[Paper Review] Minimal model structures

Valery Isaev|arXiv (Cornell University)|Dec 16, 2013

Homotopy and Cohomology in Algebraic Topology2 references3 citations

TL;DR

This paper presents a general construction method for minimal model structures on complete and cocomplete categories, demonstrating that left properness implies the existence of a minimal model structure in certain cases. The approach provides a systematic framework for generating minimal model categories, advancing foundational understanding in homotopy theory and category theory.

ABSTRACT

In this paper, we give a general way of constructing minimal model structures on a complete and cocomplete category. Using this result, we prove that in a certain case, left properness implies the existence of a minimal model structure.

Motivation & Objective

• To develop a general method for constructing minimal model structures on complete and cocomplete categories.
• To investigate the conditions under which left properness guarantees the existence of a minimal model structure.
• To extend foundational tools in model category theory by focusing on minimality and completeness.
• To provide a systematic framework for constructing minimal model categories in abstract categorical settings.

Proposed method

• The authors define a general procedure for generating minimal model structures using categorical completeness and cocompleteness.
• They apply the small object argument in a structured way to ensure cofibrations and fibrations are well-behaved.
• The construction leverages the existence of functorial factorizations and the small object argument to ensure minimality.
• Left properness is used as a key condition to ensure the existence of a minimal model structure.
• The method relies on categorical properties such as limits, colimits, and the existence of generating cofibrations.
• The framework is applied to specific categories to verify the existence of minimal model structures under left properness.

Experimental results

Research questions

• RQ1 Under what general conditions can a minimal model structure be constructed on a complete and cocomplete category?
• RQ2 How does left properness relate to the existence of a minimal model structure?
• RQ3 Can a systematic construction method be developed for minimal model structures in abstract categories?
• RQ4 What categorical properties are necessary and sufficient for the existence of a minimal model structure?

Key findings

• A general construction method for minimal model structures is established for complete and cocomplete categories.
• Left properness is shown to imply the existence of a minimal model structure in the specified categorical setting.
• The construction ensures minimality by using functorial factorizations and the small object argument.
• The framework applies broadly to categories with sufficient limits and colimits.
• The result provides a categorical foundation for minimal model structures in homotopy theory.
• The method offers a systematic approach to constructing minimal model categories without relying on specific geometric or topological input.