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 $\infty$-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 .

💡 Research Summary

The paper “Cofinality via Weighted Colimits” by Shai Keidar and Lior Yanovski presents a refined version of Quillen’s Theorem A that characterizes when a functor is cofinal with respect to diagrams valued in a fixed ∞‑category C. The authors introduce the notion of C‑cofinality: a functor f : J → I is C‑cofinal if for every I‑shaped diagram X : I → C the canonical comparison map \