Can I search on, for example, 4 categories?

When you use the term ‘categories’, do you mean you have four parent categories each with their heirarchy and children, and do you mean have four separate controls? If so, then no, you must instead make separate custom taxonomies (use a free plugin called CPT UI) for each parent. You can use a free plugin called Taxonomy switcher to move the categories to the new Taxonomies, where you will then not really need the parent, but first you would have to edit each child to make it have no parent.