The study of universal algebra, that is, the description of algebraic structures by means of symbolic expressions subject to equations, dates back to the end of the 19th century. It was motivated by the large number of fundamental mathematical structures fitting into this framework: groups, rings, lattices, and so on. From the 1970s on, the algorithmic aspect became prominent and led to the notion of term rewriting system. This chapter briefly revisits these ideas from a polygraphic viewpoint, introducing only what is strictly necessary for understanding. Term rewriting systems are introduced as presentations of Lawvere theories, which are particular cartesian categories. It is shown that a term rewriting system can also be described by a 3-polygraph in which variables are handled explicitly, i.e., by taking into account their duplication and erasure. Finally, a precise meaning is given to the statement that term rewriting systems are "cartesian polygraphs".

This chapter establishes 3-polygraphs as a notion of presentation for 2-categories. As expected, those consist in generators for 0-, 1- and 2-dimensional cells, together with relations between freely generated 2-cells, which are represented by generating 3-cells. Any 3-polygraph induces an abstract rewriting system, so that all associated general rewriting concepts make sense in this setting: confluence, termination, etc. However, more specific tools have to be adapted to this context: the notion of critical branching is defined here for 3-polygraphs, along with the proof that confluence of critical branchings implies the local confluence of the polygraph. In the case where the polygraph is terminating, local confluence implies confluence, providing a systematic method to show the convergence of a 3-polygraph. When this is the case, normal forms give canonical representatives for 2-cells modulo the congruence generated by 3-cells, and it is explained how to exploit this to show that a given 3-polygraph is a presentation of a given 2-category.

This appendix gives some examples of presentations of 2-categories by 3-polygraphs. In many examples, the presented 2-categories are in fact monoidal categories and, actually, PROs.