Groupoid scheme

In category theory, a groupoid object in a category C admitting finite fiber products is a pair of objects ${\displaystyle R,U}$ together with five morphisms ${\displaystyle s,t:R\to U,e:U\to R,m:R\times _{U,t,s}R\to R,i:R\to R}$ satisfying the following groupoid axioms

Example: 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 ${\displaystyle s(x\to y)=x,\,t(x\to y)=y}$, ${\displaystyle m(f,g)=g\circ f}$, ${\displaystyle e(x)=1_{x}}$ and ${\displaystyle i(f)=f^{-1}}$.

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