Subject: categories: Re: limits of finite sets

From: "Tom Leinster" <[email-masked]>
Date: Sat, 31 Jul 2004 13:02:27 +0100 (BST)

I asked for an explanation of the following result:

> the limit in Set of any diagram
>
>    ... ---> S_3 ---> S_2 ---> S_1
>
> of finite nonempty sets is nonempty

Thanks very much to all who replied.  I'll summarize some of the points
made to me in private replies:

1. This is called Koenig's Lemma, and is usually stated in the form
   "any finitely-branching infinite tree contains an infinite
   (positively oriented) path".

2. This also follows from a general result in topology by regarding
   each S_n as a discrete space.  The general result is that any
   "suitably-shaped" limit of nonempty compact Hausdorff spaces is
   nonempty.  For Bourbaki (General Topology), "suitably-shaped" means
   indexed by a directed poset.  More generally still, it could be any
   componentwise cofiltered limit, i.e. any limit for which each
   connected-component of the indexing category I is cofiltered (or
   equivalently, every finite connected diagram in I admits a cone).

   The proof of the general topological result specializes to give a
   nice topological proof of Koenig.  For each n, let V_n be the
   subset of the product \prod_n S_n consisting of those sequences
   whose first n terms are compatible; then \lim_n S_n is the
   intersection of the (V_n)s.  But with the discrete topology on each
   S_n, Tychonoff says that \prod_n S_n is compact, and (V_n) is a
   nested sequence of nonempty closed subsets so has nonempty
   intersection.

Tom