Include parent node ID in hierarchical recordset of child rows
It's hard to understand your requirements without seeing the definition of category - Presumably, there is a root_id column in that table?

If there is, perhaps you want this?

;WITH level_1 (category_id, category_name, root_id) AS (
    SELECT category_id, 
        category_name,
        root_id
    FROM dbo.category
    WHERE parent_category_id = 1
    UNION ALL
    SELECT c.category_id,
        category_name,
        root_id
    FROM dbo.category AS c 
    CROSS JOIN level_1 
    WHERE c.parent_category_id = level_1.category_id
)
SELECT category_id
    , category_name
    , root_id 
FROM level_1;


Just FYI, it's good practice to begin a CTE with a semicolon, to specify the schema, and avoid using SELECT *