select two record containing dates in it with two input dates
Simple answer:

If you want both date columns to be between the input dates:

SELECT category 
FROM tablename 
WHERE 'start_date' BETWEEN start_date AND end_date
  AND 'end_date' BETWEEN start_date AND end_date ;


Or if you want any of the date columns to be between the input dates:

SELECT category 
FROM tablename 
WHERE 'start_date' BETWEEN start_date AND end_date
   OR 'end_date' BETWEEN start_date AND end_date ;


Or if you want any of the ranges that are "overlapping with the range of the input dates
(seems the most likely, based on the wanted result):

SELECT category 
FROM tablename 
WHERE 'start_date' &lt;= end_date
  AND start_date &lt;= 'end_date' ;