[{"name":"item_1","category":["a","b"]},{"name":"item_2","category":["c"]},{"name":"item_3","category":["a","c"]},{"name":"item_4","category":["a"]},{"name":"item_5","category":["a"]}]

db.collection.aggregate({"$unwind":"$category"},{"$group":{"_id":"$category",count:{$sum:1}}})