  The well-known conditions for a simplicial set to be the nerve of a small
category generalize with respect to two parameters: the dimension n of the
things which compose, and the position i of the thing which is the result of
the composition. In the nerve of a small category, the dimension of the things
which compose (maps) is n=1. Compositions are 2-simplices (commutative
triangles) in which the position of the composite is the 1-simplex opposite
vertex i=1. These conditions generalize to all n>1 and i in {0 , ... , n+1}. We
call such a simplicial set an (n,i)-composer. This paper explores two aspects
of composers: models and the algebra of such generalized composition.
  Models: we develop a family of set-based models for composers analogous to
sets-and-functions models for ordinary categories. The key to obtaining
set-based models of (n,i)-composers is to observe that in the nerve of a
category of sets-and-functions, a 1-simplex is a binary relation with certain
properties on its two faces, that is, its codomain and domain. In a model of an
(n,i)-composer, each k-simplex is a (k+1)-ary relation on its faces having
certain properties, with special attention to k=n (things which compose) and
k=n+1 (a composition of things). The development of these models is guided by
requiring: (i) the usual sets-and-functions model, i.e. when n=1 and i=1,
belongs to the family; (ii) the things which compose have properties which
resemble and generalize properties possessed by ordinary functions.
  Algebra: we explore some notions which are inspired by the algebra of
ordinary composition. These include "comma-composers" generalizing comma
categories, a composer structure for function complexes, representables, and a
generalization of universal map. In its current form, this part of the paper is
provisional in content and organization.
