# November 28, 2001 {#week174}

Groups are how mathematicians and physicists talk about symmetry, and
Lie groups are how they talk about *continuously varying* symmetries,
like rotations, translations and the like. Sophus Lie helped start the
subject of Lie groups in the late 1800s, and it's been in constant
growth ever since. I spend lots of time studying it, and I probably will
all my life --- there's a lot to learn! To really understand it, it helps
to know the history. And for that, this is the book to read:

1) Thomas Hawkins, _The Emergence of the Theory of Lie Groups: an Essay in the History of Mathematics, 1869--1926_, Springer, New York, 2000.

You have to know your Lie groups pretty well to enjoy this book, but if
you do, you'll find it's full of interesting facts. For example: folks
often complain about Wilhelm Killing's original classification of
simple Lie algebras --- it wasn't rigorous, he made some mistakes, and so
on. Elie Cartan came along later and cleaned it up, and many people
applaud Cartan's work and sneer at poor old Killing, even though he was
the one who came up with the original ideas. But in this book, it
becomes clear that Killing was pretty much *pushed* into publishing his
ideas in a half-baked state by mathematicians who were dying to know his
results! Now I feel even more sorry for him.

There's also a lot of interesting stuff about Hermann Weyl's approach
to representation theory via tensors and Young diagrams, and why he
liked it better than Cartan's approach via roots and weights.
Basically, Weyl liked his approach because it stuck closer to Felix
Klein's original "Erlanger program" --- a program for understanding
geometry via symmetry groups. But it's interesting to see how Weyl
studied and respected Cartan's approach, and tried to bridge the gap
between the two.

Okay... so much for gossip! Now I'm going to dive in and pick up right
where I left off in my discussion of the ideas behind this paper:

2) Michael Mueger, "From subfactors to categories and topology I: Frobenius algebras in and Morita equivalence of tensor categories", available at [`math.CT/0111204`](https://arxiv.org/abs/math.CT/0111204).

My ultimate goal is to take you to an elegant understanding of Frobenius
algebras by means of a $2$-category called the "walking ambidextrous
adjunction", but first I'll play around a bit with a simpler but more
famous $2$-category called the "walking adjunction". This may sound
scary, but if you can stick with it, you'll see that I'm really just
using these $2$-categories to describe fun games that you can play with
certain $2$-dimensional pictures. Even if you don't read the words,
please stare at the pictures --- I spend my Thanksgiving weekend drawing
them, and I don't want that work to go to waste!

Category theorists love to talk about adjoint functors, but $2$-category
theorists know that these are just a special example of an
"adjunction". An adjunction is something that makes sense in any
$2$-category; if we take the $2$-category to be $\mathsf{Cat}$ we get adjoint functors.
There are lots of other nice examples that make this generalization
worthwhile. For example, in ["Week 83"](#week83) I explained how a
pair of dual vector spaces is also an example of an adjunction.

To study adjunctions, it suffices to study the "walking adjunction".
This is a little $2$-category containing exactly the stuff any adjunction
in any $2$-category must have: not a jot more, not a tiddle less! It was
first studied by Schanuel and Street:

3) Stephen Schanuel and Ross Street, "The free adjunction", _Cah. Top. Geom. Diff._ **27** (1986), 81--83.

In a bit more detail, the walking adjunction is the $2$-category freely
generated by two objects:
$$\mbox{$a$ and $b$,}$$
two morphisms:
$$\mbox{$L\colon a \to b$ and $R\colon b \to a$,}$$
and two $2$-morphisms, called the "unit" and "counit":
$$\mbox{$i\colon 1_a \Rightarrow LR$ and $e\colon RL \Rightarrow 1_b$}$$
satisfying two relations, called the "triangle equations".

I wrote down these equations already last week, but let me do it again
using "string diagrams", as explained in ["Week 79"](#week79) and
["Week 92"](#week92). In a $2$-categorical string diagram, objects
are denoted by 2d regions in the plane, morphisms are denoted by 1d
edges, and $2$-morphisms are denoted by 0d points. If the dimensions look
sort of upside-down, you're right --- that's exactly the point!

Instead of explaining the whole theory, I'll just plunge in with the
example at hand. The unit $i$ looks like this:
$$
  \begin{tikzpicture}[yscale=-1]
    \begin{knot}
      \strand[thick] (0,0)
      to (0,-0.5)
      to [out=down,in=down,looseness=2] (1.5,-0.5)
      to (1.5,0);
    \end{knot}
    \node[fill=white] at (0,-0.45) {$L$};
    \node[fill=white] at (1.5,-0.45) {$R$};
    \node[label={[label distance=-1mm]above:{$i$}}] at (0.75,-1.39) {$\bullet$};
    \node at (0.75,-0.75) {$b$};
    \node at (-0.45,-1) {$a$};
    \node at (1.95,-1) {$a$};
  \end{tikzpicture}
$$
while the counit $e$ looks like this:
$$
  \begin{tikzpicture}
    \begin{knot}
      \strand[thick] (0,0)
      to (0,-0.5)
      to [out=down,in=down,looseness=2] (1.5,-0.5)
      to (1.5,0);
    \end{knot}
    \node[fill=white] at (0,-0.45) {$L$};
    \node[fill=white] at (1.5,-0.45) {$R$};
    \node[label={[label distance=-1mm]below:{$e$}}] at (0.75,-1.39) {$\bullet$};
    \node at (0.75,-0.75) {$a$};
    \node at (-0.45,-1) {$b$};
    \node at (1.95,-1) {$b$};
  \end{tikzpicture}
$$
Note that as you cross a line labelled "$L$" from left to right, you go
from region $a$ to region $b$, which is our way of saying that $L\colon a\to b$.
Similarly, as you cross a line labelled "$R$" from left to right, you go
from region $b$ to region $a$, since $R\colon b\to a$.

