Editing Groupoid (section)

==== Examples ====
* If <math>f: X_0 \to Y</math> is a smooth [[Surjective function|surjective]] [[Submersion (mathematics)|submersion]] of [[smooth manifolds]], then <math>X_0\times_YX_0 \subset X_0\times X_0</math> is an equivalence relation<ref name=":0" /> since <math>Y</math> has a topology isomorphic to the [[quotient topology]] of <math>X_0</math> under the surjective map of topological spaces. If we write, <math>X_1 = X_0\times_YX_0</math> then we get a groupoid <math display=block>X_1 \rightrightarrows X_0,</math> which is sometimes called the '''banal groupoid''' of a surjective submersion of smooth manifolds.
* If we relax the reflexivity requirement and consider ''partial equivalence relations'', then it becomes possible to consider [[semidecidable]] notions of equivalence on computable realisers for sets. This allows groupoids to be used as a computable approximation to set theory, called ''PER models''. Considered as a category, PER models are a cartesian closed category with natural numbers object and subobject classifier, giving rise to the [[effective topos]] introduced by [[Martin Hyland]].