From: Steve Lack <stevel@maths.usyd.edu.au>
Date: Fri, 26 Jun 1998 19:03:16 +1000 (EST)
Subject: categories: ``cofinite sieves''

Have any topos-theorists or others come across the following
notion?

Let C be a small category, and c an object of C. If X is an 
arbitrary set of arrows with codomain c, then 
	
	R_X = { f:b-->c | there is no g:a-->b with fg in X}

clearly gives a sieve on c. Of course if X itself were a sieve
then R_X would be its complement, but I'm not assuming X is a
sieve. Say that a sieve R is _cofinite_ if it is of the form
R_X for a finite set X.

Cofinite sieves are closed under finite intersection and universal
quantification along an arbitrary arrow.

Best wishes,

Steve Lack.