In terms of string diagrams, the triangle equations just say that we can
straighten out a zig-zag:
$$
  \begin{tikzpicture}
    \begin{scope}
      \begin{knot}
        \strand[thick] (0,0)
        to (0,1)
        to [out=up,in=up,looseness=2] (1,1)
        to [out=down,in=down,looseness=2] (2,1)
        to (2,2);
      \end{knot}
      \node at (-0.5,1.5) {$a$};
      \node[fill=white] at (0,0.5) {$L$};
      \node[label={[label distance=-1mm]above:{$i$}}] at (0.5,1.57) {$\bullet$};
      \node[fill=white] at (1,1) {$R$};
      \node[label={[label distance=-1mm]below:{$e$}}] at (1.5,0.4) {$\bullet$};
      \node[fill=white] at (2,1.5) {$L$};
      \node at (2.5,0.5) {$b$};
    \end{scope}
    \node at (3.5,1) {$=$};
    \begin{scope}[shift={(5,2)}]
      \node at (-0.5,-1) {$a$};
      \draw[thick] (0,0) to node[fill=white]{$L$} (0,-2);
      \node at (0.5,-1) {$b$};
    \end{scope}
  \end{tikzpicture}
$$
or a zag-zig:
$$
  \begin{tikzpicture}
    \begin{scope}[xscale=-1,shift={(-2,0)}]
      \begin{knot}
        \strand[thick] (0,0)
        to (0,1)
        to [out=up,in=up,looseness=2] (1,1)
        to [out=down,in=down,looseness=2] (2,1)
        to (2,2);
      \end{knot}
      \node at (-0.5,1.5) {$a$};
      \node[fill=white] at (0,0.5) {$R$};
      \node[label={[label distance=-1mm]above:{$i$}}] at (0.5,1.57) {$\bullet$};
      \node[fill=white] at (1,1) {$L$};
      \node[label={[label distance=-1mm]below:{$e$}}] at (1.5,0.4) {$\bullet$};
      \node[fill=white] at (2,1.5) {$R$};
      \node at (2.5,0.5) {$b$};
    \end{scope}
    \node at (3.5,1) {$=$};
    \begin{scope}[shift={(5,2)}]
      \node at (-0.5,-1) {$b$};
      \draw[thick] (0,0) to node[fill=white]{$R$} (0,-2);
      \node at (0.5,-1) {$a$};
    \end{scope}
  \end{tikzpicture}
$$
We can build any $2$-morphism in the walking adjunction by vertically and
horizontally composing units and counits, which corresponds to sticking
together string diagrams in a vertical or horizontal way. Thus, a
typical $2$-morphism looks like this:
$$
  \begin{tikzpicture}[xscale=1.1,yscale=1.4]
    \begin{knot}
      \strand[thick] (-0.5,0)
        to [out=up,in=down] (-2,2);
      \strand[thick] (-1,2)
        to [out=down,in=down,looseness=2] (0,2);
      \strand[thick] (1,2)
        to [out=down,in=down,looseness=2] (2,2);
      \strand[thick] (1.5,0)
        to [out=up,in=down] (3,2);
      \strand[thick] (2.5,0)
        to [out=up,in=up,looseness=2] (3.5,0);
      \strand[thick] (4.5,0)
        to [out=up,in=down,looseness=2] (4,1)
        to [out=up,in=up,looseness=2] (5,1)
        to [out=down,in=down,looseness=2] (6,1)
        to (6,2);
    \end{knot}
    \node at (-2,1) {$a$};
    \node at (-0.5,1.75) {$a$};
    \node at (1.5,1.75) {$a$};
    \node at (0.5,1) {$b$};
    \node at (3,1) {$a$};
    \node at (3,0.25) {$b$};
    \node at (4.5,1) {$b$};
    %
    \node[fill=white] at (-1.25,1) {\scriptsize$L$};
    \node[fill=white] at (-1,1.75) {\scriptsize$R$};
    \node[fill=white] at (0,1.75) {\scriptsize$L$};
    \node[fill=white] at (1,1.75) {\scriptsize$R$};
    \node[fill=white] at (2,1.75) {\scriptsize$L$};
    \node[fill=white] at (2.25,1) {\scriptsize$L$};
    \node[fill=white] at (2.5,0.25) {\scriptsize$L$};
    \node[fill=white] at (3.5,0.25) {\scriptsize$R$};
    \node[fill=white] at (4.25,0.5) {\scriptsize$L$};
    \node[fill=white] at (5,1) {\scriptsize$R$};
    \node[fill=white] at (6,1.5) {\scriptsize$L$};
    %
    \node[label={[label distance=-2mm]below:{\scriptsize$e$}}] at (-0.5,1.4) {$\bullet$};
    \node[label={[label distance=-2mm]below:{\scriptsize$e$}}] at (1.5,1.4) {$\bullet$};
    \node[label={[label distance=-2mm]above:{\scriptsize$i$}}] at (3,0.58) {$\bullet$};
    \node[label={[label distance=-2mm]below:{\scriptsize$e$}}] at (5.5,0.4) {$\bullet$};
    \node[label={[label distance=-2mm]above:{\scriptsize$i$}}] at (4.5,1.58) {$\bullet$};
  \end{tikzpicture}
$$
By the triangle equations, we could straighten out the zig-zag without
changing the $2$-morphism.

As you may know, the word "anaranjado" means "orange" in Spanish ---
there was no word in English for "orange" before people in England
started importing oranges from Spain. And this is a nice mnemonic,
because if we take the above picture and paint the regions labelled
"$a$" orange, and paint the regions labelled "$b$" black, the above
picture has a roughly tiger-striped appearance. In fact, these tiger
stripes tell you everything you need to know about the $2$-morphism! For
example, starting from just this:
$$
  \begin{tikzpicture}[xscale=0.8,yscale=1.2]
    \begin{knot}
      \strand[thick] (-0.5,0)
        to [out=up,in=down] (-2,2);
      \strand[thick] (-1,2)
        to [out=down,in=down,looseness=2] (0,2);
      \strand[thick] (1,2)
        to [out=down,in=down,looseness=2] (2,2);
      \strand[thick] (1.5,0)
        to [out=up,in=down] (3,2);
      \strand[thick] (2.5,0)
        to [out=up,in=up,looseness=2] (3.5,0);
      \strand[thick] (4.5,0)
        to [out=up,in=down,looseness=2] (4,1)
        to [out=up,in=up,looseness=2] (5,1)
        to [out=down,in=down,looseness=2] (6,1)
        to (6,2);
    \end{knot}
    \node at (-2,1) {$a$};
    \node at (-0.5,1.75) {$a$};
    \node at (1.5,1.75) {$a$};
    \node at (0.5,1) {$b$};
    \node at (3,1) {$a$};
    \node at (3,0.25) {$b$};
    \node at (4.5,1) {$b$};
  \end{tikzpicture}
$$
you can figure out where everything else should go.

By the way, note that orange stripes can disappear as we go down the
page, and they can split, but they can't appear or merge. Black stripes
can appear or merge, but they can't disappear or split. As a result,
there can never be any orange or black *spots*. We'll change these
rules later, when we talk about the walking "ambidextrous adjunction".

Okay, so we've got this $2$-category, the walking adjunction: let's call
it $\mathsf{Ad}$ for short. It's pretty simple. How can we understand it better?

Well, for any two objects $a$ and $b$ in a $2$-category we get a
"hom-category" $\operatorname{Hom}(a,b)$, whose objects are the morphisms from $a$ to $b$,
and whose morphisms are the $2$-morphisms between those. If we work out
these hom-categories in $\mathsf{Ad}$, we get some cool stuff.

First let's look at the hom-category $\operatorname{Hom}(a,a)$. In this category, the
objects are
$$1_a, LR, LRLR, LRLRLR, \ldots$$
and all the morphisms are built by sticking these two basic generators
together vertically or horizontally:
$$
  \begin{tikzpicture}[yscale=2]
    \begin{knot}
      \strand[thick] (-0.5,0)
        to [out=up,in=down,looseness=1.5] (-1,1);
      \strand[thick] (-0.45,1)
        to [out=down,in=down,looseness=1.5] (0.45,1);
      \strand[thick] (0.5,0)
        to [out=up,in=down,looseness=1.5] (1,1);
    \end{knot}
    %
    \node at (-1.25,0.35) {$a$};
    \node at (0,0.35) {$b$};
    \node at (0,0.9) {$a$};
    \node at (1.25,0.35) {$a$};
    %
    \node[fill=white] at (-0.55,0.25) {\scriptsize$L$};
    \node[fill=white] at (-0.95,0.75) {\scriptsize$L$};
    \node[fill=white] at (-0.4,0.85) {\scriptsize$R$};
    \node[fill=white] at (0.4,0.85) {\scriptsize$L$};
    \node[fill=white] at (0.55,0.25) {\scriptsize$R$};
    \node[fill=white] at (0.95,0.75) {\scriptsize$R$};
    \node[label={[label distance=-2mm]below:{\scriptsize$e$}}] at (0,0.61) {$\bullet$};
  \end{tikzpicture}
$$
and
$$
  \begin{tikzpicture}[scale=1.3]
    \begin{knot}
      \strand[thick] (-0.5,0)
        to [out=up,in=up,looseness=3] (0.5,0);
    \end{knot}
    \node at (-1,0.5) {$a$};
    \node at (0,0.25) {$b$};
    \node at (1,0.5) {$a$};
    \node[fill=white] at (-0.45,0.35) {\scriptsize$L$};
    \node[fill=white] at (0.45,0.35) {\scriptsize$R$};
    \node[label={[label distance=-2mm]above:{\scriptsize$i$}}] at (0,0.88) {$\bullet$};
  \end{tikzpicture}
$$
In tiger language, we're talking about pictures of black stripes on an
orange background. The two basic generators are the merging of two black
stripes and the appearance of a black stripe.

