In category theory, a branch of mathematics, a '''groupoid object''' is both a generalization of a groupoid which is built on richer structures than sets, and a generalization of a group objects when the multiplication is only partially defined.
== Definition == A '''groupoid object''' in a category '''C''' admitting finite fiber products consists of a pair of objects <math>R, U</math> together with five morphisms :<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 is a special case of a groupoid object, where <math>R = U</math> and <math>s = t</math>. One recovers therefore topological groups by taking the category of topological spaces, or Lie groups by taking the category of manifolds, etc.
=== Groupoids === A groupoid object in the category of sets is precisely a groupoid in the usual sense: a category in which every morphism is an 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, since the maps ''s'' and ''t'' fail to satisfy further requirements (they are not necessarily submersions).
=== Groupoid schemes === A '''groupoid ''S''-scheme''' is a groupoid object in the category of schemes 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. A groupoid scheme is also called an '''algebraic groupoid''',{{sfn|Gillet|1984}} to convey the idea it is a generalization of algebraic groups and their actions.
For example, suppose an algebraic group ''G'' acts 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 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 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 from '''C''' to the category of groupoids. This way, each groupoid object determines a prestack in groupoids. This prestack is not a stack but it can be stackified 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 for a stack. More specifically, let <math>[R \rightrightarrows U]</math> be the category of <math>(R \rightrightarrows U)</math>-torsors. Then it is a category fibered in groupoids; in fact (in a nice case), a Deligne–Mumford stack. Conversely, any DM stack is of this form.
== See also == *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 | 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