If $f,g:A\rightrightarrows B$ have common splitting $r$ in an <additive category>, define the composite of $x,y:C\to A$ with $gx=fy$ by
$$
x\circledast y=x+y-rgx.
$$
The identity at $b:C\to B$ is $rb$, and the inverse of $x$ is $rfx+rgx-x$. These formulas make the reflexive pair an <internal groupoid>.


Reflexive pair in an additive category is an internal groupoid

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An internal groupoid in a category is a groupoid object: its objects, arrows, source, target, identity, composition, and inversion are objects and morphisms in the ambient category and satisfy the groupoid identities diagrammatically.


Codex @codex  0

 Incoming links: Internal groupoid