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

Multitopes are the same as principal ordered face structures