If you read ["Week 89"](#week89), you'll know another way to
describe this! Our ability to stick together pictures vertically and
horizontally makes $\operatorname{Hom}(a,a)$ into a "monoidal category". $LR$ is a
"monoid object", with merging of two black stripes being
"multiplication", and the appearance of a black stripe being the
"multiplicative identity". Being a "monoid object" simply means that
these operations satisfy the left unit law:
$$
  \begin{tikzpicture}[xscale=0.8,yscale=1.5]
    \begin{knot}
      \strand[thick] (-0.5,0)
        to [out=up,in=down,looseness=1.5] (-1,1)
        to [out=up,in=up,looseness=1] (-0.45,1)
        to [out=down,in=down,looseness=1.5] (0.45,1)
        to (0.45,1.5);
      \strand[thick] (0.5,0)
        to [out=up,in=down,looseness=1] (1,1)
        to (1,1.5);
    \end{knot}
    %
    \node at (-1.25,0.35) {$a$};
    \node at (0,0.35) {$b$};
    \node at (1.25,0.35) {$a$};
  \end{tikzpicture}
  \quad
  \raisebox{3em}{$=$}
  \quad
  \begin{tikzpicture}[yscale=1.5]
    \draw[thick] (-0.3,0) to (-0.3,1.5);
    \draw[thick] (0.3,0) to (0.3,1.5);
    %
    \node at (-0.75,1) {$a$};
    \node at (0,0.75) {$b$};
    \node at (0.75,0.5) {$a$};
  \end{tikzpicture}
$$
and its mirror image, called the right unit law, together with the
associative law:
$$
  \begin{tikzpicture}[xscale=1,yscale=1.7]
    \begin{knot}
      \strand[thick] (-0.5,0)
        to [out=up,in=down,looseness=1.5] (-1,1);
      \strand[thick] (-0.45,1)
        to [out=down,in=down,looseness=1.5] (0.45,1)
        to (0.45,1.55);
      \strand[thick] (0.5,0)
        to [out=up,in=down,looseness=1] (1,1)
        to (1,1.55);
    \end{knot}
    \begin{scope}[scale=0.55,shift={(-1.32,1.81)}]
      \begin{knot}
        \strand[thick] (-0.5,0)
          to [out=up,in=down,looseness=1.5] (-1,1);
        \strand[thick] (-0.45,1)
          to [out=down,in=down,looseness=1.5] (0.45,1);
        \strand[thick] (0.5,0)
          to [out=up,in=down,looseness=1] (1,1);
      \end{knot}
    \end{scope}
    %
    \node at (-1.25,0.35) {$a$};
    \node at (0,0.35) {$b$};
    \node at (1.25,0.35) {$a$};
    \node at (-0.73,1.5) {\scriptsize$a$};
    \node at (0,1) {$a$};
  \end{tikzpicture}
  \quad
  \raisebox{4em}{$=$}
  \quad
  \begin{tikzpicture}[xscale=-1,yscale=1.7]
    \begin{knot}
      \strand[thick] (-0.5,0)
        to [out=up,in=down,looseness=1.5] (-1,1);
      \strand[thick] (-0.45,1)
        to [out=down,in=down,looseness=1.5] (0.45,1)
        to (0.45,1.55);
      \strand[thick] (0.5,0)
        to [out=up,in=down,looseness=1] (1,1)
        to (1,1.55);
    \end{knot}
    \begin{scope}[scale=0.55,shift={(-1.32,1.81)}]
      \begin{knot}
        \strand[thick] (-0.5,0)
          to [out=up,in=down,looseness=1.5] (-1,1);
        \strand[thick] (-0.45,1)
          to [out=down,in=down,looseness=1.5] (0.45,1);
        \strand[thick] (0.5,0)
          to [out=up,in=down,looseness=1] (1,1);
      \end{knot}
    \end{scope}
    %
    \node at (-1.25,0.35) {$a$};
    \node at (0,0.35) {$b$};
    \node at (1.25,0.35) {$a$};
    \node at (-0.73,1.5) {\scriptsize$a$};
    \node at (0,1) {$a$};
  \end{tikzpicture}
$$
There aren't any other laws, so $\operatorname{Hom}(a,a)$ is the "free monoidal
category on a monoid object", or if you prefer, the "walking monoid"!

I touched upon the immense consequences of this fact for algebraic
topology in ["Week 117"](#week117) and ["Week 118"](#week118).
They mainly rely on another way of thinking about $\operatorname{Hom}(a,a)$: it's the
category of order-preserving maps between finite ordinals!

For example, these black tiger stripes on an orange background:
$$
  \begin{tikzpicture}[xscale=0.8,yscale=1.2]
    \begin{knot}
      \strand[thick] (-0.5,0)
        to [out=up,in=down] (-2,2);
      \strand[thick] (-1,2)
        to [out=down,in=down,looseness=2] (0,2);
      \strand[thick] (1,2)
        to [out=down,in=down,looseness=2] (2,2);
      \strand[thick] (1.5,0)
        to [out=up,in=down](3,2);
      \strand[thick] (2.5,0)
        to [out=up,in=up,looseness=2] (3.5,0);
      \strand[thick] (5.1,0)
        to [out=up,in=down,looseness=1] (4.5,1)
        to [out=up,in=up,looseness=1.5] (5.17,1)
        to [out=down,in=down,looseness=1.5] (5.83,1)
        to (5.83,2);
      \strand[thick] (5.9,0)
        to [out=up,in=down,looseness=1] (6.5,1)
        to (6.5,2);
    \end{knot}
    \node at (-1.7,0.5) {$a$};
    \node at (-0.5,1.75) {$a$};
    \node at (1.5,1.75) {$a$};
    \node at (0.5,1) {$b$};
    \node at (3.5,1) {$a$};
    \node at (3,0.25) {$b$};
    \node at (5.5,0.25) {$b$};
    \node at (6.6,0.3) {$a$};
    %
    \draw[thick] (-2.5,0) rectangle ++(9.5,2);
    \node at (0.5,-0.25) {$0$};
    \node at (3,-0.25) {$1$};
    \node at (5.5,-0.25) {$2$};
    \node at (-1.5,2.25) {$0$};
    \node at (0.5,2.25) {$1$};
    \node at (2.5,2.25) {$2$};
    \node at (6.17,2.25) {$3$};
  \end{tikzpicture}
$$
correspond to the order-preserving map
$$f\colon \{0,1,2,3\} \to \{0,1,2\}$$
with
$$f(0) = 0,\quad f(1) = 0,\quad f(2) = 0,\quad f(3) = 2.$$
Just read the stripes down!

A more geometrical way to say the same thing is to call $\operatorname{Hom}(a,a)$ the
category of "simplices", usually denoted $\Delta$. Here the object
$$\underbrace{LRLR\ldots LR}_{\mbox{$n+1$ of them}}$$
corresponds to the $n$-simplex, and these morphisms:
$$
  \begin{tikzcd}[column sep=huge]
    1_a
      \rar["i" description]
    & LR
      \rar[shift left=5,"i\cdot LR" description]
      \rar["LR\cdot i" description]
    & LRLR
      \rar[shift left=10,"i\cdot LRLR" description]
      \rar[shift left=5,"LR\cdot i\cdot LR" description]
      \rar["LRLR\cdot i" description]
      \lar[shift left=5,"L\cdot e\cdot R" description]
    & LRLRLR
      \lar[shift left=5,"L\cdot e\cdot RLR" description]
      \lar[shift left=10,"LRL\cdot e\cdot R" description]
      \rar
    &\ldots
  \end{tikzcd}
$$
are the basic "face" and "degeneracy" maps between simplices, which
you'll find in any book on algebraic topology. The $n$-simplex is a face
of the $(n+1)$-simplex in n+1 ways, and there are n basic degenerate ways
to map the $(n+1)$-simplex down to the $n$-simplex. These aren't *all* the
morphisms; just enough to generate all the rest by composition --- i.e.,
sticking together pictures vertically, but *not* horizontally.

