How to Pivot in PostgreSQL
This is a tricky one. crosstab() expects one (or no) value per category for each row_name.

We can work around this restriction like this:

SELECT id
     , COALESCE(cat01, max(cat01) OVER w)
     , COALESCE(cat02, max(cat02) OVER w)
     , COALESCE(cat03, max(cat03) OVER w)
FROM   crosstab(
   'SELECT id::text || row_number() OVER (PARTITION BY id, category ORDER BY value) * -1 AS ext_id
         , id, category, value
    FROM   tbl
    ORDER  BY ext_id, category, value'

   ,$$VALUES ('Cat01'::text), ('Cat02'), ('Cat03')$$
   ) AS ct (xid text, id int, cat01 text, cat02 text, cat03 text)
WINDOW w AS (PARTITION BY id);


Returns your desired result.

How?


Add an extended id: ext_id from the existing id and a row number for each value of the category for the same id. This way we ensure as many rows per id in as there are values for the most common category. We get a derived table like this to build our crosstab() on:

ext_id   | id   | category | value
---------+------+----------+-------
'1234-1' | 1234 | 'Cat01'  | 'V001'
'1234-1' | 1234 | 'Cat02'  | 'V002'
'1234-1' | 1234 | 'Cat03'  | 'V003'
'1234-2' | 1234 | 'Cat03'  | 'V004'
'1234-3' | 1234 | 'Cat03'  | 'V005'

Now we can feed it to crosstab() using the safe 2-parameter form for missing attributes. Read the basics first if you are not familiar with this:


PostgreSQL Crosstab Query


The original id is carried over as "extra column". See:


Pivot on Multiple Columns using Tablefunc

Your question leaves room for interpretation. My solution pairs the lowest values per category first and keeps filling the following rows until there are no values left. (We could combine multiple values per category any other way, it has not been defined.) If a category is short of values for a given id, the rest is filled in with NULL values.  
In the final step I replace those NULL values with the maximum value of each category per id:

COALESCE(cat01, max(cat01) OVER (PARTITION BY id, category))


which is effectively the same as:

max(cat01) OVER (PARTITION BY id, category)


I am hoping to make it slightly faster if we only default to the window function if the value is NULL.