# Maths - Cofibration

 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 fibration - lift co-fibration - extension Extension is dual to lift.
 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.
 Here the diagram has been flipped, to go from left to right, to correspond to the diagrams in Wiki and nCatLab:
 This looks complicated so lets approach it differently
 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. 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.