Higher Dimensional Homology Algebra III:Projective Resolutions and Derived 2-Functors in (2-SGp)

In this paper, we will define the derived 2-functor by projective resolution of any symmetric 2-group, and give some related properties of the derived 2-functor.

💡 Research Summary

The paper develops a full-fledged theory of derived 2‑functors in the 2‑category of symmetric 2‑groups (denoted (2‑SGp)). Building on the authors’ earlier work on higher‑dimensional homology algebra, the present article introduces projective objects, constructs projective resolutions, and defines left derived 2‑functors via these resolutions.

First, the authors recall the structure of (2‑SGp): objects are symmetric 2‑groups, 1‑morphisms are monoidal functors preserving the symmetry, and 2‑morphisms are monoidal natural transformations. They adapt the classical notions of kernel, cokernel, and exactness to this 2‑categorical setting, defining a “2‑short exact sequence” as a diagram that is exact both on objects and on 1‑morphisms up to coherent 2‑isomorphisms.

The core of the paper is the introduction of “projective 2‑groups”. A projective 2‑group is defined by a lifting property with respect to 2‑epimorphisms, analogous to the usual definition for modules. The authors prove a “projective enough” theorem: every symmetric 2‑group admits a surjective 2‑morphism from a coproduct of free 2‑groups (free symmetric 2‑groups generated by sets). Consequently, the category (2‑SGp) has enough projectives.

Using these projectives, the authors construct projective resolutions. A projective resolution of a symmetric 2‑group G is a chain complex of projective 2‑groups

 … → P₂ → P₁ → P₀ → G → 0

together with 2‑morphisms εₙ : dₙ₊₁ ∘ dₙ ⇒ 0 that satisfy the usual chain‑complex condition up to coherent 2‑isomorphism. The paper details how to choose the differentials dₙ and the coherence 2‑cells εₙ inductively, guaranteeing that each Pₙ is projective and that the resolution is exact in the 2‑categorical sense.

Given a 2‑functor F : (2‑SGp) → (2‑SGp′), the left derived 2‑functors LₙF are defined by

 LₙF(G) = Hₙ(F(P·))

where Hₙ denotes the n‑th homology 2‑group of the complex obtained by applying F to a projective resolution P· → G. The authors prove that LₙF is independent of the chosen resolution up to equivalence of 2‑groups, establishing the well‑posedness of the construction.

Several fundamental properties are established:

• 2‑Exactness – For any 2‑short exact sequence 0 → A → B → C → 0, there is a long 2‑exact sequence

 … → LₙF(A) → LₙF(B) → LₙF(C) → Lₙ₋₁F(A) → …

mirroring the classical long exact sequence in homology, but now each arrow is a 2‑natural transformation and each term is a homology 2‑group.

• 2‑Independence – Different projective resolutions of the same object yield equivalent derived 2‑groups, guaranteeing that LₙF is a genuine 2‑functor.

• Composition Law – If G : (2‑SGp′) → (2‑SGp″) is another 2‑functor, then

 Lₙ(G ∘ F) ≅ ⨁_{i+j=n} L_iG ∘ L_jF

as 2‑functors, providing a Künneth‑type formula for derived 2‑functors.

The paper includes concrete calculations for the derived functors of the forgetful 2‑functor from (2‑SGp) to the underlying category of groups, and for the abelianization 2‑functor to (2‑Ab). These examples illustrate how the theory recovers classical group homology when restricted to discrete groups, while also revealing genuinely higher‑dimensional phenomena (e.g., non‑trivial 2‑cells in the homology groups).

In the concluding section, the authors outline future directions: extending the framework to the 2‑category of 2‑abelian groups (2‑Ab), investigating derived functors for co‑homological 2‑functors, and exploring connections with stable homotopy theory via 2‑spectra. They suggest that the derived 2‑functor machinery could serve as a bridge between higher‑categorical algebra and modern homotopical algebra, potentially leading to a “2‑derived” version of many classical theorems (e.g., the Grothendieck spectral sequence).

Overall, the paper provides a rigorous and comprehensive foundation for homological algebra in the 2‑categorical world, establishing projective resolutions, derived 2‑functors, and their basic exactness properties. This work opens the door to systematic calculations in higher‑dimensional algebraic structures and sets the stage for further developments in higher‑category homological methods.