PlanetPhysics/Groupoid5

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Groupoid definitionsEdit

A groupoid Failed to parse (unknown function "\grp"): {\displaystyle \grp} is simply defined as a small category with inverses over its set of objects Failed to parse (unknown function "\grp"): {\displaystyle X = Ob(\grp)} . One often writes Failed to parse (unknown function "\grp"): {\displaystyle \grp^y_x} for the set of morphisms in Failed to parse (unknown function "\grp"): {\displaystyle \grp} from   to  .

A topological groupoid consists of a space Failed to parse (unknown function "\grp"): {\displaystyle \grp} , a distinguished subspace Failed to parse (unknown function "\grp"): {\displaystyle \grp^{(0)} = \obg \subset \grp} , called {\it the space of objects} of Failed to parse (unknown function "\grp"): {\displaystyle \grp} , together with maps

Failed to parse (unknown function "\xymatrix"): {\displaystyle r,s~:~ \xymatrix{ \grp \ar@<1ex>[r]^r \ar[r]_s & \grp^{(0)} } }

called the {\it range} and {\it source maps} respectively,

together with a law of composition

Failed to parse (unknown function "\grp"): {\displaystyle \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)] </math>s(\gamma_1 \circ \gamma_2) = r(\gamma_2)~,~ r(\gamma_1 \circ \gamma_2) = r(\gamma_1) (\gamma_1, \gamma_2) \in \grp^{(2)}Failed to parse (unknown function "\item"): {\displaystyle ~. \item[(2)] <math>s(x) = r(x) = x} ~, for all Failed to parse (unknown function "\grp"): {\displaystyle x \in \grp^{(0)}} ~.

\item[(3)] </math>\gamma \circ s(\gamma) = \gamma~,~ r(\gamma) \circ \gamma = \gamma \gamma \in \grpFailed to parse (unknown function "\item"): {\displaystyle ~. \item[(4)] } (\gamma_1 \circ \gamma_2) \circ \gamma_3 = \gamma_1 \circ (\gamma_2 \circ \gamma_3)Failed to parse (unknown function "\item"): {\displaystyle ~. \item[(5)] Each <math>\gamma} has a two--sided inverse   with </math>\gamma \gamma^{-1} = r(\gamma)~,~ \gamma^{-1} \gamma = s (\gamma)Failed to parse (unknown function "\grp"): {\displaystyle ~. Furthermore, only for topological groupoids the inverse map needs be continuous. It is usual to call <math>\grp^{(0)} = Ob(\grp)} {\it the set of objects} of Failed to parse (unknown function "\grp"): {\displaystyle \grp} ~. For Failed to parse (unknown function "\grp"): {\displaystyle u \in Ob(\grp)} , the set of arrows Failed to parse (unknown function "\lra"): {\displaystyle u \lra u} forms a group Failed to parse (unknown function "\grp"): {\displaystyle \grp_u} , called the isotropy group of Failed to parse (unknown function "\grp"): {\displaystyle \grp} 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).

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:  . Here, Failed to parse (unknown function "\grp"): {\displaystyle \grp^0 = X } , (the diagonal of   ) and  .

Therefore,   =  . When  , R is called a trivial groupoid. A special case of a trivial groupoid is      . (So every i is equivalent to every j ). Identify   with the matrix unit  . Then the groupoid   is just matrix multiplication except that we only multiply   when  , and  . We do not really lose anything by restricting the multiplication, since the pairs   excluded from groupoid multiplication just give the 0 product in normal algebra anyway. For a groupoid Failed to parse (unknown function "\grp"): {\displaystyle \grp_{lc}} to be a locally compact groupoid means that Failed to parse (unknown function "\grp"): {\displaystyle \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 Failed to parse (unknown function "\grp"): {\displaystyle \grp_{lc}^u} as well as the unit space Failed to parse (unknown function "\grp"): {\displaystyle \grp_{lc}^0} is closed in Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "/mathoid/local/v1/":): {\displaystyle \grp_{lc}} . What replaces the left Haar measure on Failed to parse (unknown function "\grp"): {\displaystyle \grp_{lc}} is a system of measures   (Failed to parse (unknown function "\grp"): {\displaystyle u \in \grp_{lc}^0} ), where   is a positive regular Borel measure on Failed to parse (unknown function "\grp"): {\displaystyle \grp_{lc}^u} with dense support. In addition, the   's are required to vary continuously (when integrated against Failed to parse (unknown function "\grp"): {\displaystyle f \in C_c(\grp_{lc}))} and to form an invariant family in the sense that for each x, the map   is a measure preserving homeomorphism from Failed to parse (unknown function "\grp"): {\displaystyle \grp_{lc}^s(x)} onto Failed to parse (unknown function "\grp"): {\displaystyle \grp_{lc}^r(x)} . Such a system   is called a left Haar system for the locally compact groupoid Failed to parse (unknown function "\grp"): {\displaystyle \grp_{lc}} .