The Azimuth Project Applied Category Theory - Chapter 1 - Puzzles (Rev #3, changes)

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Applied Category Theory

Puzzles [from the forum lectures]

• Puzzle 1. What is a “poset” according to Chapter 1 of Fong and Spivak’s book?

• Puzzle 2. How does their definition differ from the usual definition found in, e.g., Wikipedia or the nLab?

• Puzzle 3. What do mathematicians usually call the thing that Fong and Spivak call a poset?

• Puzzle 4. List some interesting and important examples of posets that haven’t already been listed in other comments in this thread.

• Puzzle 5. Why is this property called “trichotomy”?

• Puzzle 6. How do reflexivity and transitivity of ≤ follow from the rules of a category, if we have a category with at most one morphism from any object x to any object y, and we write x≤y when there exists a morphism from x to y?

• Puzzle 7. Why does any set with a reflexive and transitive relation ≤ yield a category with at most one morphism from any object x to any object y? That is: why are reflexivity and transitivity enough?

• Puzzle 10. There are many examples of monotone maps between posets. List a few interesting ones!

Definition.Show that if the monotone map Given preorders $$A,\le_A$$ and $$B,\le_B$$, a $f: A \to B$Galois connection] has an inverse is a monotone map $$f : A \to B$$ together with a monotone map $$g: B \to A$$ such that$g : B \to A$ that is also a monotone map, $g$ is both a right adjoint and a left adjoint of $f$ .

$f(a) \le_B b \textrm{ if and only if } a \le_A g(b)$
• Puzzle 12. Find a right adjoint for $f$: that is, a function $g : \mathbb{N} \to \mathbb{N}$ with $f(m) \le n \text{ if and only if } m \le g(n) \text{ for all } m,n \in mathbb{N}$. How many right adjoints can you find?

• Puzzle 13. Find a left adjoint for $f$: that is

• Puzzle 14. Find the function g:ℕ→ℕ such that g(b) is the largest possible natural number a with 2a≤b.

• Puzzle 15. Find the function g:ℕ→ℕ such that g(b) is the smallest possible natural number a with b≤2a.

• Puzzle 16. What’s going on here? What’s the pattern you see, and why is it working this way?

• Puzzle 17. Show that $f_{\ast} : PX \to PY$ is a monotone function.

• Puzzle 18. Does $f_{\ast}$ always have a left adjoint? If so, describe it. If not, give an example where it doesn’t, and some conditions under which it does have a left adjoint.

• Puzzle 19. Does $f_{\ast}$ always have a right adjoint? If so, describe it. If not, give an example where it doesn’t, and some conditions under which it does have a right adjoint.

• Puzzle 20. Does $f^{\ast}: PY \rightarrow PX$ have a right adjoint of its own?

• Puzzle TR1. Why precisely must g(b) be a least upper bound of the set?

• Puzzle 21. 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?

• 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 24 Show that $f_{!}: PX \rightarrow PY$ is the right adjoint of $f^{\ast}: PX \rightarrow PY$.

for all $$a \in A, b \in B$$. In this situation we call $$f$$ the left adjoint and $$g$$ the right adjoint.

So, the right adjoint of $$f$$ is a way of going back from $$B$$ to $$A$$ that’s related to $$f$$ in some way.

• Puzzle 11. Show that if the monotone map $$f: A \to B$$ has an inverse $$g : B \to A$$ that is also a monotone map, $$g$$ is both a right adjoint and a left adjoint of $$f$$.

Here’s one easy example to get you started. Let $$\mathbb{N}$$ be the set of natural numbers with its usual notion of $$\le$$. There’s a function $$f : \mathbb{N} \to \mathbb{N}$$ with $$f(x) = 2x$$. This function doesn’t have an inverse. But:

• Puzzle 12. Find a right adjoint for $$f$$: that is, a function $$g : \mathbb{N} \to \mathbb{N}$$ with
$f(m) \le n \textrm{ if and only if } m \le g(n)$

for all $$m,n \in \mathbb{N}$$. How many right adjoints can you find?

• Puzzle 13. Find a left adjoint for $$f$$: that is, a function $$g : \mathbb{N} \to \mathbb{N}$$ with
$g(m) \le n \textrm{ if and only if } m \le f(n)$

for all $$m,n \in \mathbb{N}$$. How many left adjoints can you find?

• Puzzle 14. Find the function g:ℕ→ℕ such that g(b) is the largest possible natural number a with 2a≤b.

• Puzzle 15. Find the function g:ℕ→ℕ such that g(b) is the smallest possible natural number a with b≤2a.

• Puzzle 16. What’s going on here? What’s the pattern you see, and why is it working this way?

• Puzzle 17. Show that $$f_{\ast} : PX \to PY$$ is a monotone function.

• Puzzle 18. Does $$f_{\ast}$$ always have a left adjoint? If so, describe it. If not, give an example where it doesn’t, and some conditions under which it does have a left adjoint.

• Puzzle 19. Does $$f_{\ast}$$ always have a right adjoint? If so, describe it. If not, give an example where it doesn’t, and some conditions under which it does have a right adjoint.

• Puzzle 20. Does $$f^{\ast}: PY \rightarrow PX$$ have a right adjoint of its own?

• Puzzle TR1. Why precisely must g(b) be a least upper bound of the set?

• Puzzle 21. 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?