question on finiteness in toposes

[categories <cat-dist@mta.ca>]

Date: Fri, 10 Jan 1997 12:57:02 MEZ
From: Thomas Streicher <streicher@mathematik.th-darmstadt.de>

One knows that for any topos E that the full subcategory of decidable K-finite 
objects forms a topos itself with 2 = 1+1 as subobject classifier.
It is also said that E_kf, the full subcat of E on K-finite objects need not
form a topos. That's what I could find out from PTJ's Topos Theory. 
The counterexample given there is E = Set^2 (where 2 = 0 -> 1). The K-finite
objects in Set^2 are the surjective maps between finite sets. It is clear that
E_kf is not closed under equalisers taken in E (!). Nevertheless, I think that
E_kf itself does have equalisers : if f,g : X -> Y then take the equaliser 
e_0 of f_0 and g_0 and take the epi-mono-factorisation of x o e_0 :

     E_0 >---> X_0
      |         |
      |  epi    | x        where  X = X_0 -> X_1
      V         V
     E_1 >---> X_1 

this clearly demonstrates that the inclusion  E_kf >--> E does not preserve 
equalisers BUT it does not show that E_kf is not a topos.
I would be interested in a reference or example where E_kf really is not a 
topos. Maybe, E = Set^2 alraedy works but it must have another defect than 
not being clossed under subobjects w.r.t. E because the decidable K-finite
objects have this "defect" as well.
Thomas Streicher