nLab Verity-Gray tensor product

Idea

The Verity-Gray tensor product or lax Gray tensor product of stratified simplicial sets is a tensor product on the category $Strat$ of stratified simplicial sets which when restricted to complicial sets, i.e. omega-nerves of strict omega-categories reproduces the Crans-Gray tensor product on strict $\omega$-categories.

Definition

Let $(X, t X)$ and $(Y, t Y)$ be stratified simplicial sets. Then their Verity-Gray tensor product $(X, t X) \otimes (Y, t Y)$ is given by

$$(X, t X) \otimes (Y, t Y) := (X \times Y, q(t X, t Y)) \,,$$

where $X \times Y$ is the cartesian product of simplicial sets (hence the standard monoidal structure on SSet), while $q(t X, t Y)$, the set of thin cells, is $tX\times tY$ for the Gray product, and for the lax-Gray product is enlarged as described in the paper.

References

definition 128 of

• Dominic Verity, Complicial sets (arXiv)

definition 59, page 32 of

• Dominic Verity, Weak complicial sets I (arXiv)

slide 60 of

• Dominic Verity, Weak complicial sets and internal quasi-categories (arXiv)