Lawrence,
I don't think isomorphism allows the kind of equivalence that one finds between orthogonally propagating fields. Because category theory is a theory of isomorphic relations, it assumes that relation is more fundamental than equivalence. I mean, specifically: an equation assumes properties of equality, reflexivity and transitivity. An isomorphic relation does not assume transitivity--consider Lee Smolin's view of a pure relational model*:
"R1: There is no background...
R2: The fundamental properties of the elementary entities consist entirely in relationships between these elementary entities...
R3: The relationships are not fixed, but evolve according to law. Time is nothing but changes in the relationships, and consists of nothing but their ordering..."
So things in relation are not necessarily equivalent.
Isomorphisms are static representations, but one would need demonstrate an evolutionary principle that obviates a background and drives sytem change in a self-organizing manner. This is the view I personally favor, and which led me to the conclusion that time (the least action principle) is the only fundamental property driving change in the universe. I agree in principle with Fotini Markopoulou that "Space does not exist, so time can." That's why my model is purely algebraic in real (Lebesgue measure) terms, crossing over from analysis by deriving from the complex plane a numerically precise epsilon term (eq. 4, ICCS 2006), that inserted into an otherwise perfectly ordered set of n-dimension Euclidean kissing spheres ("time barrier" preprint), leads to a dissipative n-dimensional system. This comports with the Jacobson-Verlinde model of entropic gravity, where gravity and (physical) information are identical. In my model, time, gravity and information are _all_ identical.
There is no background, because point-line duality and the Bekenstein-Mayo result that black holes form a 1-dimension information channel** support self organization independent of background space, requiring only a single quantum fluctuation in imaginary time, and allowing that the time metric is n-dimensional continuous, on a random, self avoiding walk.
Tom
*[2005 preprint] "The case for background independence." arXiv:hep-th/0507235v1
** "Black Holes are One-Dimensional." General Relativity
and Gravitation 33;12, December (2001)