ONT Re: Higher Order Categorical Logic -- Discussion
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
HOC. Discussion Note 11
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
I will now introduce a number of different ways of
looking at morphisms as structure preserving maps.
Let's suppose we have three functions, f : X -> Y,
G : X x X -> X, and H : Y x Y -> Y, that satisfy
the following equation for all pairs u, v in X.
f(G(u, v)) = H(f(u), f(v))
Our morphic leitmotif can be rubricized by way of the following slogan:
The image of the resultant is resultant of the images.
Here, f produces the images, G the first resultant, and H the second resultant.
Figure 1 presents a diagram of the situation in question.
o-----------------------------------------------------------o
| |
| G H |
| @ @ |
| /|\ /|\ |
| / | \ / | \ |
| / | v / | v |
| o o o o o o |
| X X X Y Y Y |
| o o o o o o |
| \ \ \ ^ ^ ^ |
| \ \ \ / / |
| \ \ / \ / / |
| \ \ \ / |
| \ / \ / \ / |
| @ @ @ |
| f f f |
| |
o-----------------------------------------------------------o
Figure 1. Structure Preserving Map f : (X, G) -> (Y, H)
Figure 1 uses arrows to indicate the relational domains at which
each of the relations f, G, H happens to be functional. That is,
it is more like the feathers of the arrows that serve to mark the
relational domains at which the relations f, G, H are functional,
but it would take yet another construction to make this precise,
as the feathers are not uniquely appointed but many splintered.
Table 2 shows the constraint matrix version of the same thing.
Table 2. f(G(u, v)) = H(f(u), f(v))
o---------o---------o---------o---------o
| % f | f | f |
o=========o=========o=========o=========o
| G % X | X | X |
o---------o---------o---------o---------o
| H % Y | Y | Y |
o---------o---------o---------o---------o
One way to read this Table is in terms of the informational redundancies
that it schematizes. In particular, it can be read to say that when one
satisfies the constraint in the G row, along with all of the constraints
in the f columns, then the constraint in the H row is automatically true.
This is the same information as the equation, f(G(u, v)) = H(f(u), f(v)).
Jon Awbrey
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
http://www.cs.bsu.edu/homepages/mighty/history.html
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o