#013: Find correct morphisms for experimental domains

Category: Mathematical foundations - Tags: General theory, Category theory, Mathematical logic, Order theory

Find the correct definition of morphism over the current definitions of experimental and theoretical domains.

Mathematical problem. Currently, the only maps we have defined over domains are the experimental relationships (see chapter 2 of the book. The idea is to find a more general definition for morphism (map preserving structures). These should, for example, recover the domain product as the categorical product. We suspect that the categorical product and coproduct will coincide.

One direct constraint is that they need to preserve the logical structure, therefore they will, for example, preserve narrowness. But the issue here is that the domain and codomain are not disconnected structures: they are embedded in a single overall structure. The domain product, for example, uses that overall structure when closing over conjunction and disjunction. Therefore the morphism should not only preserve the structures of the individual spaces but also preserve consistency with the overall structure.

The current hunch is that the map from one domain to the other maps to the narrowest broader statement. That is, if \(\mathcal{D}_X\) and \(\mathcal{D}_Y\) are two domains, a morphism \(m\) is a map \(m : \mathcal{D}_X \to \mathcal{D}_Y\) such that \(s_X \preccurlyeq m(s_X) \preccurlyeq s_Y\) for all \(s_Y \in \mathcal{D}_Y\) for which \(s_X \preccurlyeq s_Y\). It may be that this is the only requirement needed, and the preservation of the structure across the map will be a corollary.

We can also think to perform such morphism on the whole context. In that case, the morphism should be idempotent. It would basically “project” the whole context to the domain. One thing to keep in mind, therefore, is whether this type of operation will in the end be related to the projections in linear spaces.

Physical significance. The idea here is to understand what it means conceptually to map from one domain to the other. We have found that constructing a product domain means putting together two different sets of experimentally distinguishable aspects (i.e. horizontal and vertical position of a body, density of water and temperature of a mercury column in thermodynamic equilibrium with it). Here we are asking what it means to map from one domain to another.

The domain product combines the information we can experimentally obtain with different domains into a single domain. This will work differently depending on the degree of dependence of the domains. In fact, the type of product domain can be used to characterize the different cases. If \(X\) and \(Y\) are independent, for example, all combinations are possible, so the whole \(XY\) plane is possible. Conversely, if \(Y\) depends on \(X\), given a specific value of \(x\) there will be only one possible value \(y=f(x)\).

Here we are most likely trying to characterize how much information can be learned about one domain from the other. For example, if the horizontal and vertical positions of a particle are independent, then a statement about the horizontal position will tell us nothing about the vertical position. Mathematically, any statement on the horizontal position should map to the certainty on the vertical position (every case is possible). Conversely, if there is a correlation, then we should expect to map to the statement that provides the maximal amount of information given the constraint. That should map mathematically to the narrowest broader statement.