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