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Revision as of 17:42, 13 December 2007
, 17:42, 13 December 2007→Commentary Note 11.13: markup
===Commentary Note 11.13===
===Commentary Note 11.13===
As we make our way toward the foothills of Peirce's 1870 LOR, there is one piece of equipment that we dare not leave the plains without — for there is little hope that "l'or dans les montagnes là" will lie among our prospects without the ready use of its leverage and lifts — and that is a facility with the utilities that are variously called "arrows", "morphisms", "homomorphisms", "structure-preserving maps", and several other names, in accord with the altitude of abstraction at which one happens to be working, at the given moment in question.
As we make our way toward the foothills of Peirce's 1870 LOR, there
is one piece of equipment that we dare not leave the plains without --
As a middle but not too beaten track, I will lay out the definition of a morphism in the forms that we will need right off, in a slight excess of formality at first, but quickly bringing the bird home to roost on more familiar perches.
for there is little hope that "l'or dans les montagnes là" will lie
among our prospects without the ready use of its leverage and lifts --
Let's say that we have three functions ''J'', ''K'', ''L'' that have the following types and that satisfy the equation that follows:
and that is a facility with the utilities that are variously called
"arrows", "morphisms", "homomorphisms", "structure-preserving maps",
: ''J'' : ''X'' ← ''Y''
and several other names, in accord with the altitude of abstraction
at which one happens to be working, at the given moment in question.
: ''K'' : ''X'' ← ''X'' × ''X''
: ''L'' : ''Y'' ← ''Y'' × ''Y''
: ''J''(''L''(''u'', ''v'')) = ''K''(''Ju'', ''Jv'')
Our sagittarian leitmotif can be rubricized in the following slogan:
Our sagittarian leitmotif can be rubricized in the following slogan:
: The image of the ligature is the compound of the images.
Where J is the "image", K is the "compound", and L is the "ligature".
Where ''J'' is the "image", ''K'' is the "compound", and ''L'' is the "ligature".
Figure 19 presents us with a picture of the situation in question.
Figure 19 presents us with a picture of the situation in question.
<pre>
o-----------------------------------------------------------o
o-----------------------------------------------------------o
| |
| |
o-----------------------------------------------------------o
o-----------------------------------------------------------o
Figure 19. Structure Preserving Transformation J : K <- L
Figure 19. Structure Preserving Transformation J : K <- L
</pre>
Here, I have used arrowheads to indicate the relational domains
Here, I have used arrowheads to indicate the relational domains at which each of the relations ''J'', ''K'', ''L'' happens to be functional.
at which each of the relations J, K, L happens to be functional.
Table 20 gives the constraint matrix version of the same thing.
Table 20 gives the constraint matrix version of the same thing.
<pre>
Table 20. Arrow: J(L(u, v)) = K(Ju, Jv)
Table 20. Arrow: J(L(u, v)) = K(Ju, Jv)
o---------o---------o---------o---------o
o---------o---------o---------o---------o
| L # Y | Y | Y |
| L # Y | Y | Y |
o---------o---------o---------o---------o
o---------o---------o---------o---------o
</pre>
One way to read this Table is in terms of the informational redundancies
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 ''L'' row, along with all of the constraints in the ''J'' columns, then the constraint in the K row is automatically true. That is one way of understanding the equation: ''J''(''L''(''u'', ''v'')) = ''K''(''Ju'', ''Jv'').
that it schematizes. In particular, it can be read to say that when one
satisfies the constraint in the L row, along with all of the constraints
in the J columns, then the constraint in the K row is automatically true.
That is one way of understanding the equation: J(L(u, v)) = K(Ju, Jv).
===Commentary Note 11.14===
===Commentary Note 11.14===
