> **Puzzle 106**. What are some reasonable equations between morphisms that we might want to impose?

Maybe we want to limit the number of layers of management to say 4.

\[ \text{Manager} \circ \text{Manager} \circ \text{Manager} \circ \text{Manager} = 1_\text{Employee} \]

**Puzzle 106-FPE** We want to ensure management has a strict hierarchy, i.e. a tree.

What equations will enforce that?

> **Puzzle 107** How many morphisms does this category have? How is it related to the square?

The morphisms are \\( \lbrace [1_z = s \circ s \circ s \circ s], [s], [s \circ s], [s \circ s \circ s] \rbrace \\) which are 4.

If \\(s\\) represents a \\(90 \deg\\) spin of the square then we have a morphism from the initial pose to each symmetric pose of the square [without flipping/turning], of which there are 4.

\[

\begin{array}{c|c}

\text{Pose} & \text{morphism} \\\\

\hline

\tt{0 \deg} & 1_z = s \circ s \circ s \circ s \\\\

\tt{90 \deg} & s \\\\

\tt{180 \deg} & s \circ s \\\\

\tt{270 \deg} & s \circ s \circ s

\end{array}

\]

Maybe we want to limit the number of layers of management to say 4.

\[ \text{Manager} \circ \text{Manager} \circ \text{Manager} \circ \text{Manager} = 1_\text{Employee} \]

**Puzzle 106-FPE** We want to ensure management has a strict hierarchy, i.e. a tree.

What equations will enforce that?

> **Puzzle 107** How many morphisms does this category have? How is it related to the square?

The morphisms are \\( \lbrace [1_z = s \circ s \circ s \circ s], [s], [s \circ s], [s \circ s \circ s] \rbrace \\) which are 4.

If \\(s\\) represents a \\(90 \deg\\) spin of the square then we have a morphism from the initial pose to each symmetric pose of the square [without flipping/turning], of which there are 4.

\[

\begin{array}{c|c}

\text{Pose} & \text{morphism} \\\\

\hline

\tt{0 \deg} & 1_z = s \circ s \circ s \circ s \\\\

\tt{90 \deg} & s \\\\

\tt{180 \deg} & s \circ s \\\\

\tt{270 \deg} & s \circ s \circ s

\end{array}

\]