  We study Baire category for subsets of 2^omega that are downward-closed with
respect to the almost-inclusion ordering (on the power set of the natural
numbers, identified with 2^omega). We show that it behaves better in this
context than for general subsets of 2^omega. In the downward-closed context,
the ideal of meager sets is prime and b-complete (where b is the bounding
number), while the complementary filter is g-complete (where g is the groupwise
density cardinal). We also discuss other cardinal characteristics of this ideal
and this filter, and we show that analogous results for measure in place of
category are not provable in ZFC.
