Subject: factorisation of generalised geometric morphisms

From: categories <[email-masked]>
Date: Wed, 10 Jan 1996 14:30:32 -0400 (AST)

Date: Wed, 10 Jan 1996 13:55:18 MEZ
From: Thomas Streicher <[email-masked]>

I have the following question about factorisation of 
partial geometric morphisms (i.e. pullback preserving functors having right
adjoints).
It is well known that a pullback preserving functor F : A -> B between lex
categories factors as

         A ---> B/F1 ---> B   (where F_1 X = F(X->1) ) 
           F_1    Sigma_F1

where, of course, F_1 is lex. Moreover F_1 has right adjoint eta_1* o U_F1
provided F -| U.

Now my question is whether one can obtain something similar if A and B are
PARTIAL LEX, i.e. have pullbacks and binary products but not a terminal object.
In that case we can perform for every I in A the same factorisation as above

          A/I ---> B/FI ---> B   (where F_I(x:X->I) = F(x) )
              F_I    Sigma_FI

What I wonder is whether there exists a maximal factorisation of F as

         A -> C -> B
           F'   S

s.t. 

   F' preserves pullbacks and binary products and 
   S preserves pullbacks and has a right adjoint T 

and

   F' has a right adjoint if F has a right adjoint .

I have a tentative answer to that : namely take for C the pseudo-colimit of
B/F(-) : A -> Cat. 
Then everything works but the requirement that S has a right adjoint.

I'd be grateful for any pointers to literature, folklore results or ideas.

Thomas Streicher