Cofinality via Weighted Colimits

TL;DR

This paper refines Quillen's Theorem A by establishing necessary and sufficient conditions for functors to be cofinal in the context of $$-categories, using a duality for weighted colimits, with applications to algebraic structures.

Contribution

It introduces a duality for weighted colimits that refines cofinality criteria and applies it to simplify formulas for free $_ abla$-algebras in stable rational $$-categories.

Findings

Provides necessary and sufficient conditions for cofinality in $$-categories.

Establishes a duality phenomenon for weighted colimits.

Simplifies formulas for free $_ abla$-algebras in specific contexts.

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 .