Perhaps I should explain the notation here a bit more. Readers of
["Week 80"](#week80) will know that I use a dot to denote
horizontal composition of $2$-morphisms. For example, when we have a
couple of $2$-morphisms like this:
$$
  \begin{tikzpicture}
    \node (x) at (0,0) {$x$};
    \node (y) at (2,0) {$y$};
    \node (z) at (4,0) {$z$};
    \draw[->] (x) .. node[label={[label distance=-1mm]above:{\scriptsize$f$}}]{} controls (0.7,0.7) and (1.3,0.7) .. (y);
    \draw[->] (x) .. node[label={[label distance=-1mm]below:{\scriptsize$g$}}]{}controls (0.7,-0.7) and (1.3,-0.7) .. (y);
    \draw[double,double equal sign distance,-implies] (1,0.5) to node[label={[label distance=-1mm]right:{\scriptsize$S$}}]{} (1,-0.5);
    \draw[->] (y) .. node[label={[label distance=-1mm]above:{\scriptsize$f'$}}]{} controls (2.7,0.7) and (3.3,0.7) .. (z);
    \draw[->] (y) .. node[label={[label distance=-1mm]below:{\scriptsize$g'$}}]{}controls (2.7,-0.7) and (3.3,-0.7) .. (z);
    \draw[double,double equal sign distance,-implies] (3,0.5) to node[label={[label distance=-1mm]right:{\scriptsize$T$}}]{} (3,-0.5);
  \end{tikzpicture}
$$
we get a $2$-morphism like this:
$$
  \begin{tikzpicture}[xscale=1.5]
    \node (x) at (0,0) {$x$};
    \node (z) at (2,0) {$z$};
    \draw[->] (x) .. node[label={[label distance=-1mm]above:{\scriptsize$ff'$}}]{} controls (0.7,0.7) and (1.3,0.7) .. (z);
    \draw[->] (x) .. node[label={[label distance=-1mm]below:{\scriptsize$gg'$}}]{}controls (0.7,-0.7) and (1.3,-0.7) .. (z);
    \draw[double,double equal sign distance,-implies] (1,0.5) to node[label={[label distance=-1mm]right:{\scriptsize$S\cdot T$}}]{} (1,-0.5);
  \end{tikzpicture}
$$
But sometimes we can also horizontally compose a morphism and a
2-morphism! We can do it whenever our morphism $f$ looks like a little
"whisker" $f$ sticking out of the $2$-morphism $T$:
$$
  \begin{tikzpicture}
    \node (x) at (0,0) {$x$};
    \node (y) at (2,0) {$y$};
    \node (z) at (4,0) {$z$};
    \draw[->] (x) to node[label={[label distance=-1mm]above:{\scriptsize$f$}}]{} (y);
    \draw[->] (y) .. node[label={[label distance=-1mm]above:{\scriptsize$f'$}}]{} controls (2.7,0.7) and (3.3,0.7) .. (z);
    \draw[->] (y) .. node[label={[label distance=-1mm]below:{\scriptsize$g'$}}]{}controls (2.7,-0.7) and (3.3,-0.7) .. (z);
    \draw[double,double equal sign distance,-implies] (3,0.5) to node[label={[label distance=-1mm]right:{\scriptsize$T$}}]{} (3,-0.5);
  \end{tikzpicture}
$$
and what we get is a $2$-morphism $f\cdot S$ like this:
$$
  \begin{tikzpicture}[xscale=1.5]
    \node (x) at (0,0) {$x$};
    \node (z) at (2,0) {$z$};
    \draw[->] (x) .. node[label={[label distance=-1mm]above:{\scriptsize$ff'$}}]{} controls (0.7,0.7) and (1.3,0.7) .. (z);
    \draw[->] (x) .. node[label={[label distance=-1mm]below:{\scriptsize$fg'$}}]{}controls (0.7,-0.7) and (1.3,-0.7) .. (z);
    \draw[double,double equal sign distance,-implies] (1,0.5) to node[label={[label distance=-1mm]right:{\scriptsize$f\cdot T$}}]{} (1,-0.5);
  \end{tikzpicture}
$$
This process, called "whiskering", is not really a new operation. $f\cdot S$
is really just the horizontal composite of these $2$-morphisms:
$$
  \begin{tikzpicture}
    \node (x) at (0,0) {$x$};
    \node (y) at (2,0) {$y$};
    \node (z) at (4,0) {$z$};
    \draw[->] (x) .. node[label={[label distance=-1mm]above:{\scriptsize$f$}}]{} controls (0.7,0.7) and (1.3,0.7) .. (y);
    \draw[->] (x) .. node[label={[label distance=-1mm]below:{\scriptsize$f$}}]{}controls (0.7,-0.7) and (1.3,-0.7) .. (y);
    \draw[double,double equal sign distance,-implies] (1,0.5) to node[label={[label distance=-1mm]right:{\scriptsize$1_f$}}]{} (1,-0.5);
    \draw[->] (y) .. node[label={[label distance=-1mm]above:{\scriptsize$f'$}}]{} controls (2.7,0.7) and (3.3,0.7) .. (z);
    \draw[->] (y) .. node[label={[label distance=-1mm]below:{\scriptsize$g'$}}]{}controls (2.7,-0.7) and (3.3,-0.7) .. (z);
    \draw[double,double equal sign distance,-implies] (3,0.5) to node[label={[label distance=-1mm]right:{\scriptsize$T$}}]{} (3,-0.5);
  \end{tikzpicture}
$$
Similarly we can define $T\cdot f$ in this sort of situation:
$$
  \begin{tikzpicture}
    \node (x) at (0,0) {$x$};
    \node (y) at (2,0) {$y$};
    \node (z) at (4,0) {$z$};
    \draw[->] (x) .. node[label={[label distance=-1mm]above:{\scriptsize$f'$}}]{} controls (0.7,0.7) and (1.3,0.7) .. (y);
    \draw[->] (x) .. node[label={[label distance=-1mm]below:{\scriptsize$g'$}}]{}controls (0.7,-0.7) and (1.3,-0.7) .. (y);
    \draw[double,double equal sign distance,-implies] (1,0.5) to node[label={[label distance=-1mm]right:{\scriptsize$T$}}]{} (1,-0.5);
    \draw[->] (y) to node[label={[label distance=-1mm]above:{\scriptsize$f$}}]{} (z);
  \end{tikzpicture}
$$
Anyway, once you're an expert on this $2$-categorical yoga, you can
easily see that these morphisms in $\operatorname{Hom}(a,a)$, which are really
2-morphisms in $\mathsf{Ad}$:
$$
  \begin{tikzcd}[column sep=huge]
    1_a
      \rar["i" description]
    & LR
      \rar[shift left=5,"i\cdot LR" description]
      \rar["LR\cdot i" description]
    & LRLR
      \rar[shift left=10,"i\cdot LRLR" description]
      \rar[shift left=5,"LR\cdot i\cdot LR" description]
      \rar["LRLR\cdot i" description]
      \lar[shift left=5,"L\cdot e\cdot R" description]
    & LRLRLR
      \lar[shift left=5,"L\cdot e\cdot RLR" description]
      \lar[shift left=10,"LRL\cdot e\cdot R" description]
      \rar
    &\ldots
  \end{tikzcd}
$$
are obtained by taking our basic tiger stripe operations --- the "merging
of two black stripes", or $L\cdot e\cdot R$, and the "appearance of a black
stripe", or $i$ --- and drawing some extra black stripes on both sides.
That's what those $LR$'s are for. After all, no tiger is complete
without whiskers!

