Diagrammatic sets between rewriting and topology
String rewriting, diagram rewriting, presentations of higher algebras,
and more can be put under the umbrella of “higher-dimensional
rewriting theory”. This field has some intriguing connections with
combinatorial topology, which are not fully realised when rewrite
systems are modelled, as commonly done, by strict higher categories. I
will show how following these connections can lead to an alternative
framework, immune from many technical problems: diagrammatic sets.
Then, I will discuss how one can build a model of weak higher
categories on top of diagrammatic sets, to function as a semantics of
higher-dimensional rewrite systems. This may provide a link between
mainstream models from categorical homotopy theory and diagrammatic
approaches popular in Oxford. This talk is based on arXiv:1909.07639.