An $(\infty,2)$-categorical pasting theorem

TL;DR

This paper proves that in any $( abla,2)$-category, pasting diagrams have unique composites up to homotopy, by modeling the free 2-category as a homotopy colimit and providing explicit calculations in simplicial categories.

Contribution

It establishes a homotopically unique composition for pasting diagrams in $( abla,2)$-categories and demonstrates this via explicit simplicial category models.

Findings

Homotopically unique composites for pasting diagrams in $( abla,2)$-categories.

Explicit calculation of free 2-categories as homotopy colimits in simplicial models.

Model-independent proof derived from simplicial category calculations.

Abstract

We show that any pasting diagram in any $(\infty,2)$-category has a homotopically unique composite. This is achieved by showing that the free 2-category generated by a pasting scheme is the homotopy colimit of its cells as an $(\infty,2)$-category. We prove this explicitly in the simplicial categories model and then explain how to deduce the model-independent statement from that calculation.