Another approach is to use the first_value window function: http://sqlfiddle.com/#!12/7a145/14
SELECT DISTINCT
first_value("id") OVER (PARTITION BY "category" ORDER BY "date" DESC)
FROM Table1
ORDER BY 1;
... though I suspect hims056's suggestion will typically perform better where appropriate indexes are present.
A third solution is:
SELECT
id
FROM (
SELECT
id,
row_number() OVER (PARTITION BY "category" ORDER BY "date" DESC) AS rownum
FROM Table1
) x
WHERE rownum = 1;