Groupoid5

Definition 1.2 A topological groupoid consists of a space ${\mathsf{G}}$ , a distinguished subspace ${\mathsf{G}}^{(0)} = {\rm Ob(\mathsf{G)}}\subset {\mathsf{G}}$ , called the space of objects of ${\mathsf{G}}$ , together with maps

$\displaystyle r,s~:~ \xymatrix{ {\mathsf{G}}\ar@<1ex>[r]^r \ar[r]_s & {\mathsf{G}}^{(0)} }$

(1.1)

called the range and source maps respectively, together with a law of composition

$\displaystyle \circ~:~ {\mathsf{G}}^{(2)}: = {\mathsf{G}}\times_{{\mathsf{G}}^{... ...{\mathsf{G}}~:~ s(\gamma_1) = r(\gamma_2)~ \}~ {\longrightarrow}~{\mathsf{G}}~,$

(1.2)

such that the following hold :

(1): $s(\gamma_1 \circ \gamma_2) = r(\gamma_2)~,~ r(\gamma_1 \circ \gamma_2) = r(\gamma_1)$ , for all $(\gamma_1, \gamma_2) \in {\mathsf{G}}^{(2)}$ .
(2): , for all $x \in {\mathsf{G}}^{(0)}$ .
(3): $\gamma \circ s(\gamma) = \gamma~,~ r(\gamma) \circ \gamma = \gamma$ , for all $\gamma \in {\mathsf{G}}$ .
(4): $(\gamma_1 \circ \gamma_2) \circ \gamma_3 = \gamma_1 \circ (\gamma_2 \circ \gamma_3)$ .
(5): Each $\gamma$ has a two–sided inverse $\gamma^{-1}$ with $\gamma \gamma^{-1} = r(\gamma)~,~ \gamma^{-1} \gamma = s (\gamma)$ . Furthermore, only for topological groupoids the inverse map needs be continuous. It is usual to call ${\mathsf{G}}^{(0)} = Ob({\mathsf{G}})$ the set of objects of ${\mathsf{G}}$ . For $u \in Ob({\mathsf{G}})$ , the set of arrows $u {\longrightarrow}u$ forms a group ${\mathsf{G}}_u$ , called the isotropy group of ${\mathsf{G}}$ at .

Thus, as it is well kown, a topological groupoid is just a groupoid internal to the category of topological spaces and continuous maps. The notion of internal groupoid has proved significant in a number of fields, since groupoids generalise bundles of groups, group actions, and equivalence relations. For a further study of groupoids we refer the reader to Brown (2006).

As a simple, helpful example of a groupoid, consider (b) above. Thus, let R be an equivalence relation on a set X. Then R is a groupoid under the following operations: $(x, y)(y, z) = (x, z), (x, y)^{-1} = (y, x)$ . Here, ${\mathsf{G}}^0 = X$ , (the diagonal of $X \times X$ ) and $r((x, y)) = x, s((x, y)) = y$

Therefore, $R^2$

= $\left\{((x, y), (y, z)) : (x, y), (y, z) \in R \right\}$ . When $R = X \times X$ , R is called a trivial groupoid. A special case of a trivial groupoid is $R = R_n = \left\{ 1, 2, . . . , n \right\}$ $\times$ $\left\{ 1, 2, . . . , n \right\}$ . (So every i is equivalent to every j). Identify $(i,j) \in R_n$ with the matrix unit $e_{ij}$ . Then the groupoid $R_n$

is just matrix multiplication except that we only multiply $e_{ij}, e_{kl}$ when $k = j$

, and $(e_{ij} )^{-1} = e_{ji}$ . We do not really lose anything by restricting the multiplication, since the pairs $e_{ij}, {e_{kl}}$ excluded from groupoid multiplication just give the 0 product in normal algebra anyway. For a groupoid ${\mathsf{G}}_{lc}$ to be a locally compact groupoid means that ${\mathsf{G}}_{lc}$ is required to be a (second countable) locally compact Hausdorff space, and the product and also inversion maps are required to be continuous. Each ${\mathsf{G}}_{lc}^u$ as well as the unit space ${\mathsf{G}}_{lc}^0$ is closed in ${\mathsf{G}}_{lc}$ . What replaces the left Haar measure on ${\mathsf{G}}_{lc}$ is a system of measures $\lambda^u$ ( $u \in {\mathsf{G}}_{lc}^0$ ), where $\lambda^u$ is a positive regular Borel measure on ${\mathsf{G}}_{lc}^u$ with dense support. In addition, the $\lambda^u~$ 's are required to vary continuously (when integrated against $f \in C_c({\mathsf{G}}_{lc}))$ and to form an invariant family in the sense that for each x, the map $y \mapsto xy$ is a measure preserving homeomorphism from ${\mathsf{G}}_{lc}^s(x)$ onto ${\mathsf{G}}_{lc}^r(x)$ . Such a system $\left\{ \lambda^u \right\}$ is called a left Haar system for the locally compact groupoid ${\mathsf{G}}_{lc}$ .

Groupoid definitions