Introduction
This page continues from the model category page here and fibration page here.
Fibration and Co-fibration
Homotopy has the concept of:
- a fibration which has the lifting property.
- a co-fibration which has the extension property -extension is dual to lift.
Fibration |
Co-fibration (Extension Property) |
|
|---|---|---|
| Homotopy | Fibration |
Co-fibration (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.
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.
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. |
 |
| 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.
Next
- sheaves
- Fibrations and Cofibrations are used in model theory.
- Kan Extensions
Related Pages on this Site
