nLab
absolute coequalizer

Definition

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

Characterization

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

$$X\underset{f}{\overset{f}{\rightrightarrows }}Y\stackrel{1_Y}{\to }Y$$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

$$X\underset{f_1}{\overset{f_0}{\rightrightarrows }}Y\stackrel{e}{\to }Z$$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 $\mathrm{hom}(Z,-):C\to \mathrm{Set}$; that is, we have a coequalizer diagram

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

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

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

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

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

However, a coequalizer of two functions $h_0,h_1:P\to Q$ in $\mathrm{Set}$ is constructed as the quotient set of $Q$ by the equivalence relation generated by the image of $(h_0,h_1):P\to Q\times Q$. That means that we set $q\sim q\prime$ iff there is a finite sequence $p_1,\dots ,p_n$ of elements of $P$ and a finite sequence ${\epsilon }_0,\dots ,{\epsilon }_n$ with ${\epsilon }_i\in \{0,1\}$, such that $h_{{\epsilon }_1}(p_1)=q$, $h_{1-{\epsilon }_i}(p_i)=h_{{\epsilon }_{i+1}}(p_{i+1})$, and $h_{1-{\epsilon }_n}(p_n)=q\prime$. We consider $q=q\prime$ as the case $n=0$.

Therefore, since $1_Y$ and $se$ are in the same class of the equivalence relation on $\mathrm{hom}(Y,Y)$ generated by $f_0$ and $f_1$, they must be related by such a finite chain of elements of $\mathrm{hom}(Y,X)$. That is, we must have morphisms $t_1,\dots ,t_n:Y\to X$ and a sequence of binary digits ${\epsilon }_1,\dots ,{\epsilon }_n$ such that $f_{{\epsilon }_1}{t}_1=1_B$, $f_{1-{\epsilon }_i}{t}_i=f_{{\epsilon }_i}{t}_{i+1}$, and $f_{1-{\epsilon }_n}{t}_n=se$. (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).

References

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