One way to introduce cohomology is to start with homology and reverse the arrows. Can we begin with a chain complex consisting of a sequence of matrices and invert these matrices to get a cochain complex?
There are lots of problems with this:
- The matrices are not all square so to find inverse we have to choose left or right inverse if they exist.
- Matrix inverse involves division by determinant, division may not exist in integers so we would have to go to matrix over rationals.
- Determinant of boundary function tends to be zero, see case below.
As an example lets use the chain complex of a flat disk:
The determinant of this matrix is zero so we can't invert it.
So the inverse arrows are not just free groups which can be represented by matrices as with homology.
More about reversing the arrows here.
The dual vector space is the set of all s V -> R where
|V* =[ a1 a2 a3 ]||V =|||||x2||||
|where a linear functional maps a vector into its underlying real space. This can be shown as a row (covector) times a column (vector):||
|If we choose the real value we are mapping to is zero then a vector in a space will map to a plane in its cospace.|
Using representable functors
We do this by replacing the free groups by their mapping into set (the integers). This gives a contravariant mapping, see 'representable functor'.
Here we add mappings into the integers Z.
Mappings such as these are contravariant
For instance take disc as example.
In order to create a representable functor we create a map which assigns an integer value to each point. This is a map from each point to a scalar value, we can represent this as a co-vector.
From this we can derive a map which assigns an integer value to each line and so on, going in the reverse direction.
|So, for example, the values at each point might represent voltages or heights or somthing like that.||
|The lines would then represent voltage differences, or height differences etc.||ab=4
|So we introduce functions from the elements of the homology (points, lines,triangles ...) to some other quantities.|