Coho: Coho salmon, Cohomology, and Cohomosexuality

 Coho:  Coho salmon, Cohomology, and Cohomosexuality

During their ocean phase, coho salmon have silver sides and dark-blue backs. During their spawning phase, their jaws and teeth become hooked. After entering fresh water, they develop bright-red sides, bluish-green heads and backs, dark bellies and dark spots on their backs. Sexually maturing fish develop a light-pink or rose shading along the belly, and the males may show a slight arching of the back. Mature adults have a pronounced red skin color with darker backs and average 28 inches (71 cm) and 7 to 11 pounds (3.2 to 5.0 kg), occasionally reaching up to 36 pounds (16 kg). They also develop a large kype (hooked beak) during spawning.^[2] Mature females may be darker than males, with both showing a pronounced hook on the nose.^[3]

Cohomology is the other side of homology theory in mathematics. Poincare developed homology theory as a way of characterizing shapes (eg. spheres, donuts,…) in an invariant manner, independent of their peculiarities. One puts up a spanning structure that approximates the shape, usually of triangles, four-sided three dimensional figures, and so forth, and then applies an algorithm that at its end gives an account of how many holes in that shape, and in higher dimensions the equivalent punctures. And that number is invariant to how you put up the spanning structure.

Cohomology is the complementary account of shape, specifying how, say, continuous functions, might occupy the shape. Namely, homology gives an invariant account of a shape, cohomology gives an account of how something might live on that shape (say an ant crawling along on it). Cohomology also provides a deeper understanding of the advanced calculus, where one computes the gradient, divergence, and curl of functions going from scalars to vectors, from vectors to scalars, and from vectors to vectors, respectively. Cohomology also allows for an analysis of the structure of groups, such as the group of permutations, or the group of rotations. Here “shape” is to be understood in terms of how the group elements are connected to each other.

What’s crucial is that cohomology provides a way of understanding obstructions to simple addition, so that in the case of the fundamental theory of the calculus, we might expect that ∫_atob f(x) dx = F(b)-F(a), dF/dx=f, but some of the time, say in two or more dimensions, as in the complex plane,

∫_atob f(x) dx = F(b)-F(a) + h(a,b)

where h depends on the path from a to b, namely the Cauchy integral theorem where h measures the number of times you have gone around the origin. In effect, cohomology gives an account of combining stuff when the combination may not be so simple. Put differently, in combining objects their local characteristics may not be sufficiently to say how they combine with other such objects. That is, F(a) does not tell us enough about a, we need global or external information.

Homosexuality is sexual relationships between two persons of the same gender.  Much as in homology, it is a structural fact, part of standard kinship theory in anthropology. (Here sexual relationship is taken to be “marriage” in kinship theory, but clearly this is not really correct.) Cohomosexuality, in analogy to cohomology, is an account of how such relationships are instantiated in a society, how they occupy the structures of kinship in society.  In effect, it is a study of what might be called “gay life.” (Of course, there may be strictures on such relationships, so that the instantiations are hidden or enacted in peculiar ways. Namely, what might be called “obstructions.”)

Again, the problem is to develop methods of analyzing such ways of living and of discovering how obstructions affect the modes of being.  Hence, cohomosexuality must contribute to our understanding of the manifestations other sorts of sexual variety, for the obstructions are likely more pervasive than just concerning homosexuality. A society is to be known through the analogue of h(a,b), those obstructions or prejudices or legal penalties that make a’s or b’s characteristics insufficient to know how they combine.

In effect, cohomology is an account of how the world is not so modular, not just a set of LEGO blocks that might be put together willy-nilly. There are larger structural constraints. Similarly, we discover that some interactions of elementary particles are readily allowed, but others are to varying degrees forbidden. Kinship rules, global rules, give the degree of allowed-ness or forbidden-ness. Put differently, in quantum mechanics and quantum field theory, particles interact through the exchange of other particles, particles with “unreal” masses, virtual particles. Electrons repulse each other through the exchange of (virtual) photons. But some interactions are not allowed by the exchange of a particle. Still, there might be indirect paths with intermediate states, through which there might be an interaction, say of an electron and a neutral kaon. (Chemists are experts on these processes, seeking catalysts and appropriate intermediate environments.) Those indirect paths (higher orders in perturbation theory) are surpressed by the order^nth-power and by the inverse-masses of the exchanged virtual particles.

Finally, there are the coho salmon. They change colors as they mature; they reproduce employing several different sites. As in cohomology or cohomosexuality, what we are seeing is one of the ways something might occupy a space (here, riverine and oceanic). The relevant “obstruction” is that at any one time and place, they are not locally characterized. Only in having a global picture are they to be understood. Here ontogeny does not recapulate phylogeny, and in effect what we call an obstruction is the source of that fact.