# 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 \grp**Failed 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}}**
.