Category theory is famous for its founder's motto that "adjoints are everywhere". While the original context of his quote was meant to encompass mathematics, we have since seen the concept of adjoints, a.k.a. duality, a.k.a. isomorphisms, extend far beyond the realm of pure mathematics into areas such as physics, music, psychology, and more.
In viewing the genetic code from the perspective of these adjoints, we are naturally drawn to one of their identifying theorems and that is the triangle equality which is essential to establish one variable as the inverse of the other. This diagram proposes that the XYX structure required for every nucleotide to be an adjoint of every other one is present (as they mathematically must be in the possible 64 codons) but also the possibility that the genetic code encodes and makes use of these adjunctions at a fundamental level.
With four elements, the best (only?) way to map these adjunctions is as the non-cyclic Klein 4-group. This shows a more distinct physiochemical grouping than the standard row or circular visualizations, a precise symmetrical grouping of (7+5)x2 elements if we exclude the redundant UGA and UAG stops, a strong suggestion of a C12 or C24 underlying group, and a C2 "flip" in the Ramachandran plots between GG_ and the left column and CC_ and the right colum as expected from the symmetry of the Klein 4-group.
This choice of the Klein 4-group is also motivated by work in symmetric monoidal categories which have been linked to among other things, music and quantum mechanics.
In light of these alleged patterns, I'm curious if anyone knows of anyone working in category theory with a genetics perspective that I can reference or if anyone knows of anyone working category theory who might be interested in a genetics perspective.
Any reference to any work of any kind in terms of category theory and genetics would be most appreciated
(UPDATE: No joy at Math Overflow or Biology SE. Moderators from both sites have deemed this idea "unworthy" of further consideration and have not allowed its posting on their boards. As such, this is the only place on SE where this idea remains and any help, especially with applying adjuncts and category theory to the genetic code would be most helpful. Thanks for reading!)