nLab
absolute coequalizer

Definition

An absolute coequalizer in a category CC is a coequalizer which is preserved by any functor F:CDF\colon 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

XffY1 YY X \; \underoverset{f}{f}{\rightrightarrows}\; Y \overset{1_Y}{\to} Y

whenever ff is not a split epimorphism.

In fact, split coequalizers and “trivial” absolute coequalizers are the cases n=1n=1 and 00 of a general characterization of absolute coequalizers, which we now describe. Suppose that

Xf 1f 0YeZ 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,):CSethom(Z,-)\colon C \to Set; that is, we have a coequalizer diagram

hom(Z,X)f 1f 0hom(Z,Y)ehom(Z,Z) hom(Z,X)\; \underoverset{f_1\circ -}{f_0\circ -}{\rightrightarrows}\; hom(Z,Y) \overset{e\circ -}{\to} hom(Z,Z)

in SetSet. In particular, that means that ee\circ - is surjective, and so in particular there exists some s:ZYs\colon Z\to Y such that es=1 Ze s = 1_Z. In other words, ee is split epic.

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

hom(Y,X)f 1f 0hom(Y,Y)ehom(Y,Z) hom(Y,X)\; \underoverset{f_1\circ -}{f_0\circ -}{\rightrightarrows}\; hom(Y,Y) \overset{e\circ -}{\to} hom(Y,Z)

in SetSet. We also have two elements 1 Y1_Y and ses e in hom(Y,Y)hom(Y,Y) with the property that e1 Y=e=esee \circ 1_Y = e = e \circ s e (since es=1 Ze s = 1_Z).

However, a coequalizer of two functions h 0,h 1:PQh_0,h_1\colon P\to Q in SetSet is constructed as the quotient set of QQ by the equivalence relation generated by the image of (h 0,h 1):PQ×Q(h_0,h_1)\colon P\to Q\times Q. That means that we set qqq\sim q' iff there is a finite sequence p 1,,p np_1,\dots, p_n of elements of PP and a finite sequence ε 0,,ε n\varepsilon_0,\dots,\varepsilon_n with ε i{0,1}\varepsilon_i\in\{0,1\}, such that h ε 1(p 1)=qh_{\varepsilon_1}(p_1)=q, h 1ε i(p i)=h ε i+1(p i+1)h_{1-\varepsilon_i}(p_i)=h_{\varepsilon_{i+1}}(p_{i+1}), and h 1ε n(p n)=qh_{1-\varepsilon_n}(p_n)=q'. We consider q=qq=q' as the case n=0n=0.

Therefore, since 1 Y1_Y and ses e are in the same class of the equivalence relation on hom(Y,Y)hom(Y,Y) generated by f 0f_0 and f 1f_1, they must be related by such a finite chain of elements of hom(Y,X)hom(Y,X). That is, we must have morphisms t 1,,t n:YXt_1,\dots, t_n\colon Y\to X and a sequence of binary digits ε 1,,ε n\varepsilon_1,\dots,\varepsilon_n such that f ε 1t 1=1 Bf_{\varepsilon_1} t_1=1_B, f 1ε it i=f ε it i+1f_{1-\varepsilon_i} t_i = f_{\varepsilon_i} t_{i+1}, and f 1ε nt n=sef_{1-\varepsilon_n}t_n=s e. (Note that if n=1n=1 then this says precisely that we have a split coequalizer, and if n=0n=0 it is the trivial case above.) Conversely, it is easy to check that given ss and t 1,,t nt_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.

Revised on May 2, 2012 22:44:03 by Mike Shulman (71.136.234.110)