Model Structures on Preorders: Classification and Construction
Published:Dec 27, 2025 08:31
•1 min read
•ArXiv
Analysis
This paper explores model structures within the context of preorders, providing conditions for their existence and offering classification results. The work is significant because it connects abstract mathematical structures (model categories) to more concrete ones like topologies and matroids, ultimately leading to a method for constructing model structures on Boolean algebras. The detailed case studies on small Boolean algebras and their localization/colocalization relations add practical value.
Key Takeaways
- •Provides conditions for model structures on preorders.
- •Classifies model structures based on (co)fibrant objects and (co)monads.
- •Offers a construction method for model structures on Boolean algebras using topologies and matroids.
- •Includes detailed case studies on small Boolean algebras and their localization/colocalization.
- •Connects abstract model category theory to concrete mathematical objects.
Reference
“The paper provides "necessary and sufficient conditions for $\mathcal{A}$ to admit the structure of a model category whose cofibrant objects are $\mathcal{C}$ and whose fibrant objects are $\mathcal{F}$."”