Cofinality via Weighted Colimits

Abstract

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 $\infty$-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 ${\mathbb{E}}_{\infty }$-algebra on an ${\mathbb{E}}_0$-algebra in a stable rational $\infty$-category .

13 pages, omments welcome! v2 contains only minor revisions