Many of the physical sciences ultimately deal with resources. The important questions are: Can a given resource be converted into another? And if so, how? Examples abound:
- Chemistry: How can chemicals react to turn into others?
- Thermodynamics: How can work be extracted from thermal imbalance?
- Computation: How can gates be combined to form circuits computing a given function?
- Logic: How can a conclusion be derived from hypotheses?
- Quantum theory: What can entanglement achieve that couldn’t be achieved without it?
- Geometry: Which figures can be turned into others using compass and straightedge only?
This paper develops and studies a general framework intended to capture these interconversions. The eventual aim is to identify general concepts and features among them. The framework is given in three stages of increasing simplicity.
In the first stage, resources are modeled as objects in a symmetric monoidal category, whose morphisms model conversion of resources. This is very general; such a model can be constructed from any partitioned process theory.
The second stage deals with resource convertibility only. Here, one is interested only in the possibility of converting one resource into another, instead of how the conversion is implemented. The categorical structure is flattened into a commutative preordered monoid. At this level, one can for example speak of catalysis: resource A alone might not be convertible into B, but A in the presence of C could possibly be turned into B without using up C.
The third stage quantifies the cost of resources and conversions. Some resources are free (those with a morphism from the monoidal unit), whereas some are valuable. This is modeled by a monotone: a function from the monoid to the real numbers. Examples include Shannon entropy. At this level, one can, for example, handle asymptotic rates of conversion, as in currency exchange.