nLab
split coequalizer

Definition

For purposes of this page, a fork (some might say a “cofork”) in a category $C$ is a diagram of the form

$$A\underset{f}{\overset{g}{\rightrightarrows }}B\stackrel{e}{\to }C$$A \;\underoverset{f}{g}{\rightrightarrows}\; B \overset{e}{\rightarrow} C

such that $ef=eg$. A split coequalizer is a fork together with morphisms $s:C\to B$ and $t:B\to A$ such that $es=1_C$, $se=gt$, and $ft=1_B$. This is equivalent to saying that the morphism $(f,e):g\to e$ has a section in the arrow category of $C$.

Related concepts

Coequalizers and absolute coequalizers

The name “split coequalizer” is appropriate, because in any split coequalizer diagram, the morphism $e$ is necessarily a coequalizer of $f$ and $g$. For given any $h:B\to D$ such that $hf=hg$, the composite $hs$ provides a factorization of $h$ through $e$, since $hse=hgt=hft=h$, and such a factorization is unique since $e$ is (split) epic. In fact, a split coequalizer is not just a coequalizer but an absolute coequalizer: one preserved by all functors.

Contractible pairs

On the other hand, suppose we are given only $f,g:A\to B$ and $t:B\to A$ such that $ft=1_B$ and $gtf=gtg$ (which is certainly the case in any split coequalizer, since $gtf=sef=seg=gtg$). Such a situation is sometimes called a contractible pair. In this case, any coequalizer of $f$ and $g$ is split, for if $e:B\to C$ is a coequalizer of $f$ and $g$, then the equation $gtf=gtg$ implies, by the universal property of $e$, a unique morphism $s:C\to B$ such that $se=gt$, whence $ese=egt=eft=e$ and so $es=1_C$ since $e$ is epic.

Similarly, if $e:B\to C$ splits the idempotent $gt$ with section $s:C\to B$, so that $es=1$ and $se=gt$, then we have

$$eg=eseg=egtg=egtf=esef=ef$$e g = e s e g = e g t g = e g t f = e s e f = e f

and the other identities are obvious; thus $e$ is a split coequalizer of $f$ and $g$.

Split epimorphisms

Dually, if $e:B\to C$ is a split epimorphism, with a splitting $s:C\to B$, say, then $e$ is a split coequalizer of $B\underset{1}{\overset{se}{\rightrightarrows }}B$, the morphism $t$ being the identity.

Moreover, $e$ is also the split coequalizer of its kernel pair, if the latter exists. For if $A\underset{f}{\overset{g}{\rightrightarrows }}B$ is this kernel pair, then the two maps $se,1_B:B\to B$ satisfy $e\circ se=e\circ 1_B$, and hence induce a map $t:B\to A$ such that $ft=1_B$ and $gt=se$.

Examples

The “ur-example” of a split coequalizer is the following. Let $A$ be an algebra for the monad $T$ on the category $C$, with structure map $a:TA\to A$. Then the diagram

$$T^{2}A\underset{{\mu }_A}{\overset{Ta}{\rightrightarrows }}TA\stackrel{a}{\to }A,$$T^2 A \;\underoverset{\mu_A }{T a}{\rightrightarrows}\; T A \overset{a}{\rightarrow} A\, ,

called the canonical presentation of the algebra $(A,a)$, is a split coequalizer in $C$ (and a reflexive coequalizer in the category of $T$-algebras). The splittings are given by $s={\eta }_A:A\to TA$ and $t={\eta }_{TA}:TA\to T^{2}A$. (Here $\mu$ and $\eta$ are the multiplication and unit of the monad $T$.)

This split coequalizer figures prominently in Beck’s monadicity theorem.