10.3 Groupoids and Lie Algebroids

Though a groupoid can be defined as a small category whose morphisms are invertible, we follow the terminology of [287; 367].

A groupoid is a set endowed with the following two operations:

1. a partial multiplication

   
   

   where is a subset of the Cartesian product called the set of composable pairs, and

2. the inversion x ? x ^{- 1}.

The set of composable pairs consists of all the pairs ( x, x ^-1), ( x ^{- 1}, x) and the pairs ( x, y) such that x ^{- 1} x = yy ^{- 1}. Then the above mentioned operations obey the conditions:

• ( x ^-1) ^-1 = x;

• if ( x, y) ? , then ( y ^-1, x ^{- 1}) ? and ( xy) ^-1 =y ^{- 1} x ^{- 1};

• if ( x, y) ? , then ( x- ¹, xy) ? and x ^{- 1}( xy) = y;

• if ( x, y) ? , then ( xy, y ^-1) ? and ( xy) y ^{- 1} = x;

• if ( x, y), ( y, z) ? , then ( xy, z),( x, yz) ? and ( xy) z = x( yz