A regular polygraph is one whose cells have k-dimensional boundaries “shaped as k-dimensional balls”, via a suitable geometric realisation. I will describe a combinatorial approach, where regular polygraphs are defined as presheaves on a shape category, and subsequently proved to form a full subcategory of the category of polygraphs. I will also show how several operations and constructions on polygraphs – such as lax Gray products and joins – admit a sleek definition in this setting.

I will then give a non-algebraic, fully weak definition of higher category, as a regular polygraph satisfying a representability property, and sketch a complementary algebraic, semi-strict definition. Finally, I will sketch how the two are combined in a semi-strictification construction, where semi-strictness should be read in the sense of “Simpson's conjecture for regular compositions”, as in the earlier seminar entry by Simon Henry.

The programme in the second part of the talk has been fully developed in dimension 2 (arXiv:1803.06086). The first part is the subject of a paper that I will publish or circulate before the talk.