On $K_1$ of an assembler

TL;DR

This paper develops a method to construct generators and relations for the $K_1$ group of a simplicial Waldhausen category, with applications to $K_1$ of a simplicial assembler, advancing algebraic K-theory understanding.

Contribution

It introduces a new construction of generators and relations for $K_1$ in specific categories, providing tools for computing $K_1$ of assemblers.

Findings

Constructed generators and relations for $K_1$ of simplicial Waldhausen categories.

Applied these constructions to compute $K_1$ of simplicial assemblers.

Enhanced methods for algebraic K-theory calculations.

Abstract

This paper contains a construction of generators and partial relations for $K_1$ of a simplicial Waldhausen category where cofiber sequences split up to weak equivalence. The main application of these generators and relations is to produce generators for $K_1$ of a (simplicial) assembler.