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

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