PlanetPhysics/Groupoid5

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

A topological groupoid consists of a space $$\grp$$, a distinguished subspace $$\grp^{(0)} = \obg \subset \grp$$, called {\it the space of objects} of $$\grp$$, together with maps $$ r,s~:~ \xymatrix{ \grp \ar@ [r]^r \ar[r]_s & \grp^{(0)} } $$ called the {\it range} and {\it source maps} respectively, together with a law of composition $$ \circ~:~ \grp^{(2)}: = \grp \times_{\grp^{(0)}} \grp = \{ ~(\gamma_1, \gamma_2) \in \grp \times \grp ~:~ s(\gamma_1) = r(\gamma_2)~ \}~ \lra ~\grp~, $$ such that the following hold~:~

\item[(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 \grp^{(2)}$$~.

\item[(2)] ~, for all ~.

\item[(3)] $$\gamma \circ s(\gamma) = \gamma~,~ r(\gamma) \circ \gamma = \gamma$$~, for all $$\gamma \in \grp$$~.

\item[(4)] $$(\gamma_1 \circ \gamma_2) \circ \gamma_3 = \gamma_1 \circ (\gamma_2 \circ \gamma_3)$$~.

\item[(5)] Each \gamma has a two--sided inverse \gamma^{-1} with $$\gamma \gamma^{-1} = r(\gamma)~,~ \gamma^{-1} \gamma = s (\gamma)~. Furthermore, only for topological groupoids the inverse map needs be continuous. It is usual to call $$\grp^{(0)} = Ob(\grp)$$ {\it the set of objects} of $$\grp$$~. For $$u \in Ob(\grp)$$, the set of arrows $$u \lra u$$ forms a group $$\grp_u$$, called the isotropy group of $$\grp$$ at $$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:


 * (a) locally compact groups, transformation groups, and any group in general:
 * (b) equivalence relations
 * (c) tangent bundles
 * (d) the tangent groupoid
 * (e) holonomy groupoids for foliations
 * (f) Poisson groupoids
 * (g) graph groupoids.

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, $$\grp^0 = X $$, (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 $$\grp_{lc}$$ to be a locally compact groupoid means that $$\grp_{lc}$$ is required to be a (second countable) locally compact Hausdorff space, and the product and also inversion maps are required to be continuous. Each $$\grp_{lc}^u$$ as well as the unit space $$\grp_{lc}^0$$ is closed in $$\grp_{lc}$$. What replaces the left Haar measure on $$\grp_{lc}$$ is a system of measures $$\lambda^u$$ ($$u \in \grp_{lc}^0$$), where $$\lambda^u$$ is a positive regular Borel measure on $$\grp_{lc}^u$$ with dense support. In addition, the $$\lambda^u~$$ 's are required to vary continuously (when integrated against $$f \in C_c(\grp_{lc}))$$ and to form an invariant family in the sense that for each x, the map $$y \mapsto xy$$ is a measure preserving homeomorphism from $$\grp_{lc}^s(x)$$ onto $$\grp_{lc}^r(x)$$. Such a system $$\left\{ \lambda^u \right\}$$ is called a left Haar system for the locally compact groupoid $$\grp_{lc}$$.