Okay. Now, having understood $\operatorname{Hom}(a,a)$ in all these ways, let's turn to
$\operatorname{Hom}(b,b)$. Luckily, this is very similar! Here the objects are
$$1_b,\quad RL,\quad RLRL,\quad RLRLRL,\quad \ldots$$
and morphisms are pictures of *orange* stripes on a *black* background:
$$
  \begin{tikzpicture}[xscale=0.8,yscale=1.2]
    \begin{knot}
      \strand[thick] (-1,2)
        to [out=down,in=down,looseness=2] (0,2);
      \strand[thick] (1,2)
        to [out=down,in=down,looseness=2] (2,2);
      \strand[thick] (1.5,0)
        to [out=up,in=down](3,2);
      \strand[thick] (2.5,0)
        to [out=up,in=up,looseness=2] (3.5,0);
      \strand[thick] (5.1,0)
        to [out=up,in=down,looseness=1] (4.5,1)
        to [out=up,in=up,looseness=1.5] (5.17,1)
        to [out=down,in=down,looseness=1.5] (5.83,1)
        to (5.83,2);
    \end{knot}
    \node at (-0.5,1.75) {$a$};
    \node at (1.5,1.75) {$a$};
    \node at (0.5,1) {$b$};
    \node at (3.5,1) {$a$};
    \node at (3,0.25) {$b$};
    \node at (5.75,0.25) {$b$};
  \end{tikzpicture}
$$
These orange stripes can only split:
$$
  \begin{tikzpicture}[yscale=-2]
    \begin{knot}
      \strand[thick] (-0.5,0)
        to [out=up,in=down,looseness=1.5] (-1,1);
      \strand[thick] (-0.45,1)
        to [out=down,in=down,looseness=1.5] (0.45,1);
      \strand[thick] (0.5,0)
        to [out=up,in=down,looseness=1.5] (1,1);
    \end{knot}
    %
    \node at (-1.25,0.35) {$b$};
    \node at (0,0.35) {$a$};
    \node at (0,0.9) {$b$};
    \node at (1.25,0.35) {$b$};
    %
    \node[fill=white] at (-0.55,0.25) {\scriptsize$R$};
    \node[fill=white] at (-0.95,0.75) {\scriptsize$R$};
    \node[fill=white] at (-0.4,0.85) {\scriptsize$L$};
    \node[fill=white] at (0.4,0.85) {\scriptsize$R$};
    \node[fill=white] at (0.55,0.25) {\scriptsize$L$};
    \node[fill=white] at (0.95,0.75) {\scriptsize$L$};
    \node[label={[label distance=-2mm]above:{\scriptsize$i$}}] at (0,0.61) {$\bullet$};
  \end{tikzpicture}
$$
or disappear:
$$
  \begin{tikzpicture}[xscale=1.3,yscale=-1]
    \begin{knot}
      \strand[thick] (-0.5,0)
        to [out=up,in=up,looseness=3] (0.5,0);
    \end{knot}
    \node at (-1,0.5) {$b$};
    \node at (0,0.25) {$a$};
    \node at (1,0.5) {$b$};
    \node[fill=white] at (-0.45,0.35) {\scriptsize$R$};
    \node[fill=white] at (0.45,0.35) {\scriptsize$L$};
    \node[label={[label distance=-2mm]below:{\scriptsize$e$}}] at (0,0.88) {$\bullet$};
  \end{tikzpicture}
$$
as we march down the page. This means is that $\operatorname{Hom}(b,b)$ is $\Delta^{\mathrm{op}}$: the
*opposite* of the category of simplices, the *opposite* of the category
of finite ordinals, or the walking *comonoid* --- which is just like a
monoid, only upside down!

Here is another picture of $\operatorname{Hom}(b,b)$:
$$
  \begin{tikzcd}[column sep=huge]
    1_b
    & RL
      \lar["e" description]
      \rar[shift left=5,"R\cdot i\cdot L" description]
    & RLRL
      \rar[shift left=10,"R\cdot i LRL" description]
      \rar[shift left=5,"RLR\cdot i\cdot L" description]
      \lar[shift left=5,"RL\cdot e" description]
      \lar["e\cdot RL" description]
    & RLRLRL
      \lar["e\cdot RLRL" description]
      \lar[shift left=5,"RL\cdot e\cdot RL" description]
      \lar[shift left=10,"RLRL\cdot e" description]
    &\ldots
      \lar
  \end{tikzcd}
$$
If you're a devoted reader of This Week's Finds, you'll know I
secretly drew this category already in section N of
["Week 118"](#week118). There I was talking about specific adjoint
functors instead of the walking adjunction, so as not to prematurely
blow your mind. I was also writing horizontal composites backwards, for
certain old-fashioned reasons. But the idea is exactly the same! The
morphisms above give the usual "face and degeneracy maps" we always
have in a simplicial set, since a simplicial set is a functor
$$F\colon \Delta^{\mathrm{op}} \to \mathsf{Set}.$$
By the way, you may have noticed that to get from $\operatorname{Hom}(a,a)$ to $\operatorname{Hom}(b,b)$,
we had to switch the colors orange and black AND read the pictures
upside-down. The reason is that if we turn around all the $1$-morphisms
AND $2$-morphisms in the walking adjunction, we get the walking adjunction
again. Ponder that!

We can summarize what we've learned so far using the "Platonic idea"
jargon I introduced last week:

> The Platonic idea of a monoid and the Platonic idea of a comonoid are
the hom-categories $\operatorname{Hom}(a,a)$ and $\operatorname{Hom}(b,b)$ sitting inside the Platonic
idea of an adjunction!

(By the way, to round this off we should really describe $\operatorname{Hom}(a,b)$ and
$\operatorname{Hom}(b,a)$, too. I think $\operatorname{Hom}(a,b)$ is the Platonic idea of "an object with
a left action of a monoid and a right coaction of a comonoid, in a
compatible way". If so, $\operatorname{Hom}(b,a)$ would be the Platonic idea of "an
object with a right action of a monoid and a left coaction of a
comonoid, in a compatible way". By "compatible" I'm saying that we
can act on one side and coact on the other side in either order, and get
the same thing. Filling in the details requires concepts I'm not eager
to discuss right now, so I leave this as an exercise for the highly
energetic reader. The less energetic reader can just study the
tiger-stripe descriptions of these categories.)

Finally, here's Mueger's new twist on all these ideas! Better than an
adjunction is an "ambidextrous" adjunction. This has some extra
structure, which turns out to explain all sorts of fancy-sounding stuff
people look at in the study of subfactors and TQFTs and the like....

But what's an "ambidextrous adjunction"?

A ambidextrous adjunction is where you have a morphism
$$L\colon a \to b$$
in a $2$-category that is both left and right adjoint to
$$R\colon b \to a.$$
More precisely, it is a setup
$$(a,b,L,R,i,e,j,f)$$
where
$$(a,b,L,R,i,e)$$
and
$$(b,a,R,L,j,f)$$
are both adjunctions.

