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

10-Aug-2001 20:51:12 -0300,1517;000000000000-0000000a