Editing Groupoid (section)

== Relation to groups ==
{{Group-like structures}}
If a groupoid has only one object, then the set of its morphisms forms a [[group (algebra)|group]]. Using the algebraic definition, such a groupoid is literally just a group.<ref>Mapping a group to the corresponding groupoid with one object is sometimes called delooping, especially in the context of [[Homotopy|homotopy theory]], see {{cite web |url=https://ncatlab.org/nlab/show/delooping#delooping_of_a_group_to_a_groupoid |title=delooping in nLab |website=ncatlab.org |access-date=2017-10-31 }}.</ref> Many concepts of [[group theory]] generalize to groupoids, with the notion of [[functor]] replacing that of [[group homomorphism]].

Every transitive/connected groupoid - that is, as explained above, one in which any two objects are connected by at least one morphism - is isomorphic to an action groupoid (as defined above) {{tmath|1= (G, X) }}. By transitivity, there will only be one [[orbit (group theory)|orbit]] under the action.

Note that the isomorphism just mentioned is not unique, and there is no [[natural equivalence|natural]] choice. Choosing such an isomorphism for a transitive groupoid essentially amounts to picking one object {{tmath|1= x_0 }}, a [[group isomorphism]] <math>h</math> from <math>G(x_0)</math> to {{tmath|1= G }}, and for each <math>x</math> other than {{tmath|1= x_0 }}, a morphism in <math>G</math> from <math>x_0</math> to {{tmath|1= x }}.

If a groupoid is not transitive, then it is isomorphic to a [[disjoint union]] of groupoids of the above type, also called its '''connected components''' (possibly with different groups <math>G</math> and sets <math>X</math> for each connected component).

In category-theoretic terms, each connected component of a groupoid is [[equivalent categories|equivalent]] (but not [[isomorphic categories|isomorphic]]) to a groupoid with a single object, that is, a single group. Thus any groupoid is equivalent to a [[multiset]] of unrelated groups. In other words, for equivalence instead of isomorphism, one does not need to specify the sets {{tmath|1= X }}, but only the groups {{tmath|1= G }}. For example,

*The fundamental groupoid of <math>X</math> is equivalent to the collection of the [[fundamental group]]s of each [[path-connected component]] of {{tmath|1= X }}, but an isomorphism requires specifying the set of points in each component;
*The set <math>X</math> with the equivalence relation <math>\sim</math> is equivalent (as a groupoid) to one copy of the [[trivial group]] for each [[equivalence class]], but an isomorphism requires specifying what each equivalence class is;
*The set <math>X</math> equipped with an [[Group action (mathematics)|action]] of the group <math>G</math> is equivalent (as a groupoid) to one copy of <math>G</math> for each [[orbit (group theory)|orbit]] of the action, but an [[isomorphism]] requires specifying what set each orbit is.

The collapse of a groupoid into a mere collection of groups loses some information, even from a category-theoretic point of view, because it is not [[natural (category theory)|natural]]. Thus when groupoids arise in terms of other structures, as in the above examples, it can be helpful to maintain the entire groupoid. Otherwise, one must choose a way to view each <math>G(x)</math> in terms of a single group, and this choice can be arbitrary. In the example from [[topology]], one would have to make a coherent choice of paths (or equivalence classes of paths) from each point <math>p</math> to each point <math>q</math> in the same path-connected component.

As a more illuminating example, the classification of groupoids with one [[endomorphism]] does not reduce to purely group theoretic considerations. This is analogous to the fact that the classification of [[vector space]]s with one endomorphism is nontrivial.

Morphisms of groupoids come in more kinds than those of groups: we have, for example, [[fibration]]s, [[covering morphism]]s, [[universal morphism]]s, and [[quotient morphism]]s. Thus a subgroup <math>H</math> of a group <math>G</math> yields an action of <math>G</math> on the set of [[coset]]s of <math>H</math> in <math>G</math> and hence a covering morphism <math>p</math> from, say, <math>K</math> to {{tmath|1= G }}, where <math>K</math> is a groupoid with [[#Vertex groups and orbits|vertex group]]s isomorphic to {{tmath|1= H }}. In this way, presentations of the group <math>G</math> can be "lifted" to presentations of the groupoid {{tmath|1= K }}, and this is a useful way of obtaining information about presentations of the subgroup {{tmath|1= H }}. For further information, see the books by Higgins and by Brown in the References.