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# Exercises and Puzzles 7 - Chapter 1

edited May 2018

Today the workshop Applied Category Theory 2018 begins! Last night David Spivak, Bob Coecke, Jamie Vicary, Tom Leinster and others showed up at my hotel and we had a great chat. Now 60 people are listening to the welcoming address.

But the exercises must go on!

By the way, if you haven't downloaded a new copy of the book in the last week or so, you should, because David and Brendan updated it and the exercise numbers have changed:

Jerry Wedekind posted a fun exercise relating upper sets to maps between preorders:

Given a monotone function $$f: P \to Q$$ between preorders, it asks you to study a certain monotone function

$$f^{*}: \mathcal{U}(Q) \to \mathcal{U}(P)$$ from the upper sets of $$Q$$ to the upper sets of $$P$$.

I explained upper sets here:

and Exercise 61 fits nicely into the big picture that Matthew Doty and I discussed there.

There are a couple of new exercises on meets and joins in the updated version of the book. These are good for beginners:

And here are some puzzles on adjoints. As we've seen, the preimage or inverse image

$$f^{\ast}(S) = \{x \in X: f(x) \in S\}$$ gives a monotone function $$f^{\ast}: P(Y) \rightarrow P(X)$$ that is the right adjoint of $$f_{!} : P(X) \to P(Y)$$.

Puzzle 20. Does $$f^{\ast}: P(Y) \rightarrow P(X)$$ have a right adjoint of its own?

Puzzle 21. If we give $$\mathbb{N}$$ and $$\mathbb{R}$$ their usual orderings, does the monotone function $$i : \mathbb{N} \to \mathbb{R}$$ have a left adjoint? Does it have a right adjoint? If so, what are they?

Can you find an efficient way to answer puzzles like Puzzles 20 and 21, where you're asked if some monotone function has a left or right adjoint?

• Options
1.

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

If I'm not mistaken, I think that $$f^{\ast}$$ is the right adjoint of $$f_!$$. I think this is the chain of adjunctions that relates the three functions induced by $$f$$:

$$f_! \dashv f^{\ast} \dashv f_{\ast}.$$

Comment Source:> \$$f^{\ast}: P(Y) \rightarrow P(X)\$$ that is the right adjoint of \$$f_\{\ast} : P(X) \to P(Y) \$$. If I'm not mistaken, I think that \$$f^{\ast}\$$ is the right adjoint of \$$f_!\$$. I think this is the chain of adjunctions that relates the three functions induced by \$$f\$$: $$f_! \dashv f^{\ast} \dashv f_{\ast}.$$
• Options
2.

Dan - you're right. I'll fix that mistake.

Somewhere in our voluminous discussions we saw that the chain of adjunctions doesn't extend further in one direction unless $$f$$ is specially nice. We should really check out the other direction... but I know the chain of adjunctions doesn't extend further in either direction unless $$f$$ is especially nice. (If $$f$$ is a bijection, it extends infinitely in both directions.)

Comment Source:Dan - you're right. I'll fix that mistake. Somewhere in our voluminous discussions we saw that the chain of adjunctions doesn't extend further in one direction unless \$$f\$$ is specially nice. We should really check out the other direction... but I know the chain of adjunctions doesn't extend further in _either_ direction unless \$$f\$$ is especially nice. (If \$$f\$$ is a bijection, it extends infinitely in both directions.)