Internalisation
An arrow f in any category C can be turned into an arrow in some category of categories by taking the functor from the category 2 into C which picks out the arrow f. Similarly, in the category of categories, one can replace a square by a quadruple of squares. 
The arrow 0 in the diagram represents the source of an arrow, whereas the arrow 1 is the target. Similarly, a triangle is really a hexagon, and so on.
The arrow 0 in the diagram represents the source of an arrow, whereas the arrow 1 is the target. Similarly, a triangle is really a hexagon, and so on.
posted by Kea | 3:57 PM
3 Comments:
Hi Kea,
I am a little confused by fg=hk.
If O is the origin if both g and k and E is the endpoint of f and h, then are fg and hk:
a - combinationally equivalent
rather than
b - permutationally equal?
The origin and endpoints are identical.
The pathways are different two_1_vectors.
There is also one_2_vector O->E pathway that may or may not be utilized.
See Figure 1.9 of 'Plotting Complex Sinusoids as Circular Motion' which relates squares to arcs or circles and better represents my perspective.
http://ccrma.stanford.edu/~jos/filters/Plotting_Complex_Sinusoids_Circular.html
http://ccrma.stanford.edu/~jos/filters/
Plotting_Complex_Sinusoids_Circular.html
is the complete patway to my reference.
Sorry, Doug. I was just pointing out that commutative squares in 1-categories often represent algebraic equations of this nature. Yeah, it would have been better to omit it.
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