Cofibrant indecomposables in chain complex valued tame functors indexed by dimension one posets

In this work, we provide a model structure on full subcategories of tame objects in functor categories indexed by continuous realizations of posets of dimension 1. We also characterize the indecomposable cofibrant objects when the landing category is the one of bounded non-negative chain complexes. In addition, we present a general method to construct indecomposables in a functor category by a gluing technique.