Homotopy categories and fibrant model structures

TL;DR

This paper explores the structure of homotopy categories within additive categories, establishing equivalences and characterizations of fibrant model structures through fibrantly weak factorization systems.

Contribution

It introduces fibrantly weak factorization systems, constructs fibrant model structures explicitly, and establishes a correspondence between these systems and fibrant model structures.

Findings

Homotopy category is equivalent to an additive quotient of cofibrant-fibrant objects.

Fibrant model structures are characterized by trivial cofibrations and fibrations.

A one-to-one correspondence between fibrantly weak factorization systems and fibrant model structures is established.

Abstract

The homotopy category of a model structure on a weakly idempotent complete additive category is proved to be equivalent to the additive quotient of the category of cofibrant-fibrant objects with respect to the subcategory of cofibrant-fibrant-trivial objects. A model structure on pointed category is fibrant, if every object is a fibrant object. Fibrant model structures is explicitly described by trivial cofibrations, and also by fibrations. Fibrantly weak factorization systems are introduced, fibrant model structures are constructed via fibrantly weak factorization systems, and a one-one correspondence between fibrantly weak factorization systems and fibrant model structures is given. Applications are given to rediscover the $\omega$-model structures and the $\mathcal W$-model structures, and their relations with exact model structures are discussed.