Tuesday, August 4, 2009

Equivalence of Categories

We are going to start considering a hierarchy of "sameness". This is the strength of category theory really. At the top, the strongest notion of being the same is, I guess, the identity morphism. It says two things are identical. Next comes isomorphism. Then...what? First recommended reading:

And now down to business...

A Weaker Form of "Sameness"

We have that two categories $C,D$ are "isomorphic" if and only if there exists a functor $F:C\to{D}$ which is an isomorphism. We have the intuition that the categories are not identical but they are still "the same" in a sense.

Recall that our functor $F:C\to{D}$ is an isomorphism if and only if there exists a two-sided inverse functor $F^{-1}$. That is to say, $F^{-1}\circ{F}=1_{C}$ and $F\circ{F^{-1}}=1_{D}$. Note that these equations are "properties", so they're either true or false.

We want to "take it up a notch" and make them natural isomorphisms since $F^{-1}\circ{F}$, $F\circ{F^{-1}}$, $1_{C}$, and $1_{D}$ here are all functors. The only way to relate them all together is to use natural transformations, i.e. morphisms of functors!

Why is this "better"? (Well to interject, it's neither better nor worse...it's just different; kind of like apples and oranges, which is better? Well, they're different, we can't really compare them...) It's more appealing since we have more freedom with natural isomorphisms than strict equalities.

This programme of replacing boring equations with exciting isomorphisms is precisely the notion of "categorification". For a good introduction, see:

  • John Baez and James Dolan From Finite Sets to Feynman Diagrams arXiv:math/0004133 (2000)
In all seriousness, this procedure of categorification really is quite fascinating and - even if one knows nothing about Feynman diagrams - the paper is a good read.

Now, to reiterate the little explanation that was given, we will summarize it in our definition:

Definition 1. An "equivalence of categories" $C$ and $D$ consists of
  • a functor $F:C\to{D}$
such that
  • it has a "weak inverse", i.e. a functor $G:D\to{C}$ such that there exists some natural isomorphisms $\alpha:G\circ{F}\cong1_{C}$ and $\beta:1_{D}\cong F\circ{G}$.

The "orientation" of the natural isomorphisms may look odd at first, but when we get to adjoint functors it will become much clearer why this orientation is chosen.

The notion of equivalence is (hopefully) seen now as a "clearly" weaker form of "sameness". All it really says is that the composition of the functors $F\circ{G}$ is "the same as" (isomorphic to) the identity functor $1_{D}$, but how it is the same is given some freedom. We can specify some whacky relation $\beta_{d}$ for each $d\in{D}$ relating $(F\circ{G})(d)$ to $d$. It can be anything we want, even beyond our wildest dreams -- provided that $\beta_{d}$ is an isomorphism.

Similarly $G\circ{F}$ is isomorphic to $1_{C}$. It's the same as the identity, but this "sameness" is given some freedom in the sense that it "merely" has to be invertible. Of course the demand $G\circ{F}=1_{c}$ is an acceptable one, it's trivially isomorphic. But trivialities are boring.

That's precisely why we have equivalences, to avoid boredom ;)

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