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.