[[!redirects Applied Category Theory - Exercises 1]] [[Applied Category Theory]] ## Exercises [from the book] * [Ex 1.1 : property preservation](https://forum.azimuthproject.org/discussion/1871) * [Ex 1.2 : joining systems](https://forum.azimuthproject.org/discussion/1872) * [Ex 1.3 : partitions](https://forum.azimuthproject.org/discussion/1876) * [Ex 1.4 : meet join](https://forum.azimuthproject.org/discussion/1887) * [Ex 1.5 : ](https://forum.azimuthproject.org/discussion/1888) * [Ex 1.9 : ](https://forum.azimuthproject.org/discussion/1897) * [Ex 1.12 :](https://forum.azimuthproject.org/discussion/1890) * [Ex 1.15 : What is a function?]() * [Ex 1.18 : Surjection = Partion]() * [Ex 1.26 : Hasse]() * [Ex 1.28 : preorder of partions]() * [Ex 1.30 : divison preorder]() * [Ex 1.31 : total order]() * [Ex 1.35 : Simple Hasse]() * [Ex 1.37 : ](https://forum.azimuthproject.org/discussion/1895) * [Ex 1.39 : upper sets]() * [Ex 1.41 : poset product]() * [Ex 1.49 : monotone maps vs. discrete posets]() * [Ex 1.50 : monotone maps & their inverses]() * [Ex 1.52 : monotone maps between posets]() * [Ex 1.54 : skeletal dagger = poset]() * [Ex 1.57 : binary surjection on preorder on partion]() * [Ex 1.57 : pullback map]() * [Ex 1.65 : meets and joins]() * [Ex 1.65 : meets and joins](https://forum.azimuthproject.org/discussion/1915) * [Ex 1.67 : meets and joins](https://forum.azimuthproject.org/discussion/1918) * [Ex 1.72 : adjunctions]() * [Ex 1.75 : adjunctions]() * [Ex 1.77 : ](https://forum.azimuthproject.org/discussion/1911) * [Ex 1.79 : ]() * [Ex 1.80 : Galois connections](https://forum.azimuthproject.org/discussion/1910) * [Ex 1.82 : Galois connections](https://forum.azimuthproject.org/discussion/1908) * [Ex 1.83 : ]() * [Ex 1.85 : ]() * [Ex 1.87 : ]() * [Ex 1.90 : ]() * [Ex 1.91 : ]() * [Ex 1.96 : ]() * [Ex 1.97 : ](https://forum.azimuthproject.org/discussion/1892) ## Puzzles [from the forum lectures] * [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} : PX \to PY $ 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}: PY \rightarrow PX $ 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_{!}: PX \rightarrow PY $ is the right adjoint of $ f^{\ast}: PX \rightarrow PY $.