= Solution
{c}
Let $r:B\to A$ split the <reflexive pair> $f,g$. Fix $C$, and take objects $u:C\to B$ and arrows $a:C\to A$ with source $fa$ and target $ga$. Since $\mathcal C$ is an <additive category>, its hom-sets are <abelian groups>, and composition is additive.
The identity at $u$ is $ru$. For $a:u\to v$ and $b:v\to w$, define
$$
b\star a=a-rv+b.
$$
Its source is $fa-v+fb=u-v+v=u$ and its target is $ga-v+gb=v-v+w=w$. The two identity laws follow from $a-rv+rv=a$ and $ru-ru+a=a$. For $c:w\to x$, both parenthesizations of the triple composite are $a-rv+b-rw+c$, so composition is associative.
Define the inverse by
$$
a^{-1}=ru+rv-a.
$$
Its source is $u+v-u=v$ and its target is $u+v-v=u$. Moreover $a^{-1}\star a=ru$ and $a\star a^{-1}=rv$. Thus these data form a <groupoid>, proving the represented form of <reflexive pair in an additive category is an internal groupoid>:
$$
\boxed{b\star a=a-rv+b,\qquad a^{-1}=r(fa)+r(ga)-a.}
$$
Back to article page
