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. to 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

$$(X, t X) \otimes (Y, t Y) \coloneqq \big(X \times Y, q(t X, t Y)\big) \,,$$

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

Related entries

• Crans-Gray tensor product

• generalized Gray tensor product

References

• Dominic Verity, §7.2 & §11.4 in: Complicial Sets Characterising the Simplicial Nerves of Strict $\omega$-Categories, Memoirs of the AMS, 193 905 (2008) [arXiv:math/0410412, ams:memo-193-905]

• Dominic Verity, def 59 (p. 32) of: Weak complicial sets I: Basic homotopy theory, Advances in Mathematics 219 4 (2008) 1081-1149 [arXiv:math/0604414, doi:10.1016/j.aim.2008.06.003]

• Dominic Verity, around slide 60 of:: Weak complicial sets and internal quasi-categories, talk at Category Theory 2007 [pdf, pdf]