Editing Groupoid (section)

=== Subgroupoids and morphisms ===
A '''subgroupoid''' of <math>G \rightrightarrows X</math> is a [[subcategory]] <math>H \rightrightarrows Y</math> that is itself a groupoid. It is called '''wide''' or '''full''' if it is [[Wide subcategory|wide]] or [[Full subcategory|full]] as a subcategory, i.e., respectively, if <math>X = Y</math> or <math>G(x,y)=H(x,y)</math> for every {{tmath|1= x,y \in Y }}.

A '''groupoid morphism''' is simply a functor between two (category-theoretic) groupoids.

Particular kinds of morphisms of groupoids are of interest. A morphism <math>p: E \to B</math> of groupoids is called a [[fibration]] if for each object <math>x</math> of <math>E</math> and each morphism <math>b</math> of <math>B</math> starting at <math>p(x)</math> there is a morphism <math>e</math> of <math>E</math> starting at <math>x</math> such that {{tmath|1= p(e)=b }}. A fibration is called a [[covering morphism]] or [[covering of groupoids]] if further such an <math>e</math> is unique. The covering morphisms of groupoids are especially useful because they can be used to model [[covering map]]s of spaces.<ref>J.P. May, ''A Concise Course in Algebraic Topology'', 1999, The University of Chicago Press {{ISBN|0-226-51183-9}} (''see chapter 2'')</ref>

It is also true that the category of covering morphisms of a given groupoid <math>B</math> is equivalent to the category of actions of the groupoid <math>B</math> on sets.