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 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.

## Next

- sheaves
- Fibrations and Cofibrations are used in model theory.
- Kan Extensions