categories: adjoint equivalence

[baez@math.ucr.edu]

Here's some stuff James Dolan and I talked about last Friday when
discussing Paul Levy's question about the difference between an
equivalence and an adjoint equivalence.  Pardon the glitzy writing
style: this is part of my column "This Week's Finds", where I can't
count on people staying awake for category theory unless I spice it 
up a bit.  You can find the whole column here:

http://math.ucr.edu/home/baez/week173.html 

Best,
jb

...........................................................................

First, consider the "Platonic idea of an equivalence".  By this, I mean
the 2-category Equiv which is freely generated by objects a and b,
morphisms L: a -> b and R: b -> a, and isomorphisms i: 1_b => RL and 
e: LR => 1_a.   Why do I call this the "Platonic idea of an equivalence"? 
Well, any equivalence in any 2-category C is just the same as a 2-functor 

F: Equiv -> C

The functor F turns the "abstract" equivalence in Equiv into a
"concrete" equivalence in C!  This is reminiscent of Plato's theory
of ideas and how they get manifested in concrete situations.  We can
think of Equiv as the unadorned idea of an adjunction without any
contamination by accidental extra features.

I should add that James, less of an intellectual snob than I, calls
Equiv the "walking equivalence".  After all, if someone has really big
bushy eyebrows, so that when you see him walking down the street you
first notice his eyebrows and only later realize there's a person
attached, you call him a "walking pair of eyebrows".  The person is 
basically just the life support system for the eyebrows!  Similarly, in
Equiv we have a 2-category which is just the life support system for an
adjunction: no more and no less.   

Anyway, the walking equivalence is a weak 2-groupoid: a 2-category where
every 2-morphism is invertible and every morphism is invertible up to
2-isomorphism.  Weak 2-groupoids are secretly the same thing as homotopy
2-types: roughly speaking, topological spaces whose homotopy groups 
vanish above dimension 2.  And there's a pretty easy way to turn a weak
2-groupoid into a homotopy 2-type.  First you turn it into a simplicial
set, called its "nerve", and then you take the geometric realization of
that.  

Eh?  Well, I talked about geometric realization in part E of "week116",
and I talked about the nerve of a 1-category in part J of "week117", so
the only thing I need to do is say a bit about the nerve of a 2-category.
This is a simplicial set where the 0-simplices correspond to objects:


                            x


the 1-simplices correspond to morphisms:


                    x ------F-------> y


the 2-simplices correspond to 2-morphisms:


                            y
                           / \                F: x -> y 
                          /   \               G: y -> z
                         / ||  \              H: x -> z
                        F  ||   G             a: FG => H
                       /   ||a   \
                      /    \/     \
                     x------H----->z


and the higher-dimensional simplices correspond to equations, "equations
between equations", and so on.  

Anyway, if you use this trick to turn the walking equivalence into
a space, what space do you get?  

The 2-sphere!  

It's pretty easy to see... I'd draw it for you on paper if I could, but
you'll have to do it yourself.  It helps if you have a globe:

a is the North Pole, 
b is the South Pole, 
L: a -> b is the Greenwich Meridian running from north to south, 
R: b -> a is the International Date Line running from south to north, 
i: 1_a => LR is the Eastern Hemisphere, and 
e: RL => 1_b is the Western Hemisphere!

(More precisely, we just get the 2-sphere up to homotopy equivalence:
there is a whole bunch of higher-dimensional flab which I'm ignoring
here.  But that's okay, since we're doing homotopy theory.)

We can also play this game for the "walking adjoint equivalence",
AdEquiv.  This is just like the walking equivalence, except we put in
extra relations: the triangle equations.  How does this affect the space
we get?

It's very beautiful: the extra equations fill in the 2-sphere to give us
a 3-ball!  

Now, the 3-ball is contractible, so as a homotopy type it's really the
same as a point.  And a point is exactly the space we'd get from playing
the same game starting with the "walking object": the 2-category with
one object, its identity morphism, and the identity 2-morphism of that.

To the eyes of a homotopy theorist, a point and 3-ball are the same, but
the 2-sphere is not.   Similarly, to the eyes of an n-category theorist,
the walking object and the walking adjoint equivalence are "the same",
but the walking equivalence is not!   

We could make this very precise with a suitable notion of "sameness" for
2-categories.  But instead, let's jump straight to the punchline: having
an adjoint equivalence in a 2-category is "the same" as having an
object.... but having an equivalence is not!

There's even more fun to be had here.  Since every adjoint equivalence
is an equivalence, there's a 2-functor

I: Equiv -> AdEquiv

But I also said every equivalence can be massaged to obtain an adjoint
equivalence!  In fact, I said it could be done in two equally good ways.
Either of these gives a 2-functor

P: AdEquiv -> Equiv

Now, we can ask what these become when we turn them into maps between
spaces....

It turns out that I is just the inclusion of the 2-sphere into the
3-ball, while P is the map that squashes the 3-ball down to either the
eastern or western hemisphere of the sphere!  

By the way, it is irresistible to predict generalizations to higher
dimensions.  For any n, we will have weak n-groupoids called Equiv, the
"walking n-equivalence", and AdEquiv, the "walking adjoint
n-equivalence".  The geometric realization of the nerve of Equiv will be
homotopy equivalent to the n-sphere, while that of AdEquiv will be
homotopy equivalent to the (n+1)-ball.

(Note that for n = 1, Equiv will be the category with objects
a and b and isomorphisms L: a -> b, R: b -> a.   In AdEquiv, there
will be extra relations saying that R is the inverse of L.  In
this sense, it is really an adjoint equivalence rather than an
equivalence which is the proper generalization of an isomorphism!)