Dear Crowsfort,

I have a jigsaw puzzle. The pieces look like squares
\[ \begin{array}{ccc}
A & \overset{f}{\to} & B \\
g\downarrow & \Downarrow & \downarrow h\\
C & \underset{k}{\to} & D
\end{array}
\]
... actually, $f,g,h,k$ all know what their corners are, so we

*could* leave out the $A,B,C,D$, but this gets distracting. Also, the $\Downarrow$ deserves to have a name, only I can't think of a good way to make it all fit. Which particular $\Downarrow$ a square has in it makes a difference, later!

Two pieces sharing an edge fit together, so that
\[ \begin{array}{ccc}
A & \overset{f}{\to} & B \\
g\downarrow & \Downarrow & \downarrow h\\
C & \underset{k}{\to} & D \\
C & \overset{k}{\to} & D \\
g'\downarrow & \Downarrow & \downarrow h'\\
E & \underset{l}{\to} & F
\end{array}
\]
make a rectangle
\[ \begin{array}{ccc}
A & \overset{f}{\to} & B \\
g\downarrow & \Downarrow & \downarrow h\\
C & \overset{k}{\to} & D \\
g'\downarrow & \Downarrow & \downarrow h'\\
E & \underset{l}{\to} & F
\end{array}
\]
or sometimes
\[ \begin{array}{ccc}
A & \overset{f}{\to} & B \\
g\downarrow & & \downarrow h\\
C & \Downarrow & D \\
g'\downarrow & & \downarrow h'\\
E & \underset{l}{\to} & F
\end{array}
\]
and can even be squished down to a square
\[ \begin{array}{ccc}
A & \overset{f}{\to} & B \\
g'g\downarrow & \Downarrow & \downarrow h'h\\
E & \underset{l}{\to} & F
\end{array}
\]
which is handy, though we don't

*often* want to do that.

The good people who cut out my jigsaw puzzle were very nice, and provided an unlimited supply of various standard shapes, guaranteed to fit certain sorts of corners, so that if anywhere in the puzzle you find
\[ \begin{array}{ccc}
& & A \\
& & \downarrow f\\
B & \underset{g}{\to} & C
\end{array}
\]
you can add in a square
\[ \begin{array}{ccc}
P_{f,g} & \to & A \\
\downarrow & \lrcorner & \downarrow f\\
B & \underset{g}{\to} & C
\end{array}
\]
and in the same way, if you have
\[ \begin{array}{ccc}
A & \overset{f}{\to} & B \\
g \downarrow & & \\
C & &
\end{array}
\]
you can fill it in
\[ \begin{array}{ccc}
A & \overset{f}{\to} & B \\
g \downarrow & \ulcorner & \downarrow \\
C & \to & Q_{f,g}
\end{array}
\]
There is one other sort of handy square, looking like
\[\begin{array}{ccc}
A & \overset{f}{\to} & B \\
f\downarrow & = & \downarrow g \\
B & \underset{g}{\to} & C \end{array}
\]
which lets you

*go around* corners, when it looks like a good idea.
These have two further special types,
\[\begin{array}{ccc}
A & \overset{f}{\to} & B \\
f\downarrow & = & \downarrow = \\
B & \underset{=}{\to} & B \end{array}
\]
... and there is another of the similar sort that I'm sure you can guess; and there's also vertical and horizontal versions of
\[\begin{array}{ccc}
A & = & A \\
f\downarrow & = & \downarrow f \\
B & \underset{=}{\to} & B \end{array}
\]
which also happens to be a $\lrcorner$ and a $\ulcorner$.

Actually, those last two squares are special cases of these two :
\[\begin{array}{ccc}
A & \overset{f}{\to} & B \\
=\downarrow & = & \downarrow g\\
A & \underset{g f}{\to} & C
\end{array}\]
and
\[\begin{array}{ccc}
A & \overset{f}{\to} & B \\
g f \downarrow & = & \downarrow g\\
C & \underset{=}{\to} & C
\end{array}\]
or reflections of them; but most of these two

*are neithert* $\lrcorner$ nor $\ulcorner$.