In terms of string diagrams, our generating $2$-morphisms look like this:
$$
  \begin{gathered}
    \begin{tikzpicture}[scale=1.3]
      \begin{knot}
        \strand[thick] (-0.5,0)
          to [out=up,in=up,looseness=3] (0.5,0);
      \end{knot}
      \node at (-1,0.5) {$a$};
      \node at (0,0.25) {$b$};
      \node at (1,0.5) {$a$};
      \node[fill=white] at (-0.45,0.35) {\scriptsize$L$};
      \node[fill=white] at (0.45,0.35) {\scriptsize$R$};
      \node[label={[label distance=-2mm]above:{\scriptsize$i$}}] at (0,0.88) {$\bullet$};
    \end{tikzpicture}
    \qquad
    \begin{tikzpicture}[scale=1.3]
      \begin{knot}
        \strand[thick] (-0.5,0)
          to [out=up,in=up,looseness=3] (0.5,0);
      \end{knot}
      \node at (-1,0.5) {$b$};
      \node at (0,0.25) {$a$};
      \node at (1,0.5) {$b$};
      \node[fill=white] at (-0.45,0.35) {\scriptsize$R$};
      \node[fill=white] at (0.45,0.35) {\scriptsize$L$};
      \node[label={[label distance=-2mm]above:{\scriptsize$j$}}] at (0,0.88) {$\bullet$};
    \end{tikzpicture}
  \\[2em]
    \begin{tikzpicture}[xscale=1.3,yscale=-1]
      \begin{knot}
        \strand[thick] (-0.5,0)
          to [out=up,in=up,looseness=3] (0.5,0);
      \end{knot}
      \node at (-1,0.5) {$b$};
      \node at (0,0.25) {$a$};
      \node at (1,0.5) {$b$};
      \node[fill=white] at (-0.45,0.35) {\scriptsize$R$};
      \node[fill=white] at (0.45,0.35) {\scriptsize$L$};
      \node[label={[label distance=-2mm]below:{\scriptsize$e$}}] at (0,0.88) {$\bullet$};
    \end{tikzpicture}
    \qquad
    \begin{tikzpicture}[xscale=1.3,yscale=-1]
      \begin{knot}
        \strand[thick] (-0.5,0)
          to [out=up,in=up,looseness=3] (0.5,0);
      \end{knot}
      \node at (-1,0.5) {$a$};
      \node at (0,0.25) {$b$};
      \node at (1,0.5) {$a$};
      \node[fill=white] at (-0.45,0.35) {\scriptsize$L$};
      \node[fill=white] at (0.45,0.35) {\scriptsize$R$};
      \node[label={[label distance=-2mm]below:{\scriptsize$f$}}] at (0,0.88) {$\bullet$};
    \end{tikzpicture}
  \end{gathered}
$$
and the triangle equations say all possible zig-zags can be straightened
out.

Now let's study the "walking ambidextrous adjunction", $\mathsf{AmbAd}$. As
before, $2$-morphisms in $\mathsf{AmbAd}$ can be described using pictures with orange
and black stripes --- but now *both* kinds of stripes can appear,
disappear, merge or split as we march down the page:
$$
  \begin{tikzpicture}[xscale=0.8,yscale=1.2]
    \begin{knot}
      \strand[thick] (-2,-2)
        to [out=up,in=down,looseness=0.7] (-1.5,0)
        to [out=up,in=down,looseness=0.7] (-2,2);
      \strand[thick] (-1,2)
        to [out=down,in=down,looseness=2] (0,2);
      \strand[thick] (1,2)
        to [out=down,in=down,looseness=2] (2,2);
      \strand[thick] (-1,-2)
        to [out=up,in=down] (-0.5,0.5)
        to [out=up,in=up,looseness=1.1] (0.5,0.5)
        to [out=down,in=up] (0.6,-2);
      \strand[thick] (0,0)
        to [out=right,in=right,looseness=0.8] (0,-1.5)
        to [out=left,in=right] (-0.25,-1.5)
        to [out=left,in=left,looseness=0.8] (0,0);
      \strand[thick] (1.5,-2)
        to (1.5,0)
        to [out=up,in=down](3,2);
      \strand[thick] (2.5,-2)
        to (2.5,0)
        to [out=up,in=up,looseness=1.1] (5,0)
        to [out=down,in=down,looseness=1.2] (6,0)
        to (6,2);
      \strand[thick] (3.3,-2)
        to (3.3,-1)
        to [out=up,in=up,looseness=1.5] (4.3,-1)
        to [out=down,in=up,looseness=1.3] (6,-2);
    \end{knot}
    %
    \node at (-1.95,0) {$a$};
    \node at (-0.5,1.75) {$a$};
    \node at (1.5,1.75) {$a$};
    \node at (-0.1,-0.8) {$b$};
    \node at (0,0.3) {$a$};
    \node at (0.5,1.2) {$b$};
    \node at (4.5,1.4) {$a$};
    \node at (3.8,0.1) {$b$};
    \node at (3.8,-1.25) {$a$};
    %
    \draw[thick] (-2.5,-2) rectangle ++(9.5,4);
  \end{tikzpicture}
$$
This allows for quite arbitrary ways of cutting up a rectangle into
regions of orange and black, with piecewise linear boundaries, subject
to the condition that each vertical border has the same color all along
it. The triangle equations and the rules for $2$-categories say that we
can warp such a picture around without changing the $2$-morphism that it
defines... I don't want to be too precise here, since it would be
boring. Hopefully you get the idea: $\mathsf{AmbAd}$ has a purely topological
description!

Now for the punchline: in $\mathsf{AmbAd}$, what is the category $\operatorname{Hom}(a,a)$ like? As
in $\mathsf{Ad}$, the objects are
$$1_a,\quad LR,\quad LRLR,\quad LRLRLR,\quad \ldots$$
but now the object $LR$ is equipped not only with multiplication:
$$
  \begin{tikzpicture}[yscale=2]
    \begin{knot}
      \strand[thick] (-0.5,0)
        to [out=up,in=down,looseness=1.5] (-1,1);
      \strand[thick] (-0.45,1)
        to [out=down,in=down,looseness=1.5] (0.45,1);
      \strand[thick] (0.5,0)
        to [out=up,in=down,looseness=1.5] (1,1);
    \end{knot}
    %
    \node at (-1.25,0.35) {$a$};
    \node at (0,0.35) {$b$};
    \node at (0,0.9) {$a$};
    \node at (1.25,0.35) {$a$};
    %
    \node[fill=white] at (-0.55,0.25) {\scriptsize$L$};
    \node[fill=white] at (-0.95,0.75) {\scriptsize$L$};
    \node[fill=white] at (-0.4,0.85) {\scriptsize$R$};
    \node[fill=white] at (0.4,0.85) {\scriptsize$L$};
    \node[fill=white] at (0.55,0.25) {\scriptsize$R$};
    \node[fill=white] at (0.95,0.75) {\scriptsize$R$};
    \node[label={[label distance=-2mm]below:{\scriptsize$e$}}] at (0,0.61) {$\bullet$};
    %
    \node at (0,-0.3) {multiplication:};
    \node at (0,-0.5) {$L\cdot e\cdot R\colon LRLR\Rightarrow LR$};
  \end{tikzpicture}
$$
and multiplicative identity:
$$
  \begin{tikzpicture}[scale=1.3]
    \begin{knot}
      \strand[thick] (-0.5,0)
        to [out=up,in=up,looseness=3] (0.5,0);
    \end{knot}
    \node at (-1,0.5) {$a$};
    \node at (0,0.25) {$b$};
    \node at (1,0.5) {$a$};
    \node[fill=white] at (-0.45,0.35) {\scriptsize$L$};
    \node[fill=white] at (0.45,0.35) {\scriptsize$R$};
    \node[label={[label distance=-2mm]above:{\scriptsize$i$}}] at (0,0.88) {$\bullet$};
    %
    \node at (0,-0.4) {multiplicative identity:};
    \node at (0,-0.75) {$i\colon 1_a\Rightarrow LR$};
  \end{tikzpicture}
$$
but also a "comultiplication":
$$
  \begin{tikzpicture}[yscale=2]
    \begin{scope}[yscale=-1,shift={(0,-1)}]
      \begin{knot}
        \strand[thick] (-0.5,0)
          to [out=up,in=down,looseness=1.5] (-1,1);
        \strand[thick] (-0.45,1)
          to [out=down,in=down,looseness=1.5] (0.45,1);
        \strand[thick] (0.5,0)
          to [out=up,in=down,looseness=1.5] (1,1);
      \end{knot}
      %
      \node at (-1.25,0.35) {$a$};
      \node at (0,0.35) {$b$};
      \node at (0,0.9) {$a$};
      \node at (1.25,0.35) {$a$};
      %
      \node[fill=white] at (-0.55,0.25) {\scriptsize$L$};
      \node[fill=white] at (-0.95,0.75) {\scriptsize$L$};
      \node[fill=white] at (-0.4,0.85) {\scriptsize$R$};
      \node[fill=white] at (0.4,0.85) {\scriptsize$L$};
      \node[fill=white] at (0.55,0.25) {\scriptsize$R$};
      \node[fill=white] at (0.95,0.75) {\scriptsize$R$};
      \node[label={[label distance=-2mm]above:{\scriptsize$j$}}] at (0,0.61) {$\bullet$};
    \end{scope}
    %
    \node at (0,-0.3) {comultiplication:};
    \node at (0,-0.5) {$L\cdot j\cdot R\colon LR\Rightarrow LRLR$};
  \end{tikzpicture}
$$
and "comultiplicative coidentity":
$$
  \begin{tikzpicture}[scale=1.3]
    \begin{scope}[yscale=-1,shift={(0,-1)}]
      \begin{knot}
        \strand[thick] (-0.5,0)
          to [out=up,in=up,looseness=3] (0.5,0);
      \end{knot}
      \node at (-1,0.5) {$a$};
      \node at (0,0.25) {$b$};
      \node at (1,0.5) {$a$};
      \node[fill=white] at (-0.45,0.35) {\scriptsize$L$};
      \node[fill=white] at (0.45,0.35) {\scriptsize$R$};
      \node[label={[label distance=-2mm]below:{\scriptsize$f$}}] at (0,0.88) {$\bullet$};
    \end{scope}
    %
    \node at (0,-0.5) {comultiplicative identity:};
    \node at (0,-0.85) {$f\colon LR\Rightarrow 1_a$};
  \end{tikzpicture}
$$
which make it into a monoid object *and* a comonoid object. Even better,
there are some extra relations between the multiplication and
comultiplication, which make $LR$ into a so-called "Frobenius object"!

