nLab 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 StratStrat 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 ω\omega-categories.


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

(X,tX)(Y,tY):=(X×Y,q(tX,tY)), (X, t X) \otimes (Y, t Y) := (X \times Y, q(t X, t Y)) \,,

where X×YX \times Y is the cartesian product of simplicial sets (hence the standard monoidal structure on SSet), while q(tX,tY)q(t X, t Y), the set of thin cells, is tX×tYtX\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)

Last revised on November 11, 2014 at 07:29:42. See the history of this page for a list of all contributions to it.