It's time for more exercises!

Monotone functions between discrete posets:

* [Exercise 50 - Chapter 1](https://forum.azimuthproject.org/discussion/1928/)

Pullbacks of partitions, with a nice picture by [[Frederick Eisele]]:

* [Exercise 52 - Chapter 1](https://forum.azimuthproject.org/discussion/1917)

Showing that there's a category of preorders:

* [Exercise 54 - Chapter 1](https://forum.azimuthproject.org/discussion/1955)

Showing that skeletal dagger preorders are discrete:

* [Exercise 56 - Chapter 1](https://forum.azimuthproject.org/discussion/1972)

Showing that a certain map from a poset of partitions to the Boolean poset \$$\mathbb{B}\$$ is monotone:

* [Exercise 58 - Chapter 1](https://forum.azimuthproject.org/discussion/1956)

Next, some puzzles about adjoints. For any set \$$X\$$ let \$$P(X)\$$ be the **[power set](https://en.wikipedia.org/wiki/Power_set)** of set, namely the set of all subsets of \$$X\$$. It's easy to see that the subset relation \$$\subseteq\$$ makes \$$P(X)\$$ into a poset. Suppose we have any function between sets

$$f : X \to Y$$

This gives a function

$$f_{!} : P(X) \to P(Y)$$

sending each subset \$$S \subseteq X\$$ to its **[image](https://en.wikipedia.org/wiki/Image_(mathematics)#Image_of_a_subset)** under \$$f\$$:

$$f_{!}(S) = \\{ y \in Y: \; y = f(x) \textrm{ for some } x \in S \\} .$$

**Puzzle 17.** Show that \$$f_{!} : P(X) \to P(Y) \$$ is a monotone function.

**Puzzle 18.** Does \$$f_{!} \$$ 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_{!}\$$ 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.

For answers, see the [discussion on Lecture 6](https://forum.azimuthproject.org/discussion/comment/16490/#Comment_16490). One of the adjoints of \$$f_{!}\$$ only exists when \$$f\$$ obeys a certain condition!

To see _all_ the exercises in Chapter 1, [go here](http://www.azimuthproject.org/azimuth/show/Applied+Category+Theory#chapter_1).