Verity-Gray tensor product

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.

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.

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)

Revised on November 11, 2014 07:29:42
by Tim Porter
(2.31.52.236)