categories: Re: charact. of absolute equalisers

[pare@mscs.cs.dal.ca (Robert Pare)]

Dear Martin,

These were studied in my PhD thesis "Absoluteness Properties in Category Theory"
McGill 1969. See "Absolute Coequalizers" Springer Lecture Notes 86 (1969), 
132-145, and "On Absolute Colimits" J. Alg. 19 (1971) 80-95.

The proof I give shows that for colimits it is sufficient to test
preservation by the Yoneda functor (or all representables). So while the
Yoneda embedding preserves all limits, it preserves no colimits execpt
those it absolutely has to.

It's interesting that you came up with the same name for them.

Bob
> 
> Dear categorists,
> 
> I wonder whether the following result is known:
> 
> Call an equaliser in a category C *absolute* if it is preserved by all
> functors. 
> 
> Proposition: An equaliser e:A->B, u,v:B->C is absolute if and only if
> there are maps p:B->A, h1,h2,...hn:C->B such that 
> 
> pe=id, 
> ep=h1 u
> h1 u = h2 v
> h2 u = h3 v
> ...
> hn v = id
> 
> Proof: An equaliser endowed with such maps is obviously preserved by
> any functor since whenever we have maps e,u,v,p,h1...hn such that
> ue=ve together with the equations listed above then e equalises u and v.
> 
> For the converse consider the functor which maps X to C(B,X)/~ where ~
> is the left congruence generated by u~v, i.e., f~g iff there are h1...hn
> such that f=h1 u h1 v = h2 u ...  hn v = g
> 
> If this functor preserves the equaliser then, since Fu([id]) =
> Fv([id]) we obtain p:B->A such that ep~id. Thus we have maps h1..hn
> with the desired properties. To show pe=id we calculate as follows:
> epe=h1 u e = h1 v e = h2 u e = ... = hn v e = e, so pe=id since e is a
> mono.
> 
> Best wishes, 
>  Martin Hofmann
> 
>