Reflexive pair

A reflexive pair is a pair $f,g:A\rightrightarrows B$ with a common splitting $r:B\to A$, so $fr=gr=1_B$.

A reflexive coequalizer is a coequalizer of a reflexive pair, meaning a parallel pair $r,s:X\rightrightarrows Y$ with a common section $t:Y\to X$ satisfying $rt=st=1_Y$. A functor preserves reflexive coequalizers when it takes every such coequalizer to a coequalizer of the image pair.

For a reflexive pair $f,g:A\rightrightarrows B$ with common section $r$, form the pushout in a category of $f$ and $g$, with maps $j_1,j_2:B\to P$. The equation $j_{1}f=j_{2}g$, composed with $r$, gives $j_1=j_2=q$. This common map is a coequalizer of $f,g$: every map equalizing them defines a commuting pair into the pushout and hence factors uniquely through $q$.

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.

For a reflexive pair $f,g:A\rightrightarrows B$ with section $r$ in a preadditive category, each hom-set diagram defines a groupoid. Arrows $a,b:C\to A$ compose when $ga=fb$, by $b\circ a=a+b-rga$. Identity at $x:C\to B$ is $rx$, and the inverse of $a$ is $rfa+rga-a$. Bilinearity proves the endpoint, identity and associativity laws and compatibility with precomposition in $C$. In the hom-set formulation this does not assume that the composable-arrow pullback exists.

Let $f,g:A\rightrightarrows B$ be a reflexive pair in an additive category with finite limits, with common section $r:B\to A$. Regard $f$ as source and $g$ as target. For arrows $x,y:C\to A$ satisfying $gx=fy$, defineThe identity at $b:C\to B$ is $rb$, and the inverse of $x$ is $rfx+rgx-x$. These operations define an internal groupoid. The finite limits hypothesis supplies the objects of composable arrows needed for the groupoid diagrams.

The source of the composite is $fx+fy-gx=fx$, and its target is $gx+gy-gx=gy$. The common-section identities prove both identity laws and both inverse laws. For a composable triple $x,y,z$, either parenthesization gives $x+y+z-rgx-rgy$, proving associativity. Bilinearity makes these formulas compatible with precomposition in $C$.

The limit hypothesis cannot simply be discarded. Take the full subcategory of finite-dimensional vector spaces whose dimensions are $3a+5b$, for nonnegative integers $a,b$. A projection $k^5\to k^3$ has a common section with itself, but its composable-pair pullback in a category in vector spaces has dimension $7$, which is not of the allowed form. Testing the pullback in a category property on $k^3$ would force that same dimension, so the smaller category has no such pullback in a category. The reflexive-pair groupoid formula in a preadditive category still makes sense on generalized arrows without asserting existence of this object.