Cofinality via Weighted Colimits

We prove a refinement of Quillen’s Theorem A, providing necessary and sufficient conditions for a functor to be cofinal with respect to diagrams valued in a fixed $\mathrm{\infty }$-category. We deduce this from a general duality phenomenon for weighted colimits, which is of independent interest. As a sample application, due to Betts and Dan-Cohen, we describe a simplified formula for the free ${\mathbb{E}}_{\mathrm{\infty }}$-algebra on an ${\mathbb{E}}_0$-algebra in a stable rational $\mathrm{\infty }$-category .