Example 202 Both small-scale and large-scale projects, such as those in research or development, require resources. Resource allocation (through grants, investment funding, contracts, etc.) requires a detailed plan (for how those resources are to be spent), especially as the project increases in scale. If Rsrc is a category consisting of relevant resources, so that objects are resources (e.g., for simplicity, different-sized checks or bags of money) and morphisms are given by a natural relation between those resources (e.g., ≤ in the case of a uniform money-valuation of the different resource objects), and if ProjPlan is a category consisting of project tasks, given some natural ordering (e.g., by order of priority in the carrying out of the plan), then we might consider the functor
$$ V: \text{Rsrc} \to \text{Projplan} $$
that maps resource $r$ to the collection of project plans that are viable given that resource, and the functor
$$N : \text{Projplan} \to \text{Rsrc}$$
taking a project task p to all those resources that are necessary to complete the task (which, depending on how Rsrc is structured, say in a simple case of “costs,” might just amount to returning an interval bounded by the least cost for which the task could be carried out, and including all other more ample amounts)
We would probably not expect V and N to construct strict inverses to one another, for we do not expect that, for any given resource r, a list of necessary resources for those plans that are deemed viable given r would be equal to r. Though we might expect that, among the resources, r ≤ NV(r). Similarly, we would not expect that, for a given project task p, the result of applying N to p and then V to N(p), would always equal the same task p. Yet we would expect VN(p) ≤ p in ProjPlan. This suggests that we have an adjunction,
Sheaf Theory by Rosiak, Pg-238
I have a few questions about the following explanation.
What is meant by uniform money evaluation? Google tells me no good results.
I tried to explain this to a friend by considering a real life example of a category of resource and a category of plans, and a trouble in any example that I could construct is that, a more expesnive resource would always be able to take care of the resources having less value than it could. As in, anything that a 10 \$ bill could buy, could be bought by a 100 \$ bill. So I don't get how exactly it is meant that, this functor "maps resource r to the collection of project plans that are viable given that resource"