[This is a joint project with Tim Campion and Chris Kapulkin.]
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Some (n+1)-cubes in a cubical ω-category witness equalities between n-cubes while others are "genuine" (n+1)-dimensional morphisms. It has been shown by Steiner that, if we "mark" the (n+1)-cubes of the former kind, then the ω-category structure can be recovered from the underlying marked cubical set. In particular, the composition operations correspond to certain open boxes admitting unique marked fillers. One would expect dropping the uniqueness condition (and adding other suitable conditions) to lead to a model for weak ω-categories (aka (∞,∞)-categories), and our project aims to establish various expected properties of this model. Our emphasis is on the (lax
and pseudo) Gray tensor products and how they relate to Verity's complicial model.
A Cubical Model For Weak Ω-Categories