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Let’s start by laying out some common components of all these conversions. They all involve transforming a set of resources into a set of products. Every resource has some products that it can make and others that it cannot make. These transformations should be composable, that is if we can turn A into B and B into C, then we should be able to turn A into C through multiple steps. All of these ideas can be modelled by a symmetric monoidal category. That’s a complicated expression, let’s see what that is and give an example.
We define a symmetric monoidal category as (S, >, I, *). This is a lot of structures, I’ll go through each in turn.
S is simply all the set of all objects that we are interested in. If we are applying this structure to a chemical problem, it may be all the chemicals we have access to, as well as the ones we wish to create. This is just like the standard mathematical set.
> defines an order on S. I could simply list the properties of the order, but I think it’s more intuitive to give an example.
How do we interpret this diagram in terms of >? By looking at Figure 1, we see that there is an arrow from A to B, so A > B. We can also compose arrows, so A > C and A > D. I didn’t include them in Figure 1, but every point also has an arrow going to itself, so A > A, B > B, etc. It is also possible for A > B and B > A if I had drawn in an arrow going from B to A.
What is the interpretation of this? It’s pretty simple, if A > B, then we can turn A into B by a process. Notice that C and D cannot be turned into anything (besides themselves), they are stuck in their current state. Since A > A, we can turn A into A by a trivial process. Since the arrows can be combined, we know that A > B and B > C means A > C. This makes sense when we think about composition. To summarize, the objects contained in > tell us what objects in S can be turned into other objects in S.
Now let’s turn to I and *. These parts tell us about the actual act of performing a process to convert elements into another. * is a binary operation that acts as A * B = C. This operation represents actually turning A and B into C. I is just is the “neutral resource” where A * I = I * A = A for all elements. Again, there are a bunch of properties needed for this operation, but there is one that is significantly more important and serves to connect * and >.
This property is called monotonicity, and is defined as A1 > B1, A2 > B2, means that A1 * A2 > B1 * B2. We can think of this property as “if we can turn A1 into B1 and A2 into B2, then we can turn the combination of A1 and A2 into the combination of B1 into B2.” Thinking this way is intuitive for Resource Theory, but needs to be formalized in the math.
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