An $(\infty,n)$-categorical pasting theorem

Abstract

We identify a reasonably large class of pushouts of strict $n$-categories which are preserved by the "inclusion" functor from strict $n$-categories to weak $(\infty,n)$-categories. These include the pushouts used to assemble from its generating cells any object of Joyal's category $\Theta$, any of Street's orientals, any lax Gray cube, and more generally any "regular directed CW complex." More precisely, the theorem applies to any \emph{torsion-free complex} in the sense of Forest -- a corrected version of Street's \emph{parity complexes}. This result may be regarded as partial progress toward Henry's conjecture that the pushouts assembling any non-unital computad are similarly preserved by the "inclusion" into weak $(\infty,n)$-categories. In future work we shall apply this result to give new models of $(\infty,n)$-categories as presheaves on torsion-free complexes, and to construct the Gray tensor product of weak $(\infty,n)$-categories. This result is deduced from an \emph{$(\infty,n)$-categorical pasting theorem}, in the spirit of Power's 2-categorical and $n$-categorical pasting theorems, and the $(\infty,2)$-categorical pasting theorems of Columbus and of Hackney, Ozornova, Riehl, and Rovelli. This says that, when assembling a "pasting diagram" from its generating cells, the space of "composite cells which can be pasted together from all of the generators" is contractible.