Chapter 1 Generative effects: Posets and adjunctions 1.1 Motivation Your brain as a whole is conscious. On the other hand, your brain is made up of neurons, none of which are conscious. Where does the consciousness come from? A generative effect is a property of a interconnected system that cannot be explained solely in terms of properties of the constituent subsystems. As such, we will begin our foray into category theory by talking not about compositionality per se, but about the failure of compositionality: how new effects can be generated when systems are joined. The use of category theory to explore generative effects follows work by Adam [Adam:2017a]. Much of this chapter, however, can also be considered a classical, order-theoretic warm up for the full-fledged category theory to come. To explore the notion of a generative effect we must first provide at least a rudi- mentary formalization of systems, including how they are joined and what it means to observe properties. To get some intuition for how we do this, let’s take a simpler example. Consider three points; we’ll call them , and . • ◦ ∗ In this example, a system will simply be a way of connecting these points together. We might think of our points as sites on a power grid, with a system describing connection by power lines, or as people susceptible to some disease, with a system describing possible contagion. As an abstract example of a system, there is a system where and are connected, but neither are connected to . We shall draw this like • ◦ ∗ so: • ∗ ◦ Connections are symmetric, so if a is connected to b, then b is connected to a. Con- nections are also transitive, meaning that if a is connected to b, and b is connected to 1 2 CHAPTER 1. GENERATIVE EFFECTS: POSETS AND ADJUNCTIONS c, then a is connected to c—that is, all a, b, and c are connected. Friendship is not transitive—my friend’s friend is not necessarily my friend—but word-of-mouth is. Let’s depict two more systems. Here is a system in which none of the points are connected: • ∗ ◦ And here is the system is which all three points are connected: • ∗ ◦ The property of these systems that we will measure is whether is connected to • . A measurement will be an assignment of either true or false to a system; we will ∗ assign true if is connected to , and false otherwise. We thus assign the value true • ∗ to the systems: • ∗ • ∗ ◦ ◦ and we will assign the value false to the remaining systems: • ∗ • ∗ • ∗ ◦ ◦ ◦ We have thus decided what we shall mean, in this simple case, by system and property. But what does it mean to join systems? We shall join two systems A and B simply by combining their connections. That is, we shall say the joined system A B of A and B has point x connected to point y _ if there are some points z1;:::; zn such that, in at least one of A or B, it is true that a is connected to z1, zi is connected to zi+1, and zn is connected to y. In a three-point system, the above definition is overkill, but we want to say something that works in any number of elements. In our three-element, it means for example that • ∗ • ∗ • ∗ _ ◦ ◦ ◦ and • ∗ • ∗ • ∗ (1.1) _ ◦ ◦ ◦ 1.1. MOTIVATION 3 Exercise 1.1. What is the result of joining the following two systems? 11 12 13 11 12 13 • • • • • • _ 21 22 23 21 22 23 • • • • • • ♦ Let’s return to the second example above, Eq. (1.1). Here we can see a generative effect. Let’s write Φ for the above function measuring whether is connected to . • ∗ • ∗ • ∗ Now Φ ◦ Φ ◦ false. That is, for both the systems on the left hand side, is ¹ º ¹ º • not connected to . ∗ • ∗ • ∗ On the other hand, when we join these two systems, we see that Φ ◦ ◦ • ∗ ¹ _ º Φ ◦ true: in the joined system, is connected to . Thus the joined system has a ¹ º • ∗ property that neither of the constituent subsystems has—this property is a generative effect. Such effects, or at least their potential to exist, can be incredibly important to deduce. For example, the two constituent systems could be the views of two local authorities on possibly contagion between an infected person and a vulnerable • person . In this case, each authority would conclude from their own model that no ∗ infection can occur, but combining their information would show the opposite. 