Cofinality Theorems of Infinity Categories and Algebraic K-Theory

TL;DR

This paper proves a general cofinality theorem for infinity categories, showing when inclusion morphisms are weak homotopy equivalences, and applies it to algebraic K-theory, providing new proofs and insights.

Contribution

It introduces a new cofinality theorem for infinity categories and offers an alternative proof of Barwick's cofinality theorem in algebraic K-theory.

Findings

Cofinal full inclusion functors of ()-categories are weak homotopy equivalences

Established a criterion for when an inclusion morphism between simplicial sets is a weak homotopy equivalence

Provided a new proof of Barwick's cofinality theorem in algebraic K-theory

Abstract

In this paper, we establish a theorem that proves a condition when an inclusion morphism between simplicial sets becomes a weak homotopy equivalence. Additionally, we present two applications of this result. The first application demonstrates that cofinal full inclusion functors of (\infty)-categories are weak homotopy equivalences. For our second application, we provide an alternative proof of Barwick's cofinality theorem of algebraic (K)-theory.