Equivalences in diagrammatic sets

TL;DR

This paper introduces an internal notion of equivalence in diagrammatic sets, a topologically sound alternative to strict $$-categories, and proves key properties of these equivalences, advancing the understanding of weak invertibility.

Contribution

It develops an algebraic calculus for natural equivalences in diagrammatic sets, establishing their fundamental properties and differentiating this framework from strict $$-categories.

Findings

Equivalences include all degenerate cells.

Equivalences are closed under 2-out-of-3.

Enwrapping diagrams with equivalences is invertible up to higher equivalence.

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.