## 3 Methods
### The Conceptual System Architecture as a Category  
[PERSON]'s definition of _systems_[44], in conjunction with the conventional representation of system architecture as a combination of structure and behavior [3], gives rise to a System category, \(SYS\), with the following types: Structure, Input-Output (or interchangeably, Onput [81]), Resource, and State. Onputs and Resources can be referred to as Operands. The morphisms include Behaviors (which transform Operands) and Relations (which map types to themselves and to each other). A morphism \(f:\ Op\otimes St\to Op\otimes St\) is a behavior of the system at any level, which transforms a combination (denoted as a Cartesian product) of operands (\(Op\)) and the system's state (\(St\)) to another combination of operands and state. The morphism \(f\) is a composition of input\(\rightarrow\)state and state\(\rightarrow\)output:  
\[f_{in}:\ Op\otimes St\to St\;;\;f_{out}:\ St\to Op\otimes St\;;f=f_{out} \circ f_{in}\]  
A system _sys_ is an instantiation of the category \(SYS\). [PERSON]'s system morphism is a _functor_ that maps a category to itself, or endfunctor, denoted as \(F_{Wymore}:SYS\to SYS\).  
Cyber-physical systems (CPS) reside concurrently in the physical and cybernetic spaces, and constitute a primary concern of systems engineering. In our categorization, CPS behaviors/functions are morphisms, and system components (sensors, processors, actuators, etc.) and operands (Currency, Data, Energy, and Matter--CDEM) are types. CPS components carry out functions that convert input operands to output operands. Some examples are shown in Table 3. CPS functions can be composed to create higher-level functionalities, for example, sensing, then actuating: \(Matter\to Energy\to Data\to Energy\). We can also define functions as Cartesian products, e.g., a plant's function is:Input data (e.g., commands), matter (e.g., raw material), and energy (e.g., electrical power) are converted to output data (statuses, reports), matter (finished goods, byproducts), and energy (heat). Some sequences may not be valid within particular system domains. For instance, in logistics, manufacturing a device (an actuator or sensor) is possible, but in an operational system such as an aircraft or autonomous vehicles, manufacturing a new part may not make sense. On the other hand, sensor \"manufacturing\" of images from signals may be valid.  
Our categorical formulation views systems as behavior-enabling structures, which affect operands. Operands are also types. In fact, a system can be an operand, and an operand can be a system. Imagine a vehicle exiting the production line as an operand, and emerging as a system, or software delivered as a file, i.e., as an operand, and emerging as an executable, i.e., a system.  
Conceptual architecting usually specifies the function before the form, while physical architecting typically does the opposite. Attributing a functional-behavioral morphism \(f_{C}\) to a structural entity \(C\) implies or entails a relational morphism \(r_{f\to C}\colon f_{C}\to C\) that assigns a function to a function-performing system component, and a dual \(r_{C\to f}\colon C\to f_{C}\) that allocates a component to perform a function. This mapping of morphism to type and type to morphism implies a type-morphism duality of system behavior.