categories: charact. of absolute equalisers

[Martin Hofmann <mhofmann@mathematik.tu-darmstadt.de>]

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