2.1: Getting from a to b

You can’t make an omelette without breaking an egg. To obtain the things we want requires resources, and the process of transforming what we have into what we want is often an intricate one. In this chapter, we will discuss how monoidal preorders can help us think about this matter.

Consider the following three questions you might ask yourself:

• Given what I have, is it possible to get what I want?
• Given what I have, what is the minimum cost to get what I want?
• Given what I have, what is the set of ways to get what I want?

These questions are about resources those you have and those you want but perhaps more importantly, they are about moving from have to want: possibility of, cost of, and ways to.

Such questions come up not only in our lives, but also in science and industry. In chemistry, one asks whether a certain set of compounds can be transformed into another set, how much energy such a reaction will require, or what methods exist for making it happen. In manufacturing, one asks similar questions.

From an external point of view, both a chemist and an industrial firm might be regarded as store-houses of information on the above subjects. The chemist knows which compounds she can make given other ones, and how to do so; the firm has stored knowledge of the same sort. The research work of the chemist and the firm is to use what they know in order to derive or discover new knowledge.

This is roughly the first goal of this chapter: to discuss a formalism for expressing recipes methods for transforming one set of resources into another and for deriving new recipes from old. The idea here is not complicated, neither in life nor in our mathematical formalism. The value added then is to simply see how it works, so we can build on it within the book, and so others can build on it in their own work.

We briefly discuss the categorical approach to this idea—namely that of monoidal preorders for building new recipes from old. The following wiring diagram shows, assuming one knows how to implement each of the interior boxes, how to implement the preparation of a lemon meringue pie:

The wires show resources: we start with prepared crust, lemon, butter, sugar, and egg resources, and we end up with an unbaked pie resource. We could take this whole method and combine it with others, e.g. baking the pie:

In the above example we see that resources are not always consumed when they are used. For example, we use an oven to convert or catalyze the transformation of an unbaked pie into a baked pie, and we get the oven back after we are done. It’s a nice feature of ovens! To use economic terms, the oven is a “means of production” for pies.

String diagrams are important mathematical objects that will come up repeatedly in this book. They were invented in the mathematical context more specifically in the context of monoidal categories by Joyal and Street [JS93], but they have been used less formally by engineers and scientists in various contexts for a long time.

As we said above, our first goal in this chapter is to use monoidal preorders, and the corresponding wiring diagrams, as a formal language for recipes from old. Our second goal is to discuss something called V-categories for various monoidal preorders V.

A V-category is a set of objects, which one may think of as points on a map, where V somehow “structures the question” of getting from point a to point b. The examples of monoidal preorders V that we will be most interested in are called Bool and Cost. Roughly speaking, a Bool-category is a set of points where the question of getting from point a to point b has a true / false answer. A Cost-category is a set of points where the question of getting from a to b has an answer d \(\in\) [0, ∞], a cost.

This story works in more generality than monoidal preorders. Indeed, in Chapter 4 we will discuss something called a monoidal category, a notion which generalizes monoidal preorders, and we will generalize the definition of V-category accordingly. In this more general setting, V-categories can also address our third question above, describing methods of getting between points. For example a Set-category is a set of points where the question of getting from point a to point b has a set of answers (elements of which might be called methods).

We will begin in Section 2.2 by defining symmetric monoidal preorders, giving a few preliminary examples, and discussing wiring diagrams. We then give many more examples of symmetric monoidal preorders, including both some real-world examples, in the form of resource theories, and some mathematical examples that will come up again throughout the book. In Section 2.3 we discuss enrichment and V-categories how a monoidal preorder V can “structure the question” of getting from a to b and then give some important constructions on V-categories (Section 2.4), and analyze them using a sort of matrix multiplication technique (Section 2.5).