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Exercises and Puzzles 8 - Chapter 1

edited May 2018

Here are four puzzles from Lecture 9:

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.

Puzzle 23. Suppose $$f : X \to Y$$ is a function between sets. For any $$S \subseteq X$$ define

$$f_{!}(S) = \{y \in Y: \; y = f(x) \textrm{ for some } x \in S\}$$ and

$$f_{\ast}(S) = \{y \in Y: x \in S \textrm{ for all } x \textrm{ such that } y = f(x)\} .$$ I showed in the lecture that $$f_{\ast}: P(X) \to P(Y)$$ is the left adjoint of the inverse image map $$f^{\ast}: P(Y) \to P(X)$$. Show that $$f_{\ast}: P(X) \to P(Y)$$ is the right adjoint of this map.

Puzzle 24. 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. 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"?

I believe these were all answered in the lengthy discussion of Lecture 9, so if you get stuck browse through that!