Dennis trace map for certain $K$-groups of categories with cofibrations

TL;DR

This paper introduces a new framework for defining K-theory and Hochschild homology groups for categories with cofibrations using ordered finite simplicial sets, and constructs the Dennis trace map between them.

Contribution

It defines K-theory and Hochschild homology for categories with cofibrations based on simplicial sets and constructs the Dennis trace map connecting these groups.

Findings

Defined K-theory and Hochschild homology for categories with cofibrations.

Constructed the Dennis trace map between these groups.

Established a new approach using ordered finite simplicial sets.

Abstract

Let $\mathcal C$ be a small category with cofibrations. In this paper, we define the $K$-theory and Hochschild homology groups of $\mathcal C$ of order $Y$, where $Y$ is an ordered finite simplicial set with basepoint. Further, we construct the Dennis trace map between these groups.