From: Tom Leinster <leinster@dpmms.cam.ac.removethisbit.uk>
Subject: Re: Category construction question
Date: 9 Aug 2000 12:30:03 -0500
Newsgroups: sci.math.research
Summary: [missing]

Bill Halchin's question is all about finding a left adjoint to the inclusion

  G: (posets) ----> (pre-ordered sets).

In other words, given a pre-ordered set, what is the "universal" way of
turning it into a genuine ordered set?  The answer is: if P is a pre-ordered
set, define F(P) to be the poset whose elements are equivalence classes of
elements of P, where two elements p, p' of P are declared equivalent if p <=
p' <= p.  The functor F defined in this way is left adjoint to G.

Another way of expressing the fact that G has a left adjoint is to say that
for each pre-ordered set P, the category constructed by Bill Halchin (in his
Prop I) has an initial object. The initial object is, in fact, the quotient
map

  P ----> GF(P).

This equivalent formulation of adjointness is laid out in section V.6 of
`Categories for the Working Mathematician' (Mac Lane); Halchin's category is
called (P \downarrow G) there.  

As for (P \downarrow G) having a terminal (final) object, I think that's
trivial: it's just

  P ----> 1,

where 1 is the one-element ordered set.


Tom Leinster


Bill Halchin wrote: 

% Hello,
% 
% In the commutative diagram below,
% 
% - P is a fixed(!) pre-order
% 
% - Q and R are posets
% 
% - f and g are order-preserving functions from
%       a pre-order to a poset.
% 
% - h  : Q -> R is a poset order-preserving function,
%         i.e. a poset homomorphism such that
% 
%         f = h . g,
% 
%         where '.' is the composition operator
% 
% 
% 
% 
%                     g
%     (P, <=) ---------------------> (Q, <=)
%      \                           |
%        \                         |
%          \                       |
%            \                     |
%              \                   |
%                \                 | h
%                  \               |
%           f        \             |
%                      \           |
%                        \         |
%                          \       |
%                            \     |
%                              \   |
%                                \ v
%                                 (R, <=)
% 
% 
% 
% 
% Proposition I: The following construction is a category:
% 
%        objects: all order-preserving functions from
%                   the pre-order (P, <=) to posets
% 
%        morphisms: poset homomorphisms that lead to the
%                      commutative diagram above
% 
% 
% 
% 
% Question: If my Proposition I is correct, then what
%              properties does an initial object have
%              if it exists? What properties would a
%              final object have?
% 
% 
% Regards,
% 
% Bill Halchin