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In the book of Gaitsgory and Rozenblyum Ch A.1 Basics of $(\infty,2)$-categories, 3.2, they define the Gray tensor product of $(\infty,2)$-categories. Here are my questions:

  1. What is an explicit description of $\mathrm{Seq}_\bullet ([m]\otimes [n])$, where $\otimes$ is the Gray product, and $\mathrm{Seq}_\bullet: \text{2-Cat}\rightarrow(\text{1-Cat})^{\Delta^{op}}$ is their model of $(\infty,2)$-categories?

  2. In Lurie's paper $(\infty,2)$-Categories and the Goodwillie Calculus I, he defines several models for the homotopy theory of $(\infty,2)$-categories. I was wondering whether it is possible to define Gray product on these model categories, e.g. $\mathrm{Set}_\Delta^{\mathrm{sc}}$, $\mathrm{Fun}(\Delta^{op}, \mathrm{Set}_\Delta^+)$, and whether there are explicit definitions?

Thanks in advance!

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1 Answer 1

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For question (2), there is actually a left Quillen bifunctor $$ \times_{\mathrm{gr}}: \mathrm{Set}_\Delta^{\mathrm{sc}} \times \mathrm{Set}_\Delta^{\mathrm{sc}} \to \mathrm{Set}_\Delta^{\mathrm{sc}} $$ which models the Gray tensor product. If $(X,T),(Y,S)$ are scaled simplicial sets then $(X,T) \times_{\mathrm{gr}} (Y,S) = (X \times Y,R)$ where the set $R$ of thin triangles consists of those triangles $\sigma: \Delta^2 \to X \times Y$ such that:

1) The image of $\sigma$ in both $X$ and $Y$ is thin.

2) Either the images of $\sigma$ and $\sigma|_{\Delta^{\{1,2\}}}$ in $X$ are degenerate or the images of $\sigma$ and $\sigma|_{\Delta^{\{0,1\}}}$ in $Y$ are degenerate.

For example, $\Delta^1 \times_{\mathrm{gr}} \Delta^1$ is a square in which one of the two triangles is thin and the other is not. This data is essentially the same as a square which commutes up to a non-invertible $2$-cell. Note however that even if $(X,T)$ and $(Y,S)$ are both fibrant then $(X,T) \times_{\mathrm{gr}} (Y,S)$ will generally not be fibrant (because it can have invertible $2$-cells which are not designated as thin).

This construction features prominently in a forthcoming preprint of mine concerning limits and colimits in $(\infty,2)$-categories, which I will hopefully put on the arXiv soon.

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