= Construction of finite colimits from coproducts and reflexive coequalizers
For a finite <diagram in a category> $D:\mathcal J\to\mathcal C$, set $X=\coprod_jD(j)$ and $Y=\coprod_{a:i\to j}D(i)$. The two maps $s,t:Y\rightrightarrows X$ on the $a$-summand are the injection of $D(i)$ and the injection of $D(j)$ after $D(a)$. Their <coequalizer> is the <colimit> of $D$. The pair $[s,1_X],[t,1_X]:Y\amalg X\rightrightarrows X$ is a <reflexive pair> with the same coequalizer. Thus finite <coproduct in a category> constructions and coequalizers of reflexive pairs suffice, including the empty coproduct for the empty diagram. Finite products cannot replace coproducts here: the poset with elements $0,a,b,u_0,u_1,\ldots$, order $0<a,b<u_{n+1}<u_n$, and incomparable $a,b$ has finite meets and top $u_0$, hence finite categorical products. Every reflexive parallel pair is an equal pair and has its identity as coequalizer, but $a,b$ have no least upper bound and therefore no coproduct.
Back to article page
