On The Combinatorics of the Gray Cylinder

TL;DR

This paper explores the combinatorial properties of the lax Gray cylinder in strict omega-categories, establishing relationships with other categorical constructs through span constructions.

Contribution

It introduces new combinatorial insights into the Gray cylinder and relates it to the Cartesian cylinder and shift functor via span constructions.

Findings

Established combinatorial properties of the lax Gray cylinder.

Constructed a span relating Gray cylinder, Cartesian cylinder, and shift functor.

Enhanced understanding of the structure of strict omega-categories.

Abstract

This paper develops some combinatorics of the lax Gray cylinder on the cells of {\Theta} understood as a full subcategory of the category of strict {\omega}-categories. More, we construct a span relating the Cartesian cylinder, the Gray cylinder, and the shift functor.