Higher Dimensional Syntax

The aim of this paper is to present the notion of higher-dimensional syntax, which is a hierarchy of languages. Each term of a n-dimensional language will be typed by terms of the underlying n \Gamma 1-dimensional language. This is an application of the emerging higher-dimensional category notions. On the lower dimensions live notions of syntax which are already known. So dimension zero corresponds to words, dimension one corresponds to equationfree algebraic theories and dimension two corresponds to algebraic data types available in programming languages like ML and Haskell. In this paper I show that this hierarchy continues for higher dimensions and contributes new notions of syntax. As expected syntax comes with a structural induction principle which allows definition of several operation. The operations of substitution, unification and type inference have been investigated. Further the operation of recursive iteration schemes like generalised map and fold algorithms ha..