They were also kind enough to suggest a few ways to get started, using a special corner called "$*$", or "the point", though it's not really the point of all this. Still, there's always exactly one edge $A\to *$, no matter what $A$ is, and you can also draw it vertically:
\[ \begin{array}{c}
A\\ \downarrow \\ *
\end{array} \]
The corner $*$ also has another nifty feature, that the collection of edges $ * \to A$ might as well be called $A$. There's only one corner, $\{\}$ to which you can't draw an arrow from $*$; but on the other hand, there's always exactly one arrow

*from* $\{\}$ to any other corner $A$,

*including to $*$*! So, for instance, there's a nice corner
\[ \begin{array}{ccc}
\{\} & \to & * \\
\downarrow & & \\
*
\end{array} \]
and because of the $\ulcorner$ pieces, this gets filled-in as
\[ \begin{array}{ccc}
\{\} & \to & * \\
\downarrow & \Box & \downarrow\\
* & \to & * + *
\end{array} \]
although it's more common, among my fellow puzzlers, to call that new thing $\mathbb{S}^0$. It has two points, as you can see. Oh! this one tile happens to be of *both* sorts: it's the standard tile to fill-in those two edges $*\to \mathbb{S}^0$ as well as the standard tile to fill-in the edge $\{\}\to*$ drawn twice from a single copy of $\{\}$.
Sometimes it's fun just to look at the special pieces
\[ \begin{array}{ccc}
A & \to & * \\
\downarrow & \ulcorner & \downarrow \\
* & \to & \Sigma A \end{array}
\]
which highlight a fascinating sequence of corner labels $A, \Sigma A, \Sigma^2 A, \ldots$ --- the ones you get starting with $\mathbb{S}^0$ are called the

*spheres* (or

*homotopy spheres*) and have the special names $\mathbb{S}^n = \Sigma^n \mathbb{S}^0 $.
Going in the other direction --- if you have a favourite arrow $ * \overset{a}{\to} A $, the special square you get is labelled
\[ \begin{array}{ccc}
\Omega_a A & \to & * \\
\downarrow & \lrcorner & \downarrow a\\
* & \underset{a}{\to} & A \end{array}
\]
... to tell you how one is supposed to keep going after that, I have to tell you one last thing about the special squares labelled $\lrcorner$ and $\ulcorner$; given any square at all
\[ \begin{array}{ccc}
A & \overset{f}{\to} & B \\
g\downarrow & \Downarrow & \downarrow h\\
C & \underset{k}{\to} & D
\end{array}
\]
there are of course the standard two squares
\[ \begin{array}{ccc}
P_{h,k} & \to & B \\
\downarrow & \lrcorner & \downarrow h\\
C & \underset{k}{\to} & D
\end{array}
\]
and the other one to $Q_{f,g}$; in essence,

*what it means to be a* $\lrcorner$ is that, there's essentially just one edge $ A \overset{w}\to P_{h,k} $ that fits into this puzzle
\[ \begin{array}{ccccc}
A & \overset{=}{\to} & A & \overset{f}{\to} & B \\
=\downarrow & = & w \downarrow & \Downarrow & \downarrow = \\
A & \underset{w}{\to} & P_{h,k} & \to & B \\
g\downarrow & \Downarrow & \downarrow & \lrcorner & \downarrow h \\
C & \underset{=}{\to} & C & \underset{k}{\to} & D
\end{array} \]
There's a similar story about unique edges $ Q_{f,g} \to D $ that fit in another puzzle --- try it and see!
But particularly, since we always have

*this* square
\[\begin{array}{ccc}
* & \to & * \\
\downarrow & = & \downarrow a\\
* & \underset{a}{\to} & A
\end{array}\]
there's exactly one $ * \to \Omega_a A$ that fits in all the necessary puzzles, and

*this* is what lets us keep going to make new spaces $ \Omega^2_a A, \cdots $. Here's a

*puzzle* for you: come up with a good edge $ A \to \Omega_{?} \Sigma A$! This entails finding a way to fill-in that $?$; you should be able to think of perhaps-two.

There are lots of things I haven't mentioned, but of course, that will always be true, even if I say all the things that

*should come first!* You're welcome to play with the jigsaw, too; we'll never run out of pieces!

__the joiner__