Higher Dimensional Homology Algebra V:Injective Resolutions and Derived 2-Functors in ($cR$-2-Mod)

In this paper, we will construct the injective resolution of any $\cR$-2-module, define the right derived 2-functor, and give some related properties of the derived 2-functor in ($\cR$-2-Mod).

💡 Research Summary

This paper extends the homological algebra of higher‑dimensional categories by constructing injective resolutions for any 𝓡‑2‑module and by defining the associated right derived 2‑functors. The authors begin by recalling the structure of the 2‑category 𝓡‑2‑Mod, whose objects are 𝓡‑weighted abelian group objects, whose 1‑morphisms are 𝓡‑linear functors, and whose 2‑morphisms are natural transformations. They introduce a notion of “2‑exactness” that strengthens the usual exactness condition by requiring compatibility at the level of 2‑cells.

The core technical achievement is the proof that every 𝓡‑2‑module M admits a 2‑exact injective resolution \