absolute coequalizer

An **absolute coequalizer** in a category $C$ is a coequalizer which is preserved by *any* functor $F\colon C \to D$. This is a special case of an absolute colimit.

Intuitively, an absolute coequalizer is a diagram that is a coequalizer “purely for diagrammatic reasons.” The most common example is a split coequalizer. A trivial example of an absolute coequalizer that is not split is a diagram of the form

$X \; \underoverset{f}{f}{\rightrightarrows}\; Y \overset{1_Y}{\to} Y$

whenever $f$ is not a split epimorphism.

In fact, split coequalizers and “trivial” absolute coequalizers are the cases $n=1$ and $0$ of a general characterization of absolute coequalizers, which we now describe. Suppose that

$X\; \underoverset{f_1}{f_0}{\rightrightarrows}\; Y \overset{e}{\to} Z$

is an absolute coequalizer. Then it must be preserved, in particular, by the hom-functor $hom(Z,-)\colon C \to Set$; that is, we have a coequalizer diagram

$hom(Z,X)\; \underoverset{f_1\circ -}{f_0\circ -}{\rightrightarrows}\; hom(Z,Y) \overset{e\circ -}{\to} hom(Z,Z)$

in $Set$. In particular, that means that $e\circ -$ is surjective, and so in particular there exists some $s\colon Z\to Y$ such that $e s = 1_Z$. In other words, $e$ is split epic.

Now the given coequalizer must also be preserved by the hom-functor $hom(Y,-)$, so we have another coequalizer diagram

$hom(Y,X)\; \underoverset{f_1\circ -}{f_0\circ -}{\rightrightarrows}\; hom(Y,Y) \overset{e\circ -}{\to} hom(Y,Z)$

in $Set$. We also have two elements $1_Y$ and $s e$ in $hom(Y,Y)$ with the property that $e \circ 1_Y = e = e \circ s e$ (since $e s = 1_Z$).

However, a coequalizer of two functions $h_0,h_1\colon P\to Q$ in $Set$ is constructed as the quotient set of $Q$ by the equivalence relation generated by the image of $(h_0,h_1)\colon P\to Q\times Q$. That means that we set $q\sim q'$ iff there is a finite sequence $p_1,\dots, p_n$ of elements of $P$ and a finite sequence $\varepsilon_0,\dots,\varepsilon_n$ with $\varepsilon_i\in\{0,1\}$, such that $h_{\varepsilon_1}(p_1)=q$, $h_{1-\varepsilon_i}(p_i)=h_{\varepsilon_{i+1}}(p_{i+1})$, and $h_{1-\varepsilon_n}(p_n)=q'$. We consider $q=q'$ as the case $n=0$.

Therefore, since $1_Y$ and $s e$ are in the same class of the equivalence relation on $hom(Y,Y)$ generated by $f_0$ and $f_1$, they must be related by such a finite chain of elements of $hom(Y,X)$. That is, we must have morphisms $t_1,\dots, t_n\colon Y\to X$ and a sequence of binary digits $\varepsilon_1,\dots,\varepsilon_n$ such that $f_{\varepsilon_1} t_1=1_Y$, $f_{1-\varepsilon_i} t_i = f_{\varepsilon_i} t_{i+1}$, and $f_{1-\varepsilon_n}t_n=s e$. (Note that if $n=1$ then this says precisely that we have a split coequalizer, and if $n=0$ it is the trivial case above.) Conversely, it is easy to check that given $s$ and $t_1,\dots, t_n$ satisfying these equations, the given fork must be a coequalizer, for essentially the same reason that any split coequalizer is a coequalizer. Thus we have a complete characterization of absolute coequalizers.

This characterization is essentially a special case of the characterization of absolute colimits (in unenriched categories).

- Robert Pare,
*Absolute coequalizers*, Lecture Notes in Math. 86 (1969), 132-145.