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 R , U {\displaystyle R,U} together with five morphisms
s , t : R → U , e : U → R , m : R × U , t , s R → R , i : R → R {\displaystyle s,t:R\to U,\ e:U\to R,\ m:R\times _{U,t,s}R\to R,\ i:R\to R}
satisfying the following groupoid axioms
s ∘ e = t ∘ e = 1 U , s ∘ m = s ∘ p 1 , t ∘ m = t ∘ p 2 {\displaystyle s\circ e=t\circ e=1_{U},\,s\circ m=s\circ p_{1},t\circ m=t\circ p_{2}} where the p i : R × U , t , s R → R {\displaystyle p_{i}:R\times _{U,t,s}R\to R} are the two projections, (associativity) m ∘ ( 1 R × m ) = m ∘ ( m × 1 R ) , {\displaystyle m\circ (1_{R}\times m)=m\circ (m\times 1_{R}),}
(unit) m ∘ ( e ∘ s , 1 R ) = m ∘ ( 1 R , e ∘ t ) = 1 R , {\displaystyle m\circ (e\circ s,1_{R})=m\circ (1_{R},e\circ t)=1_{R},}
(inverse) i ∘ i = 1 R {\displaystyle i\circ i=1_{R}} , s ∘ i = t , t ∘ i = s {\displaystyle s\circ i=t,\,t\circ i=s} , m ∘ ( 1 R , i ) = e ∘ s , m ∘ ( i , 1 R ) = e ∘ t {\displaystyle m\circ (1_{R},i)=e\circ s,\,m\circ (i,1_{R})=e\circ t} .
Examples
Group objects A group object is a special case of a groupoid object, where R = U {\displaystyle R=U} and s = t {\displaystyle s=t} . 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 s ( x → y ) = x , t ( x → y ) = y {\displaystyle s(x\to y)=x,\,t(x\to y)=y} , m ( f , g ) = g ∘ f {\displaystyle m(f,g)=g\circ f} , e ( x ) = 1 x {\displaystyle e(x)=1_{x}} and i ( f ) = f − 1 {\displaystyle 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).
Groupoid schemes A groupoid S-scheme is a groupoid object in the category of schemes over some fixed base scheme S. If U = S {\displaystyle U=S} , then a groupoid scheme (where s = t {\displaystyle 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 algebraic groups and their actions. For example, suppose an algebraic group G acts from the right on a scheme U. Then take R = U × G {\displaystyle R=U\times G} , s the projection, t the given action. This determines a groupoid scheme.
… excerpt ends here. Continue reading the full article.
