Groupoid object

 ( Algebraic Geometry ) 

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 when the multiplication is only partial function.

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Definition A groupoid object in a category C admitting finite fiber products consists of a pair of objects $R,\; U$ together with five

$s,\; t:\; R\; \backslash to\; U,\; \backslash \; e:\; U\; \backslash to\; R,\; \backslash \; m:\; R\; \backslash times\_\{U,\; t,\; s\}\; R\; \backslash to\; R,\; \backslash \; i:\; R\; \backslash to\; R$

satisfying the following groupoid axioms

1. $s\; \backslash circ\; e\; =\; t\; \backslash circ\; e\; =\; 1\_U,\; \backslash ,\; s\; \backslash circ\; m\; =\; s\; \backslash circ\; p\_1,\; t\; \backslash circ\; m\; =\; t\; \backslash circ\; p\_2$ where the $p\_i:\; R\; \backslash times\_\{U,\; t,\; s\}\; R\; \backslash to\; R$ are the two projections,
2. (associativity) $m\; \backslash circ\; (1\_R\; \backslash times\; m)\; =\; m\; \backslash circ\; (m\; \backslash times\; 1\_R),$
3. (unit) $m\; \backslash circ\; (e\; \backslash circ\; s,\; 1\_R)\; =\; m\; \backslash circ\; (1\_R,\; e\; \backslash circ\; t)\; =\; 1\_R,$
4. (inverse) $i\; \backslash circ\; i\; =\; 1\_R$, $s\; \backslash circ\; i\; =\; t,\; \backslash ,\; t\; \backslash circ\; i\; =\; s$, $m\; \backslash circ\; (1\_R,\; i)\; =\; e\; \backslash circ\; s,\; \backslash ,\; m\; \backslash circ\; (i,\; 1\_R)\; =\; e\; \backslash circ\; t$.

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Examples

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Group objects A group object is a special case of a groupoid object, where $R\; =\; U$ and $s\; =\; t$. One recovers therefore topological groups by taking the category of topological spaces, or by taking the category of manifolds, etc.

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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 $s(x\; \backslash to\; y)\; =\; x,\; \backslash ,\; t(x\; \backslash to\; y)\; =\; y$, $m(f,\; g)\; =\; g\; \backslash circ\; f$, $e(x)\; =\; 1\_x$ and $i(f)\; =\; f^\{-1\}$. 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).

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Groupoid schemes A groupoid S-scheme is a groupoid object in the category of schemes over some fixed base scheme S. If $U\; =\; S$, then a groupoid scheme (where $s\; =\; t$ are necessarily the structure map) is the same as a group scheme. A groupoid scheme is also called an algebraic groupoid, to convey the idea it is a generalization of and their actions.

For example, suppose an algebraic group G acts from the right on a scheme U. Then take $R\; =\; U\; \backslash times\; G$, s the projection, t the given action. This determines a groupoid scheme.

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Constructions Given a groupoid object ( R, U), the equalizer of $R\; \backslash ,\backslash overset\{s\}\backslash underset\{t\}\backslash rightrightarrows\backslash ,\; U$, 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 stackification to yield a stack.

The main use of the notion is that it provides an atlas for a stack. More specifically, let $R$ be the category of $(R\; \backslash rightrightarrows\; U)$-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

• Simplicial scheme

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Notes

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