Abelian and model structures on tame functors

TL;DR

This paper explores how categories of tame functors can be endowed with abelian and model category structures, providing a structure theorem and a technique for generating indecomposables in these categories.

Contribution

It introduces conditions for abelian and model structures on tame functors, a structure theorem for cofibrant objects, and a method to generate indecomposables in functor categories.

Findings

Category of tame functors inherits abelian structure with minimal resolutions.

Category of tame functors admits a model structure with minimal cofibrant replacements.

A technique to generate indecomposable objects in functor categories.

Abstract

In this paper, we discuss certain circumstances in which the category of tame functors inherits an abelian category structure with minimal resolutions and a model category structure with minimal cofibrant replacements. We also present a structure theorem for cofibrant objects in the category of tame functors indexed by realizations of posets of dimension $1$ with values in the category of chain complexes in an abelian category whose all objects are projectives. Moreover, we introduce a general technique to generate indecomposable objects in the abelian category of functors indexed by finite posets.