On weak higher dimensional categories I: Part 1

Inspired by the concept of opetopic set introduced in a recent paper by John C. Baez and James Dolan, we give a modified notion called multitopic set. The name reflects the fact that, whereas the Baez/Dolan concept is based on operads, the one in this paper is based on multicategories. The concept of multicategory used here is a mild generalization of the same-named notion introduced by Joachim Lambek in 1969. Opetopic sets and multitopic sets are both intended as vehicles for concepts of weak higher dimensional category. Baez and Dolan define weak n-categories as (n+1)-dimensional opetopic sets satisfying certain properties. The version intended here, multitopic n-category, is similarly related to multitopic sets. Multitopic n-categories are not described in the present paper; they are to follow in a sequel. The present paper gives complete details of the definitions and basic properties of the concepts involved with multitopic sets. The category of multitopes, analogs of opetopes of Baez and Dolan, is presented in full, and it is shown that the category of multitopic sets is equivalent to the category of set-valued functors on the category of multitopes.