Categories and Functors

In my last post, when describing the setup for the Bost-Connes system, I displayed my lack of understanding of groupoids. In the next few posts I want to clear that up, describe Étale groupoids, and provide some more details on the proofs of the groupoid isomorphisms in Connes' monograph that I mentioned in the last construction. First let me talk about categories.


Categories and Functors
To describe a groupoid, it's good to have the language of category theory in your toolkit. Category theory abstracts the notion of functions, homomorphisms, et cetera and allow you to make statements about those relations. More formally, by a category (actually, a small category) $\mathcal{C}$ I mean a set objects $\mathcal{C}^0$, together with a set of "morphisms" or "arrows" between these objects $\mathcal{C}^1$. For each arrow $f \in \mathcal{C}^1$ we have the domain $d(f) \in \mathcal{C}^0$ and the range $r(f) \in \mathcal{C}^0$. We write $f: A \to B$ to denote that $A=d(f)$ and $B=r(f)$. The arrows are functions on the objects, and we can compose them. Formally, if $f : A\to B$ and $g: B \to C$, then there is an arrow $g \circ f : A \to C$. Additionally, for every $A \in \mathcal{C}^0$ there is an identity array $1_A : A \to A$ such that $f \circ 1_A = f = 1_B \circ f$. Composition is, of course, associative.

As an example, lets pretend we have a well-defined notion of the set of all groups, call it $\mathcal{G}^0$. Between any two groups there exist some homomorphism $\phi$. $\phi$ may be trivial, sending everything to to the identity, but it exists nevertheless. Collect all such $\phi$'s into the set $\mathcal{G}^1$. Then $\mathcal{G}$ is a category.

We can also look at a single group, $G$, and consider it as a category. The object set, $G^0$ contains only the identity element $e$. All the fun happens in the set $G^1$, which contains all the other elements of of $G$. Then "arrow" composition is not only associative, but also has an inverse and the identity is $1_e$. This example will make more sense after I discuss groupoids, but keep it in mind!

One can think of categories as being purely algebraic structures. And like any algebraic structure, we'll want some structure-preserving map between them. This brings us to functors. A functor $F: \mathcal{C} \to \mathcal{D}$ between categories $\mathcal{C}$ and $\mathcal{D}$ is a mapping of objects to objects and arrows to arrows such that it preserves all the expected notions. To be more explicit, a functor meets:

• $F(f:A \to B) = F(f) : F(A) \to F(B)$
• $F(1_A) = 1_{F(A)}$
and finally
• $F(g \circ f) = F(g) \circ F(f)$

So in the case of the category of groups and homomorphisms between them, a functor preserves the groups themselves, but also all homomorphisms of those groups. In the case of the category of a group, a functor is actually just a group isomorphism.

Sources
Noncommutative Geometry, Quantum Fields, Motives. Connes
Lectures on Arithmetic Noncommutative Geometry. Marcolli (the usual two)
A Homology Theory for Étale groupoids. Marius Crainic and Ieke Moerdijk
Abstract Algebra Dummit, Foote (appendix on category theory.)
Category Theory Steve Awodey