Homotopy theory of normed sets II. Model categories

This paper is a continuation of the paper by the same author published in no. 2017:6 of this journal, where the foundations of the theory of normed and graded sets, and other algebraic structures were laid out. Here these foundations are used to present a homotopy theory of normed and graded sets, and other algebraic structures, by introducing combinatorial model structures on categories of relevant simplicial objects. We also construct a homotopy theory of metric spaces, which turns out to be deeply related to that of normed sets.