groupoid

Groupoid definitions

Definition 1.1   A groupoid $ {\mathsf{G}}$ is simply defined as a small category with inverses over its set of objects $ X = Ob({\mathsf{G}})$ . One often writes $ {\mathsf{G}}^y_x$ for the set of morphisms in $ {\mathsf{G}}$ from $ x$ to $ y$ .

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)
$ s(x) = r(x) = x$  , 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 $ u$ .

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).

Several examples of groupoids are:

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}$ .



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As of this snapshot date, this entry was owned by bci1.