Lecture 9 - Adjoints and the Logic of Subsets

# Lecture 9 - Adjoints and the Logic of Subsets

We've seen that classical logic is closely connected to the logic of subsets. For any set $$X$$ we get a poset $$P(X)$$, the power set of $$X$$, whose elements are subsets of $$X$$, with the partial order being $$\subseteq$$. If $$X$$ is a set of "states" of the world, elements of $$P(X)$$ are "propositions" about the world. Less grandiosely, if $$X$$ is the set of states of any system, elements of $$P(X)$$ are propositions about that system.

This trick turns logical operations on propositions - like "and" and "or" - into operations on subsets, like intersection $$\cap$$ and union $$\cup$$. And these operations are then special cases of things we can do in other posets, too, like join $$\vee$$ and meet $$\wedge$$.

We could march much further in this direction. I won't, but try it yourself!

Puzzle 22. What operation on subsets corresponds to the logical operation "not"? Describe this operation in the language of posets, so it has a chance of generalizing to other posets. Based on your description, find some posets that do have a "not" operation and some that don't.

I want to march in another direction. Suppose we have a function $$f : X \to Y$$ between sets. This could describe an observation, or measurement. For example, $$X$$ could be the set of states of your room, and $$Y$$ could be the set of states of a thermometer in your room: that is, thermometer readings. Then for any state $$x$$ of your room there will be a thermometer reading, the temperature of your room, which we can call $$f(x)$$.

This should yield some function between $$P(X)$$, the set of propositions about your room, and $$P(Y)$$, the set of propositions about your thermometer. It does. But in fact there are three such functions! And they're related in a beautiful way!

The most fundamental is this:

Definition. Suppose $$f : X \to Y$$ is a function between sets. For any $$S \subseteq Y$$ define its inverse image under $$f$$ to be

[ f^{\ast}(S) = \{x \in X: \; f(x) \in S\} . ]

The pullback is a subset of $$X$$.

The inverse image is also called the preimage, and it's often written as $$f^{-1}(S)$$. That's okay, but I won't do that: I don't want to fool you into thinking $$f$$ needs to have an inverse $$f^{-1}$$ - it doesn't. Also, I want to match the notation in Example 1.89 of Seven Sketches.

The inverse image gives a monotone function

[ f^{\ast}: P(Y) \to P(X), ]

since if $$S,T \in P(Y)$$ and $$S \subseteq T$$ then

[ f^{\ast}(S) = \{x \in X: \; f(x) \in S\}
\subseteq \{x \in X:\; f(x) \in T\} = f^{\ast}(T) . ]

Why is this so fundamental? Simple: in our example, propositions about the state of your thermometer give propositions about the state of your room! If the thermometer says it's 35°, then your room is 35°, at least near your thermometer. Propositions about the measuring apparatus are useful because they give propositions about the system it's measuring - that's what measurement is all about! This explains the "backwards" nature of the function $$f^{\ast}: P(Y) \to P(X)$$, going back from $$P(Y)$$ to $$P(X)$$.

Propositions about the system being measured also give propositions about the measurement apparatus, but this is more tricky. What does "there's a living cat in my room" tell us about the temperature I read on my thermometer? This is a bit confusing... but there is an answer because a function $$f$$ really does also give a "forwards" function from $$P(X)$$ to $$P(Y)$$. Here it is:

Definition. Suppose $$f : X \to Y$$ is a function between sets. For any $$S \subseteq X$$ define its image under $$f$$ to be

[ f_{!}(S) = \{y \in Y: \; y = f(x) \textrm{ for some } x \in S\} . ]

The image is a subset of $$Y$$.

The image is often written as $$f(S)$$, but I'm using the notation of Seven Sketches, which comes from category theory. People pronounce $$f_{!}$$ as "$$f$$ lower shriek".

The image gives a monotone function

[ f_{!}: P(X) \to P(Y) ]

since if $$S,T \in P(X)$$ and $$S \subseteq T$$ then

[ f_{!}(S) = \{y \in Y: \; y = f(x) \textrm{ for some } x \in S \}
\subseteq \{y \in Y: \; y = f(x) \textrm{ for some } x \in T \} = f_{!}(T) . ]

But here's the cool part:

Theorem. $$f_{!}: P(X) \to P(Y)$$ is the left adjoint of $$f^{\ast}: P(Y) \to P(X)$$.

Proof. We need to show that for any $$S \subseteq X$$ and $$T \subseteq Y$$ we have

[ f_{!}(S) \subseteq T \textrm{ if and only if } S \subseteq f^{\ast}(T) . ]

David Tanzer gave a quick proof in Puzzle 19. It goes like this: $$f_{!}(S) \subseteq T$$ is true if and only if $$f$$ maps elements of $$S$$ to elements of $$T$$, which is true if and only if $$S \subseteq \{x \in X: \; f(x) \in T\} = f^{\ast}(T)$$. $$\quad \blacksquare$$

This is great! But there's also another way to go forwards from $$P(X)$$ to $$P(Y)$$, which is a right adjoint of $$f^{\ast}: P(Y) \to P(X)$$. This is less widely known, and I don't even know a simple name for it. Apparently it's less useful.

Definition. Suppose $$f : X \to Y$$ is a function between sets. For any $$S \subseteq X$$ define

[ f_{\ast}(S) = \{y \in Y: x \in S \textrm{ for all } x \textrm{ such that } y = f(x)\} . ]

This is a subset of $$Y$$.

Puzzle 23. Show that $$f_{\ast}: P(X) \to P(Y)$$ is the right adjoint of $$f^{\ast}: P(Y) \to P(X)$$.

What's amazing is this. Here's another way of describing our friend $$f_{!}$$. For any $$S \subseteq X$$ we have

[ f_{!}(S) = \{y \in Y: x \in S \textrm{ for some } x \textrm{ such that } y = f(x)\} . ]

This looks almost exactly like $$f_{\ast}$$. The only difference is that while the left adjoint $$f_{!}$$ is defined using "for some", the right adjoint $$f_{\ast}$$ is defined using "for all". In logic "for some $$x$$" is called the existential quantifier $$\exists x$$, and "for all $$x$$" is called the universal quantifier $$\forall x$$. So we are seeing that existential and universal quantifiers arise as left and right adjoints!

This was discovered by Bill Lawvere in this revolutionary paper:

• F. Willam Lawvere, Adjointness in foundations, Dialectica 23 (1969). Reprinted with author commentary in Theory and Applications of Categories 16 (2006), 1-16.

By now this observation is part of a big story that "explains" logic using category theory.

Two more puzzles! Let $$X$$ be the set of states of your room, and $$Y$$ the set of states of a thermometer in your room: that is, thermometer readings. Let $$f : X \to Y$$ map any state of your room to the thermometer reading.

Puzzle 24. What is $$f_{!}(\{\text{there is a living cat in your room}\})$$? How is this an example of the "liberal" or "generous" nature of left adjoints, meaning that they're a "best approximation from above"?

Puzzle 25. What is $$f_{\ast}(\{\text{there is a living cat in your room}\})$$? How is this an example of the "conservative" or "cautious" nature of right adjoints, meaning that they're a "best approximation from below"?

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