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 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} .