Groupoids

I'm continuing a series of posts working out some details regarding some details in the groupoids used to construct the Bost-Connes system.

Groupoids

A groupoid is essentially a group, but one where the binary operation is not defined for all pairs of elements. That is, a groupoid is set G such that for all $g \in G$, there exist an inverse $g^{-1}$, together with a partial function $ \ast : G \times G \to G$ ($\ast$ is not necessarily a binary operation; it needn't be defined for all ordered pairs.) and with the following properties:

• Associativity: If $a * b$ and $b * c$ are defined, then $(a*b)*c=a*(b*c)$ is also defined
• Inverse: $a*a^{-1}=a^{-1}*a$ is always defined.
• Units: If $a*b$ is defined, then $a*b*b^{-1}=a$ and $a^{-1}*a*b=b$

While the above definition of a groupoid is familiar to one who has any experience with groups, it's sometimes more useful to think of it as a category: Let $G$ be a groupoid, and define $G^0$ to be the set of all elements of the form $g * g^{-1}$. Now let $x, y \in G^0$. If $f \in G$ is an element such that $y * f * x$ is defined, then we say $f \in G^1$ and $f: x \to y$.

In this case, it's not to hard to see that composition works: if we have $f:x \to y$ and $g:y \to z$, then $gf$ is defined and so is $z*gf*x$. Additionally, note that for every $x \in G^0$, the identity arrow $1_x$ is simply $x$. One oddity we have is that the inverse map $x * f^{-1} * y$ is defined (remember that since $x, y \in G^0$, we have $x=a*a^{-1}$ for some $a$, so that $x^{-1}=x$.) so that we have an inverse arrow $f^{-1} : y \to x$. Thus any groupoid can be considered as a category where all arrows have inverses.

Moreover, any category where all arrows have inverses is also a groupoid, so the two characterizations are actually equivalent definitions. Here's how: Given a groupoid $C$ in the category theoretic sense (id est, we have objects $C^0$ and arrows $C^{-1}$), we can denote all arrows $f: x \to y$ by $G(x,y)$. Let G be the disjoint union of all such $G(x,y)$. Then inverses are defined for all elements, and arrow composition because the partially defined groupoid operation.

Now lets take a groupoid $G$ (with objects, now called units $G^0$ and arrows/morphisms $G^{1}$). Recall the notation $G(x,y) = \{ f \in G^1 : f:x \to y. x,y \in G^0\}$. Let $x \in G^0$. Then $G(x,x)$ is a group (recall the example of a category of a group from earlier!) We call it the isotropy group at $x$ and denote it by $I_x$.

Before we continue, lets give an example. Let $GL(\mathbb{R})$ denote all the real invertible matrices under matrix multiplication. then $GL(\mathbb{R})^0$ is in bijective correspondence with the natural numbers, as for each natural number $n$ we have the corresponding $n \times n$ identity matrix. $G(n,m)$ is empty when $n \neq m$, but $G(n,n)$ is the general linear group $GL_n(\mathbb{R})$ of all $n \times n$ invertible matrices.

Finally, we say a groupoid $G$ is a topological groupoid when $G^0$ and $G^1$ are topological spaces, and all the corresponding maps on them (e.g. for $f \in G^1$ we have the domain $d(f)$, $d : G^1 \to G^0$, arrow composition $ \circ : G^1 \times G^1 \to G^1$, et cetera) must be continuous. Or equivalently, inversion and the groupoid operation must be continuous maps. A topological groupoid $G$ is said to be Étale if the domain map $d : G^1 \to G^0$ is a local homeomorphism (and thus all the other maps too), or equivalently, if inversion and the groupoid operation are local homeomorphism. Examples of these will come in the next post.

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