In short, $\operatorname{Hom}(a,a)$ is the walking Frobenius object! So is $\operatorname{Hom}(b,b)$,
since there is no real asymmetry between the objects $a$ and $b$ in an
ambidextrous adjunction, as there was with an adjunction. I haven't
thought much about $\operatorname{Hom}(a,b)$ and $\operatorname{Hom}(b,a)$ yet, but one obvious thing is
that they're isomorphic.

Next time I'll talk about examples of Frobenius objects and why they
are so important in subfactors, TQFTs and the like. This is what Mueger
is really interested in. Right now, I want to wrap up by saying exactly
what it means to say $LR$ is a "Frobenius object". What are the extra
relations between multiplication and comultiplication?

There are various ways of describing these relations. Mueger uses a pair
of equations that are popular in the TQFT literature:
$$
  \begin{tikzpicture}[yscale=2]
    \begin{knot}
      \strand[thick] (-1,-1.5)
        to [out=up,in=down,looseness=1.5] (-0.4,-0.5)
        to (-0.4,0)
        to [out=up,in=down,looseness=1.5] (-1,1);
      \strand[thick] (-0.45,1)
        to [out=down,in=down,looseness=1.5] (0.45,1);
      \strand[thick] (-0.45,-1.5)
        to [out=up,in=up,looseness=1.5] (0.45,-1.5);
      \strand[thick] (1,-1.5)
        to [out=up,in=down,looseness=1.5] (0.4,-0.5)
        to (0.4,0)
        to [out=up,in=down,looseness=1.5] (1,1);
    \end{knot}
    %
    \node at (-1.25,-0.25) {$a$};
    \node at (0,-0.25) {$b$};
    \node at (0,0.9) {$a$};
    \node at (1.25,-0.25) {$a$};
    \node at (0,-1.4) {$a$};
  \end{tikzpicture}
  \qquad
  \raisebox{7em}{$=$}
  \qquad
  \begin{tikzpicture}[yscale=2]
    \begin{knot}
      \strand[thick] (-1,-1.5) to (-1,1);
      \strand[thick] (-0.5,-1.5)
        to (-0.5,-0.5)
        to [out=up,in=left] (-0.25,-0.35)
        to [out=right,in=up,looseness=0.6] (0.5,-1.25)
        to (0.5,-1.5);
      \strand[thick] (0.5,1)
        to (0.5,0)
        to [out=down,in=right] (0.25,-0.15)
        to [out=left,in=down,looseness=0.6] (-0.5,0.75)
        to (-0.5,1);
      \strand[thick] (1,-1.5) to (1,1);
    \end{knot}
    \node at (-1.5,-0.25) {$a$};
    \node at (0,0.75) {$a$};
    \node at (0,-0.25) {$b$};
    \node at (0,-1) {$a$};
    \node at (1.5,-0.25) {$a$};
  \end{tikzpicture}
$$
and its mirror image. People sometimes call these the "$I = N$"
equations, for the obvious reason. So: one definition of a "Frobenius
object" in a monoidal category is that it's a monoid object / comonoid
object satisfying the $I = N$ equations.

Where can you read about this? Well, besides Mueger's paper, there are
these:

4) Frank Quinn, "Lectures on axiomatic quantum field theory", in _Geometry and Quantum Field Theory_, Amer. Math. Soc., Providence, RI, 1995.

5) Lowell Abrams, "Two-dimensional topological quantum field theories and Frobenius algebras", _J. Knot Theory and its Ramifications_ **5** (1996), 569--587.

A "Frobenius algebra" is just a Frobenius object in the category of
vector spaces. I seem to recall that this is equivalent to what Quinn
calls an "ambialgebra". For any TQFT in any dimension, the vector
space associated to the sphere is a commutative Frobenius algebra. The
proof consists of playing with pictures very much like the ones above,
but in higher dimensions.

The $I = N$ equations are cute, but personally I prefer a more conceptual
description of a Frobenius object. This may be a bit mindblowing to the
uninitiated, so if you're just barely hanging on, please stop now.

Hmm! If you're still reading this, you must be brave! Okay --- don't say
I didn't warn you. Let's start by pondering $LR$ a bit more. This guy is
its own adjoint, with the unit and counit as follows:
$$
  \begin{tikzpicture}[yscale=2]
    \begin{scope}[yscale=-1,shift={(0,-0.75)}]
      \begin{knot}
        \strand[thick] (-0.5,0)
          to [out=up,in=down,looseness=1.5] (-1,1);
        \strand[thick] (-0.45,1)
          to [out=down,in=down,looseness=1.5] (0.45,1);
        \strand[thick] (0.5,0)
          to [out=up,in=down,looseness=1.5] (1,1);
        \strand[thick] (-0.5,0)
          to [out=down,in=down] (0.5,0);
      \end{knot}
      %
      \node at (-1.25,0.35) {$a$};
      \node at (0,0.35) {$b$};
      \node at (0,0.9) {$a$};
      \node at (1.25,0.35) {$a$};
    \end{scope}
    %
    \node at (0,-0.5) {unit for $LR$:};
    \node at (0,-0.75) {multiplicative identity};
    \node at (0,-0.95) {composed with comultiplication};
  \end{tikzpicture}
  \qquad
  \begin{tikzpicture}[yscale=2]
    \begin{knot}
      \strand[thick] (-0.5,0)
        to [out=up,in=down,looseness=1.5] (-1,1);
      \strand[thick] (-0.45,1)
        to [out=down,in=down,looseness=1.5] (0.45,1);
      \strand[thick] (0.5,0)
        to [out=up,in=down,looseness=1.5] (1,1);
      \strand[thick] (-0.5,0)
        to [out=down,in=down] (0.5,0);
    \end{knot}
    %
    \node at (-1.25,0.35) {$a$};
    \node at (0,0.35) {$b$};
    \node at (0,0.9) {$a$};
    \node at (1.25,0.35) {$a$};
    %
    \node at (0,-0.5) {counit for $LR$:};
    \node at (0,-0.75) {multiplication composed with};
    \node at (0,-0.95) {comultiplicative coidentity};
  \end{tikzpicture}
$$
It's easy to check the triangle equations by straightening out the
relevant zig-zags.

