# Groupoid object

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In [category theory](/source/category_theory), a branch of [mathematics](/source/mathematics), a '''groupoid object''' is both a generalization of a [groupoid](/source/groupoid) which is built on richer structures than sets, and a generalization of a [group object](/source/group_object)s when the multiplication is only [partially defined](/source/partial_function).

== Definition ==
A '''groupoid object''' in a [category](/source/category_(mathematics)) '''C''' admitting finite [fiber product](/source/pullback_(category_theory))s consists of a pair of [objects](/source/Object_(category_theory)) <math>R, U</math> together with five [morphism](/source/morphism)s
:<math>s, t: R \to U, \ e: U \to R, \ m: R \times_{U, t, s} R \to R, \ i: R \to R</math>
satisfying the following groupoid axioms
# <math>s \circ e = t \circ e = 1_U, \, s \circ m = s \circ p_1, t \circ m = t \circ p_2</math> where the <math>p_i: R \times_{U, t, s} R \to R</math> are the two projections,
# (associativity) <math>m \circ (1_R \times m) = m \circ (m \times 1_R),</math>
# (unit) <math>m \circ (e \circ s, 1_R) = m \circ (1_R, e \circ t) = 1_R,</math>
# (inverse) <math>i \circ i = 1_R</math>, <math>s \circ i = t, \, t \circ i = s</math>, <math>m \circ (1_R, i) = e \circ s, \, m \circ (i, 1_R) = e \circ t</math>.<ref>{{harvnb|Algebraic stacks|loc=Ch 3. § 1.}}</ref>

== Examples ==

=== Group objects ===
A [group object](/source/group_object) is a special case of a groupoid object, where <math>R = U</math> and <math>s = t</math>. One recovers therefore [topological group](/source/topological_group)s by taking the [category of topological spaces](/source/category_of_topological_spaces), or [Lie group](/source/Lie_group)s by taking the [category of manifolds](/source/category_of_manifolds), etc.

=== Groupoids ===
A groupoid object in the [category of sets](/source/category_of_sets) is precisely a [groupoid](/source/groupoid) in the usual sense: a category in which every morphism is an [isomorphism](/source/isomorphism). Indeed, given such a category '''C''', take ''U'' to be the set of all objects in '''C''', ''R'' the set of all morphisms in '''C''', the five morphisms given by <math>s(x \to y) = x, \, t(x \to y) = y</math>, <math>m(f, g) = g \circ f</math>, <math>e(x) = 1_x</math> and <math>i(f) = f^{-1}</math>. When the term "groupoid" can naturally refer to a groupoid object in some particular category in mind, the term '''groupoid set''' is used to refer to a groupoid object in the category of sets.

However, unlike in the previous example with Lie groups, a groupoid object in the category of manifolds is not necessarily a [Lie groupoid](/source/Lie_groupoid), since the maps ''s'' and ''t'' fail to satisfy further requirements (they are not necessarily [submersion](/source/Submersion_(mathematics))s).

=== Groupoid schemes ===
A '''groupoid ''S''-scheme''' is a groupoid object in the category of [scheme](/source/scheme_(mathematics))s over some fixed base scheme ''S''. If <math>U = S</math>, then a groupoid scheme (where <math>s = t</math> are necessarily the structure map) is the same as a [group scheme](/source/group_scheme). A groupoid scheme is also called an '''algebraic groupoid''',{{sfn|Gillet|1984}} to convey the idea it is a generalization of [algebraic group](/source/algebraic_group)s and their actions.

For example, suppose an algebraic group ''G'' [acts](/source/Group-scheme_action) from the right on a scheme ''U''. Then take <math>R = U \times G</math>, ''s'' the projection, ''t'' the given action. This determines a groupoid scheme.

== Constructions ==
Given a groupoid object (''R'', ''U''), the [equalizer](/source/equaliser_(mathematics)) of <math>R \,\overset{s}\underset{t}\rightrightarrows\, U</math>, if any, is a group object called the '''inertia group''' of the groupoid. The [coequalizer](/source/coequalizer) of the same diagram, if any, is the quotient of the groupoid.

Each groupoid object in a category '''C''' (if any) may be thought of as a [contravariant functor](/source/contravariant_functor) from '''C''' to the category of groupoids. This way, each groupoid object determines a [prestack](/source/prestack) in groupoids. This prestack is not a [stack](/source/Stack_(mathematics)) but it can be [stackified](/source/stackification) to yield a stack.<!--For example, the stackification of an algebraic group is the classifying stack BG of ''G''.-->

The main use of the notion is that it provides an [atlas](/source/atlas_(stack)) for a stack. More specifically, let <math>[R \rightrightarrows U]</math> be the category of [<math>(R \rightrightarrows U)</math>-torsors](/source/torsor_under_a_groupoid). Then it is a [category fibered in groupoids](/source/category_fibered_in_groupoids); in fact (in a nice case), a [Deligne–Mumford stack](/source/Deligne%E2%80%93Mumford_stack). Conversely, any DM stack is of this form.

== See also ==
*[Simplicial scheme](/source/Simplicial_scheme)

== Notes ==
{{reflist}}

== References == 
*{{citation |ref={{harvid|Algebraic stacks}} |url=http://www.math.unizh.ch/index.php?pr_vo_det&key1=1287&key2=580&no_cache=1 |first1=Kai |last1=Behrend |author-link1=Kai Behrend |first2=Brian |last2=Conrad |author-link2=Brian Conrad |first3=Dan |last3=Edidin |first4=William |last4=Fulton |author-link4=William Fulton (mathematician) |first5=Barbara |last5=Fantechi |author-link5=Barbara Fantechi |first6=Lothar |last6=Göttsche |author-link6=Lothar Göttsche |first7=Andrew |last7=Kresch |author-link7=Andrew Kresch |year=2006 |title=Algebraic stacks |access-date=2014-02-11 |archive-url=https://web.archive.org/web/20080505043444/http://www.math.unizh.ch/index.php?pr_vo_det&key1=1287&key2=580&no_cache=1 |archive-date=2008-05-05 |url-status=dead}}
* {{citation
 | last = Gillet | first = Henri | author-link = Henri Gillet
 | department = Proceedings of the Luminy conference on algebraic {{mvar|K}}-theory (Luminy, 1983)
 | doi = 10.1016/0022-4049(84)90036-7
 | issue = 2-3
 | journal = [Journal of Pure and Applied Algebra](/source/Journal_of_Pure_and_Applied_Algebra)
 | mr = 772058
 | pages = 193–240
 | title = Intersection theory on algebraic stacks and {{mvar|Q}}-varieties
 | url = https://scholar.archive.org/work/d7ssf5njzjdqla4qxvmvmj5ylu
 | volume = 34
 | year = 1984}}

Category:Algebraic geometry
Category:Scheme theory
Category:Category theory

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Adapted from the Wikipedia article [Groupoid object](https://en.wikipedia.org/wiki/Groupoid_object) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Groupoid_object?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.