Multitopes are the same as principal ordered face structures

We show that the category of principal ordered face structures is equivalent to the category of multitopes. We show that the category of principal ordered face structures is equivalent to the category of multitopes. On the way we introduce the notion of a graded tensor theory to state the abstract properties of the category of ordered face structures and show how it fits into the recent work of T. Leinster and M. Weber concerning the nerve construction.