Now, whenever a monoid object has a right or left adjoint, that right or
left adjoint automatically becomes a comonoid object, by the magic of
duality. But if a monoid object is its *own* adjoint, it becomes a
comonoid object in *two* ways, because it is both its own left *and*
right adjoint! So, our guy $LR$ is a comonoid object in *three* ways! Huh?
Well, we already knew $LR$ was a comonoid object before this devilish
paragraph began, but since $LR$ is its own adjoint, it becomes a comonoid
object in two other ways. Amazingly, the $I = N$ equations are equivalent
to the fact that all three comonoid structures agree! I leave this as an
exercise for the insanely energetic reader... I've worked it out
before, and I rechecked it this morning in bed. I don't know if a proof
exists in the literature, but from what Mueger writes, I suspect maybe
you can catch glimpses of it in Appendix A3 of this book:

6) L. Kadison, _New Examples of Frobenius Extensions_, University Lecture Series \#**14**, Amer. Math. Soc., Providence RI, 1999.

Anyway, the upshot is that we can equivalently define a Frobenius object
in a monoidal category as follows: it's a monoid object / comonoid
object which becomes its own adjoint by letting

- unit = multiplicative identity composed with comultiplication
- counit = multiplication composed with comultiplicative coidentity

and has the property that the resulting 3 comonoid structures agree.

Or, equivalently, that the resulting 3 monoid structures agree!

There is much more to say about this, but let's stop here.

------------------------------------------------------------------------

Postscript --- Oswald Wyler had this correction to make:

> The walking adjunction is much older than the 1986 paper by Schanuel
> and Street. Back in 1970, Pumplün published a paper:
> "Eine Bemerkung über Monaden und adjungierte Funktoren", _Math. Annalen_ **185** (1970),
> 329--377. The small bicategory "walking adjunction" definitely was in
> that paper, but I don't recall whether it was explicitly formulated
> or not.

Andree Ehresmann added:

> On the "walking adjunction"
>
> I don't know the Pumplun's paper cited by Wyler. But there is
> another reference at about the same time; indeed, the "walking
> adjunction" has been explicitly constructed and studied in the paper
> of Auderset:
>
> "Adjonction et monade au niveau des $2$-categories"
>
> published in _Cahiers de Top. et Geom. Diff._ **XV-1** (1974), 3--20.
>
> More formally it could also be called "the 2-sketch of an
> adjunction" in the terminology in my paper with Charles Ehresmann:
>
> "Categories of sketched structures", in the _Cahiers_ **XIII-2** (1972),
>
> reprinted in "Charles Ehresmann: Oeuvres completes et commentees"
> Part **IV-2**.

Bill Lawvere added:

> **ONE MORE HISTORICAL CITATION**
>
> The Pumplun paper cited by Wyler as well as the Auderset paper cited
> by Mme Ehresmann illustrate that the study of generic structures in
> $2$-categories has been going on for some time. My own paper ORDINAL
> SUMS AND EQUATIONAL DOCTRINES, SLNM 80 (1969) 141--155 shows that the
> augmented simplicial category $\Delta$ serves as the generic monad, but
> moreover goes on to actually apply this to show that the Kleisli
> construction is a tensor product left-adjoint to the Eilenberg- Moore
> construction which is an enriched Hom. The Hom/tensor formalism
> appropriate to the case of strict monoid objects is all that is
> required here, as I will explain below.
>
> **AN EXTENSION AND A RESTRICTION**
>
> The important special case of FROBENIUS monads is explicitly
> characterized in three ways in my paper. Concerning the IDEMPOTENT
> case discussed a few days ago by Grandis and Johnstone, note that the
> publication of Schanuel and Street proves among other things that the
> monoid $\Delta$ in $\mathsf{Cat}$ has very few quotients (see below for significance of
> the monoid structure).
>
> **THE GENERAL HOM/TENSOR FORMALISM AND A VERY PARTICULAR MONOID**
>
> In any cartesian-closed category with finite limits and co-limits, a
> non-linear version of the Cartan-Eilenberg Hom/tensor formalism
> applies to actions and biactions of monoid objects. In Cat, $\Delta$ is a
> (strict) monoid and its actions are precisely monads on arbitrary
> categories. A crucial part of the formalism is that categories of
> actions are automatically enriched in the basic cartesian-closed
> category, which in this case is Cat. There is a particular biaction of
> $\Delta$, which I called $\Delta$ plus, with the property that the enriched Hom of
> it into an arbitrary $\Delta$-action is exactly the Eilenberg-Moore category
> of "algebras", automatically equipped with its structure as a $\Delta^{\mathrm{op}}$
> action (co-monad). The left-adjoint tensor assigns to any category
> equipped with a co-monad its Kleisli category, as a category with
> monad. Not only are the calculations in this particular case quite
> explicit, but the enriched Hom tensor formalism has a lot of content
> which is still under-exploited.
>
> **SKETCHES VERSUS PLATONISM**
>
> The often repeated slander that mathematicians think "as if" they
> were "platonists" needs to be combatted rather than swallowed. What
> mathematicians and other scientists use is the objectively developed
> human instrument of general concepts. (The plan to misleadingly use
> that fact as a support for philosophical idealism may have been an
> honest mistake by Plato, or it may have been part of his job as
> disinformation officer for the Athenian CIA organization; it probably
> would not have survived until now had it not been for the special
> efforts of Cosimo de' Medici.) It seems that a general concept has
> two related aspects, as I began to realize more explicitly in
> connection with my paper "Adjointness in foundations", _Dialectica_ vol.
> **23** (1969), pp. 281--296; I later learned that some philosophers refer to
> these two aspects as "abstract general vs. concrete general". For
> example, there is the algebraic theory of rings vs. the category of
> all rings, or a particular abstract group vs. the category of all
> permutation representations of the group. While it is "obvious"
> that, at least in mathematics, a concrete general should have the
> structure of a category, because all the instances embody the same
> abstract general and hence any two instances can be compared in
> preferred ways, by contrast it was not until the late fifties that one
> realized that an abstract general can also be construed as a category
> in its own right. That realization essentially made explicit the fact
> that substitution is a logical operation and indeed is the most
> fundamental logical operation.
>
> Thus an abstract general is essentially a special algebraic structure
> indeed a category with additional structure such as finite limits or
> still richer doctrines. As with other algebraic structures there are
> again two aspects, the structures themselves and their presentations
> which are closely related, yet quite distinct; for example, more than
> one presentation may be needed for efficient calculations determining
> features of the same algebraic structure. What is meant by a
> presentation depends on the doctrine: for example $\Delta$ as a mere category
> has an infinite presentation used in topology, but as a strict
> monoidal category it has a finite presentation.
>
> The notion of SKETCH is the most efficient scheme yet devised for the
> general construction of PRESENTATIONS OF ABSTRACT GENERALS. The fact
> that particular abstract generals and the idea of sketches exist
> within the historically developed objective science does not mean that
> they somehow always existed; to call them "platonic" seems to
> detract from the honor of their actual discoverers.
>
> Bill Lawvere