Pulling back data along a functor
We’ll migrate data between the graph-indexing schema Gr and the loop schema, called DDS (for discrete-dynamical system).
We’ll write down an example instance \(\mathbf{DDS}\xrightarrow{I}\mathbf{Set}\)
| State | Next |
|---|---|
| 1 | 4 |
| 2 | 4 |
| 3 | 5 |
| 4 | 5 |
| 5 | 5 |
| 6 | 7 |
| 7 | 6 |
We want to convert this state information into a graph that will let us visualize our machine.
Use the following functor \(F\):
src is sent to identity
Can now generate a graph using the composite functor \(\mathbf{Gr}\xrightarrow{F}\mathbf{DDS}\xrightarrow{I}\mathbf{Set}\)
| Arr | src | tar |
|---|---|---|
| 1 | 1 | 4 |
| 2 | 2 | 4 |
| 3 | 3 | 5 |
| 4 | 4 | 5 |
| 5 | 5 | 5 |
| 6 | 6 | 7 |
| 7 | 7 | 6 |
\(Vert = \bar{7}\)
We can now draw the graph:
This procecure can be called “pulling back data along a functor". We pulled back data \(I\) along functor \(F\) (via functor composition).