[Paper Review] Higher Dimensional Homology Algebra II:Projectivity

Fang Huang, Shao-Han Chen+2 more|arXiv (Cornell University)|Jun 24, 2010

Homotopy and Cohomology in Algebraic Topology5 references3 citations

TL;DR

This paper proves that the 2-categories of symmetric 2-groups (2-SGp) and R-2-modules (R-2-Mod) have enough projective objects, extending classical homological algebra to higher dimensions. By leveraging the known projective resolution property of R-Mod and constructing discrete symmetric 2-groups from R-modules, the authors establish a lifting mechanism via the π₀ functor and essential surjectivity, thereby confirming a conjecture by Bourn and Vitale on projectivity in 2-SGp.

ABSTRACT

In this paper, we will prove that the 2-category (2-SGp) of symmetric 2-groups and 2-category ($\cR$-2-Mod) of $\cR$-2-modules(\cite{5}) have enough projective objects, respectively.

Motivation & Objective

• To resolve a conjecture by Bourn and Vitale regarding the existence of enough projective objects in the 2-category of symmetric 2-groups (2-SGp).
• To extend the foundational projective resolution property of 1-dimensional homological algebra (R-Mod) to higher categorical structures.
• To establish that the 2-category R-2-Mod also has enough projective objects, enabling derived category constructions in higher homological algebra.
• To provide a categorical framework for developing (co)homology theories in 2-dimensional homological algebra.

Proposed method

• Construct discrete symmetric 2-groups M_dis from R-modules M, where objects are elements of M and only identity morphisms exist.
• Use the π₀ functor to extract the abelian group of isomorphism classes of objects in a symmetric 2-group, forming a bridge to classical module theory.
• Leverage the fact that R-Mod has enough projective objects to lift a projective R-module P to a projective R_dis-2-module P_dis.
• Establish that any essentially surjective R_dis-homomorphism F: M_dis → N_dis induces a surjective group homomorphism π₀(F): π₀(M_dis) → π₀(N_dis).
• Construct a lifting G': P_dis → M for any given G: P_dis → N and essentially surjective F: M → N using the projectivity of P in R-Mod and the isomorphism lifting property.
• Verify that the constructed morphism F: P_dis → M is essentially surjective by using the fullness of the comparison functor H: M → (π₀(M))_dis and composition of isomorphisms.

Experimental results

Research questions

• RQ1 Does the 2-category of symmetric 2-groups (2-SGp) have enough projective objects, as conjectured by Bourn and Vitale?
• RQ2 Can the projective resolution property of R-Mod be lifted to the 2-categorical setting of R-2-Modules?
• RQ3 How do the π₀ and discrete constructions (·)_dis relate symmetric 2-groups and R-2-modules to classical abelian groups and R-modules?
• RQ4 What conditions ensure that a morphism in R-2-Mod is essentially surjective, and how can this be used to construct projective resolutions?
• RQ5 Can the derived category framework of classical homological algebra be generalized to 2-categories via projective objects?

Key findings

• The 2-category (2-SGp) of symmetric 2-groups has enough projective objects, confirming the conjecture of Bourn and Vitale.
• For any R-module M, the discrete symmetric 2-group M_dis is an R_dis-2-module, establishing a categorical bridge between 1- and 2-dimensional algebra.
• Any surjective R-module homomorphism f: M → N induces an essentially surjective R_dis-homomorphism F: M_dis → N_dis.
• The π₀ functor induces a surjective group homomorphism π₀(F): π₀(M) → π₀(N) for any essentially surjective F: M → N in (R-2-Mod).
• If P is a projective R-module, then P_dis is a projective object in (R_dis-2-Mod), via lifting of homomorphisms and isomorphism data.
• The 2-category (R-2-Mod) has enough projective objects: for any M in (R-2-Mod), there exists a projective P_dis and an essentially surjective R_dis-homomorphism F: P_dis → M.