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