## Idea [[Applied Category Theory]] is an online course being taught by [[John Baez]] based on this free book: * Brendan Fong and David Spivak, _[Seven Sketches in Compositionality](http://math.mit.edu/~dspivak/teaching/sp18/7Sketches.pdf). See also the [website with videos](http://math.mit.edu/~dspivak/teaching/sp18/). Here are the puzzles for Chapter 2 of this book, often with solutions. ## Puzzles * [Puzzle 1](https://forum.azimuthproject.org/discussion/comment/15878/#Comment_15878). What is a "poset" according to Chapter 1 of Fong and Spivak's book? * [Puzzle 2](https://forum.azimuthproject.org/discussion/comment/15878/#Comment_15878). How does their definition differ from the usual definition found in, e.g., Wikipedia or the nLab? * [Puzzle 3](https://forum.azimuthproject.org/discussion/comment/15878/#Comment_15878). What do mathematicians usually call the thing that Fong and Spivak call a poset? * [Puzzle 4](https://forum.azimuthproject.org/discussion/comment/15878/#Comment_15878). List some interesting and important examples of posets that haven't already been listed in other comments in this thread. * [Puzzle 5](https://forum.azimuthproject.org/discussion/comment/15954/#Comment_15954). Why is this property called "trichotomy"? * [Puzzle 6](https://forum.azimuthproject.org/discussion/comment/16077/#Comment_16077). 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](https://forum.azimuthproject.org/discussion/comment/16077/#Comment_16077). 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](https://forum.azimuthproject.org/discussion/1828). There are many examples of monotone maps between posets. List a few interesting ones! * [Puzzle 11](https://forum.azimuthproject.org/discussion/1828). 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 $f$ . * [Puzzle 12](https://forum.azimuthproject.org/discussion/1828). 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](https://forum.azimuthproject.org/discussion/1828). Find a left adjoint for $f$: that is * [Puzzle 14](https://forum.azimuthproject.org/discussion/1845). Find the function g:ℕ→ℕ such that g(b) is the largest possible natural number a with 2a≤b. * [Puzzle 15](https://forum.azimuthproject.org/discussion/1845). Find the function g:ℕ→ℕ such that g(b) is the smallest possible natural number a with b≤2a. * [Puzzle 16](https://forum.azimuthproject.org/discussion/1845). What's going on here? What's the pattern you see, and why is it working this way? * [Puzzle 17](https://forum.azimuthproject.org/discussion/comment/16490/#Comment_16490). Show that $ f_{\ast} : P X \to P Y $ is a monotone function. * [Puzzle 18](https://forum.azimuthproject.org/discussion/comment/16490/#Comment_16490). 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](https://forum.azimuthproject.org/discussion/comment/16490/#Comment_16490). 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](https://forum.azimuthproject.org/discussion/comment/16509/#Comment_16509). Does $f^{\ast}: P Y \rightarrow P X $ have a right adjoint of its own? * [Puzzle TR1](https://forum.azimuthproject.org/discussion/comment/16509/#Comment_16509). Why precisely must g(b) be a least upper bound of the set? * [Puzzle 21](https://forum.azimuthproject.org/discussion/1909). 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](https://forum.azimuthproject.org/discussion/1931) 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](https://forum.azimuthproject.org/discussion/1931) Show that $ f_{!}: P X \rightarrow P Y $ is the right adjoint of $ f^{\ast}: P X \rightarrow P Y $. * [Puzzle 28](https://forum.azimuthproject.org/discussion/1963) Show that if $P$ is a partition of a set $X$, and we define a relation $\sim_P$ on $X$ * [Puzzle 29](https://forum.azimuthproject.org/discussion/1963) Show that if $\sim$ is an equivalence relation on a set $X$, we can define a partition $P_\sim$ on $X$ * [Puzzle 31](https://forum.azimuthproject.org/discussion/1963) Show that the previous two puzzles give a one-to-one correspondence between partitions of $X$ and equivalence relations on $X$. * [Puzzle 32](https://forum.azimuthproject.org/discussion/1963) Proposition 1.11 of _[Seven Sketches] * [Puzzle 33](https://forum.azimuthproject.org/discussion/1963) Is an equivalence relation always a preorder? category: courses [[!redirects Applied Category Theory - Chapter 1 - Puzzles]]