Groupoid object: Difference between revisions
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In [[category theory]], a branch of mathematics, a '''groupoid object''' |
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 object]]s when the multiplication is only partially defined. |
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== Definition == |
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A '''groupoid object''' in a [[category (mathematics)|category]] ''C'' admitting finite [[pullback (category theory)|fiber product]]s consists of a pair of [[Object (category theory)|objects]] <math>R, U</math> together with five [[morphism]]s |
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:<math>s, t: R \to U, \ e: U \to R, \ m: R \times_{U, t, s} R \to R, \ i: R \to R</math> |
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satisfying the following groupoid axioms |
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# <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, |
# <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, |
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# (associativity) <math>m \circ (1_R \times m) = m \circ (m \times 1_R),</math> |
# (associativity) <math>m \circ (1_R \times m) = m \circ (m \times 1_R),</math> |
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# (unit) <math>m \circ (e \circ s, 1_R) = m \circ (1_R, e \circ t) = 1_R,</math> |
# (unit) <math>m \circ (e \circ s, 1_R) = m \circ (1_R, e \circ t) = 1_R,</math> |
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# (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> |
# (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> |
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A [[group object]] is a special case of a groupoid object. |
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== Examples == |
== Examples == |
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⚫ | |||
=== Group objects === |
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Incidentally, one can consider a notion of a semigroupoid (unital semigroup = a category with a single object); but, according to this example, that is nothing but a category; so a groupoid object is really a special case of a "category object", better known as a [[stack (mathematics)|stack]] (or [[prestack]]). |
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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 group]]s by taking the [[category of topological spaces]], or [[Lie group]]s by taking the [[category of manifolds]], etc. |
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=== Groupoids === |
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⚫ | A '''groupoid ''S''-scheme''' is a groupoid object in the category of [[scheme (mathematics)|scheme]]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]]. A groupoid scheme is also called an '''algebraic groupoid''', |
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⚫ | 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 arrows 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. |
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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 [[Submersion (mathematics)|submersions]]). |
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⚫ | |||
== |
=== Groupoid schemes === |
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⚫ | A '''groupoid ''S''-scheme''' is a groupoid object in the category of [[scheme (mathematics)|scheme]]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]]. A groupoid scheme is also called an '''algebraic groupoid''',{{sfn|Gillet|1984}} to convey the idea it is a generalization of [[algebraic group]]s and their actions. |
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⚫ | |||
⚫ | |||
⚫ | 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 [[stackification|stackified]] to yield a stack.<!--For example, the stackification of an algebraic group is the classifying stack BG of ''G''.--> |
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== Constructions == |
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⚫ | The main use of the notion is that it provides an [[atlas (stack)|atlas]] for a |
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⚫ | |||
⚫ | 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 (mathematics)|stack]] but it can be [[stackification|stackified]] to yield a stack.<!--For example, the stackification of an algebraic group is the classifying stack BG of ''G''.--> |
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⚫ | The main use of the notion is that it provides an [[atlas (stack)|atlas]] for a stack. More specifically, let <math>[R \rightrightarrows U]</math> be the category of [[torsor under a groupoid|<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. |
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== See also == |
== See also == |
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{{reflist}} |
{{reflist}} |
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== References == |
== References == |
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*{{citation|url=http://www.math.unizh.ch/index.php?pr_vo_det&key1=1287&key2=580&no_cache=1|first1=Kai|last1=Behrend|first2=Brian|last2=Conrad|first3=Dan|last3=Edidin|first4=William|last4=Fulton|first5=Barbara|last5=Fantechi|first6=Lothar|last6=Göttsche|first7=Andrew|last7=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 |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 |first2=Brian |last2=Conrad |first3=Dan |last3=Edidin |first4=William |last4=Fulton |first5=Barbara |last5=Fantechi |first6=Lothar |last6=Göttsche |first7=Andrew |last7=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}} |
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* {{citation |
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*H. Gillet, [http://ac.els-cdn.com/0022404984900367/1-s2.0-0022404984900367-main.pdf?_tid=a96230a2-515a-11e7-8661-00000aab0f27&acdnat=1497483728_c3a14f29c251aa4dc2b55b3abfa6ba9e Intersection theory on algebraic stacks and Q-varieties], J. Pure Appl. Algebra 34 (1984), 193–240, Proceedings of the Luminy conference on algebraic K-theory (Luminy, 1983). |
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| last = Gillet | first = Henri | author-link = Henri Gillet |
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| department = Proceedings of the Luminy conference on algebraic {{mvar|K}}-theory (Luminy, 1983) |
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| doi = 10.1016/0022-4049(84)90036-7 |
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| issue = 2-3 |
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| journal = [[Journal of Pure and Applied Algebra]] |
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| mr = 772058 |
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| pages = 193–240 |
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| title = Intersection theory on algebraic stacks and {{mvar|Q}}-varieties |
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| url = https://scholar.archive.org/work/d7ssf5njzjdqla4qxvmvmj5ylu |
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| volume = 34 |
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| year = 1984}} |
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[[Category:Algebraic geometry]] |
[[Category:Algebraic geometry]] |
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[[Category:Scheme theory]] |
[[Category:Scheme theory]] |
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[[Category:Category theory]] |
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{{algebraic-geometry-stub}} |
Latest revision as of 05:18, 4 July 2024
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
[edit]A groupoid object in a category C admitting finite fiber products consists of a pair of objects together with five morphisms
satisfying the following groupoid axioms
- where the are the two projections,
- (associativity)
- (unit)
- (inverse) , , .[1]
Examples
[edit]Group objects
[edit]A group object is a special case of a groupoid object, where and . One recovers therefore topological groups by taking the category of topological spaces, or Lie groups by taking the category of manifolds, etc.
Groupoids
[edit]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 arrows in C, the five morphisms given by , , and . 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
[edit]A groupoid S-scheme is a groupoid object in the category of schemes over some fixed base scheme S. If , then a groupoid scheme (where are necessarily the structure map) is the same as a group scheme. A groupoid scheme is also called an algebraic groupoid,[2] 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 , s the projection, t the given action. This determines a groupoid scheme.
Constructions
[edit]Given a groupoid object (R, U), the equalizer of , 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.
The main use of the notion is that it provides an atlas for a stack. More specifically, let be the category of -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
[edit]Notes
[edit]- ^ Algebraic stacks, Ch 3. § 1.
- ^ Gillet 1984.
References
[edit]- Behrend, Kai; Conrad, Brian; Edidin, Dan; Fulton, William; Fantechi, Barbara; Göttsche, Lothar; Kresch, Andrew (2006), Algebraic stacks, archived from the original on 2008-05-05, retrieved 2014-02-11
- Gillet, Henri (1984), "Intersection theory on algebraic stacks and Q-varieties", Proceedings of the Luminy conference on algebraic K-theory (Luminy, 1983), Journal of Pure and Applied Algebra, 34 (2–3): 193–240, doi:10.1016/0022-4049(84)90036-7, MR 0772058