# Idea

The Verity-Gray tensor product or lax Gray tensor product of stratified simplicial sets is a tensor product on the category $\mathrm{Strat}$ 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 $\left(X,tX\right)$ and $\left(Y,tY\right)$ be stratified simplicial sets. Then their Verity-Gray tensor product $\left(X,tX\right)\otimes \left(Y,tY\right)$ is given by

$\left(X,tX\right)\otimes \left(Y,tY\right):=\left(X×Y,q\left(tX,tY\right)\right)\phantom{\rule{thinmathspace}{0ex}},$(X, t X) \otimes (Y, t Y) := (X \times Y, q(t X, t Y)) \,,

where $X×Y$ is the cartesian product of simplicial sets (hence the standard monoidal structure on SSet), while $q\left(tX,tY\right)$, the set of thin cells, is …

# 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)
Created on April 29, 2009 15:43:58 by Urs Schreiber (134.100.222.156)