Let’s start by laying out some common components of all these conversions. All of them involve **transforming **a set of **resources** right into a set of **products. **Every resource has some products that it *can* make and others that it *cannot *make. These transformations must be composable, that is that if we are able to turn **A **into **B **and **B **into **C**, then we should always give you the option to show **A **into **C **through multiple steps. All of those ideas will be modelled by a **symmetric monoidal category**. That’s a sophisticated expression, let’s see what that’s and provides an example.

We define a symmetric monoidal category as (**S**, **>**, **I**, *). That is numerous structures, I’ll undergo each in turn.

**S** is solely all of the set of all objects that we’re enthusiastic about. If we’re applying this structure to a chemical problem, it might be all of the chemicals now we have access to, in addition to those we want to create. That is similar to the usual mathematical set.

**>** defines an *order* on **S**. I could simply list the properties of the order, but I feel it’s more intuitive to provide an example.

How can we interpret this diagram by way of **>**? By taking a look at Figure 1, we see that there’s an arrow from **A** to **B**, so **A** **>** **B**. We also can 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. Additionally it is possible for **A** **>** **B** and **B** **>** **A** if I had drawn in an arrow going from **B** to **A**.

What’s the interpretation of this? It’s pretty easy, if **A** **>** **B**, then we are able to turn **A** into **B** by a process. Notice that **C** and **D** can’t be become anything (besides themselves), they’re stuck of their current state. Since **A** **>** **A**, we are able to turn **A** into **A **by a trivial process. For the reason that arrows will be combined, we all know that **A** **>** **B** and **B** **>** **C** means **A** **>** **C**. This is smart after we take into consideration composition. To summarize, the objects contained in **>** tell us what objects in **S **will be become other objects in **S.**

Now let’s turn to **I** and *. These parts tell us concerning the actual act of performing a process to convert elements into one other. * 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’s one which is significantly more essential and serves to attach * and **>**.

This property is named *monotonicity*, and is defined as **A1** > **B1**, **A2** > **B2**, implies that **A1 *** **A2** > **B1 *** **B2**. We will consider this property as “if we are able to turn **A1** into **B1** and **A2** into **B2**, then we are able to turn the mix of **A1** and **A2** into the mix of **B1** into **B2**.” Pondering this fashion is intuitive for Resource Theory, but must be formalized in the maths.