Equivalences in diagrammatic sets

Abstract

We show that diagrammatic sets, a topologically sound alternative to polygraphs and strict $\omega$-categories, admit an internal notion of equivalence in the sense of coinductive weak invertibility. We prove that equivalences have the expected properties: they include all degenerate cells, are closed under 2-out-of-3, and satisfy an appropriate version of the "division lemma", which ensures that enwrapping a diagram with equivalences at all sides is an invertible operation up to higher equivalence. On the way to this result, we develop methods, such as an algebraic calculus of natural equivalences, for handling the weak units and unitors which set this framework apart from strict $\omega$-categories.