  We define a family of structures called "opetopic algebras", which are
algebraic structures with an underlying opetopic set. Examples of such are
categories, planar operads, and Loday's combinads over planar trees. Opetopic
algebras can be defined in two ways, either as the algebras of a "free pasting
diagram" parametric right adjoint monad, or as models of a small projective
sketch over the category of opetopes. We define an opetopic nerve functor that
fully embeds each category of opetopic algebras into the category of opetopic
sets. In particular, we obtain fully faithful opetopic nerve functors for
categories and for planar coloured Set-operads.
  This paper is the first in a series aimed at using opetopic spaces as models
for higher algebraic structures.
