Yes, this is what foreign key constraints are for. They also allow for rules on how to handle deletes -- for example, allowing a delete to cascade through dependent records, which is of course very dangerous and not what you would want in this situation. A simple basic constraint will do.
HOWEVER!!!! This is the important thing to understand about mysql. The default mysql engine (myisam) has absolutely no support for foreign key constraints. You need to use an engine that supports them -- most commonly innodb.
If you specify a constraint when you're generating your DDL, a myisam table will accept the constraint but simply ignore it, so make sure all your related tables are setup/altered to be innodb tables before you add your constraint(s).
How do you add a constraint?
ALTER TABLE `event` ADD CONSTRAINT `category_event`
FOREIGN KEY (`category_id`) REFERENCES `category` (`category_id`);
In this example, it assumes your event table has the foreign key category_id in it, to create your linkage. After adding this constraint, if you attempt to delete a row from the category table, and an existing event row contains that key, mysql will disallow the DELETE and return an error. This is discussed in great detail here: http://dev.mysql.com/doc/refman/5.6/en/innodb-foreign-key-constraints.html