicial complex has the HEP with respect to any subcomplex. This allows homotopies to be constructed piece by piece.
simplicial complex is discussed on the page here. Each face (above dimension 0) will contain multiple subfaces so this starts the look like a many:one relationship. 
face maps 
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:  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:
 0 means subset relationship does not exist.
 1 means subset relationship exists.
Next
 sheaves
 Fibrations and Cofibrations are used in model theory.
 Kan Extensions
Related Pages on this Site