1.1.1 Ordering systems We can obtain the notion of joining systems from a more basic structure: order. Namely, we note that the systems themselves are ordered in a hierarchy. Given systems A and B, we say that A B if whenever x is connected to y in A then x is ≤ connected to y in B. For example, • ∗ • ∗ ≤ ◦ ◦ 4 CHAPTER 1. GENERATIVE EFFECTS: POSETS AND ADJUNCTIONS This leads to the diagram • ∗ ◦ • ∗ • ∗ • ∗ ◦ ◦ ◦ • ∗ ◦ (1.2) where we draw an arrow from system A to system B if A B. Such diagrams are ≤ known as Hasse diagrams. Now the notion of join, , is derived from this order: A B is the smallest system _ _ that is bigger than both A and B. That is, A A B and B A B , and for any ≤ ¹ _ º ≤ ¹ _ º C, if A C and B C then A B C. ≤ ≤ ¹ _ º ≤ Exercise 1.2. Try it yourself. 1. Write down all the partitions of a two element set ; , order them as above, f• ∗g and draw the Hasse diagram. 2. Now do the same thing for a four element-set, say 1; 2; 3; 4 . There should be f g 15 partitions. Choose any two systems in your 15-element Hasse diagram, call them A and B. 3. What is A B? _ 4. Find all the systems C with A C and B C. ≤ ≤ 5. Is A B under all such C, i.e. is it true that A B C? _ ¹ _ º ≤ ♦ The properties true and false also have an order, false true: ≤ true false For any A; B in true; false , we can again write A B to mean the least element f g _ that is greater than both A and B. 1.2. POSETS 5 Exercise 1.3. In the order true; false , where false true, what is: f g ≤ 1. true false _ 2. false true _ 3. true true _ 4. false false? _ ♦ Let’s return to our systems with , , and , and our “ is connected to ” function, • ◦ ∗ • ∗ which we called Φ. It takes any such system and returns either true or false. Note that the map Φ preserves the order: if A B and there is a connection between ≤ ≤ • and in A, then there is such a connection in B too. The existence of the generative ∗ effect, however, is captured in the equation Φ A Φ B , Φ A B : (1.3) ¹ º _ ¹ º ¹ _ º That is, we have two systems with no connection and , which obtain the property • ∗ when they are combined. Equation (1.3) is how we define a generative effect of Φ. These ideas capture the most basic ideas in category theory. In particular, we see hints of the notions of category, functor, colimit, and adjunction. In this chapter we will explore these ideas in the elementary setting of ordered sets. 1.2 Posets We will not give a definition of set here, but informally we will think of it as a collection of things, known as elements. These things could be all the leaves on a tree, or the names of your favourite fruits, or simply some symbols a, b, c. For example, we write X h; 1 to denote the set, called X, that contains exactly two elements, one called f g h and one called 1. If x is element of X, we write x X; so we have h X and 1 X. 2 2 2 Given a set X, we call a collection, possibly empty, of elements of X a subset of X. Given two sets X and Y, the product X Y of X and Y is the set of pairs x; y , where × ¹ º x X and y Y. 2 2 Exercise 1.4. Let X h; 1 and Y 1; 2; 3 . There are six elements in X Y; write them out. ♦ f g f g × A particularly important sort of subset is a subset of the product of a set with itself. Definition 1.5. Let X be a set. A (binary) relation on X is a subset R X X. ⊆ × In Section 1.1, we several times used the symbol to denote a sort of order. Here ≤ is a formal definition of ordered set. Definition 1.6. A preorder on a set X is a binary relation R on X, where we write x y ≤ if x; y R, such that ¹ º 2 1. x x and ≤ 6 CHAPTER 1. GENERATIVE EFFECTS: POSETS AND ADJUNCTIONS 2. if x y and y z, then x z. ≤ ≤ ≤ The first condition is called reflexivity and the second is called transitivity. We call a set with a preorder a poset, although we note that this term often denotes a partially ordered set, in which the additional condition holds that if x y and y x ≤ ≤ then x y. The difference is a fairly minor one. Example 1.7. Let X be the set of partitions of ; ; ; it has five elements, shown in f• ◦ ∗g Eq. (1.2).