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!)