Abstract rewriting internalized

TL;DR

This paper develops a unified internal rewriting theory within categories, generalizing classical and algebraic rewriting, and proves an analogue of Newman's Lemma applicable in this broad setting.

Contribution

It introduces an internal rewriting framework in categories satisfying mild conditions, unifying and extending classical and algebraic rewriting theories with new proofs.

Findings

Recovered classical rewriting results in Set and Vect categories

Proved an analogue of Newman's Lemma in the internal setting

Connected rewriting theory to decreasing diagrams

Abstract

In traditional rewriting theory, one studies a set of terms up to a set of rewriting relations. In algebraic rewriting, one instead studies a vector space of terms, up to a vector space of relations. Strikingly, although both theories are very similar, most results (such as Newman's Lemma) require different proofs in these two settings. In this paper, we develop rewriting theory internally to a category $\mathcal C$ satisfying some mild properties. In this general setting, we define the notions of termination, local confluence and confluence using the notion of reduction strategy, and prove an analogue of Newman's Lemma. In the case of $\mathcal C= \operatorname{Set}$ or $\mathcal C = \operatorname{Vect}$ we recover classical results of abstract and algebraic rewriting in a slightly more general form, closer to von Oostrom's notion of decreasing diagrams.