# Equational reasoning with 2-dimensional diagrams

## Abstract

The significance of the 2-dimensional calculus, which goes back to Penrose, has already been pointed out by Joyal and Street. Independently, Burroni has introduced a general notion of *n*-dimensional presentation and he has shown that the equational logic of terms is a special case of 2-dimensional calculus.

Here, we propose a combinatorial definition of 2-dimensional diagrams and a simple method for proving that certain monoidal categories are finitely 2-presentable. In particular, we consider Burroni's presentation of finite maps and we extend it to the case of finite relations.

This paper should serve as a reference for our future work on symbolic computation, including a theory of 2-dimensional rewriting and the design of software for interactive diagrammatic reasoning.

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