The factor set of Gr-categories of the type $(Pi,A)$

Any $\Gamma$-graded categorical group is determined by a factor set of a categorical group. This paper studies the factor set of the group $\Gamma$ with coefficients in the categorical group of the type $(\Pi,A).$ Then, an interpretation of the notion of $\Gamma-$operator $3-$cocycle is presented and the proof of cohomological classification theorem for the a $\Gamma-$graded Gr-category is also presented.

💡 Research Summary

The paper investigates the structure of Γ‑graded Gr‑categories whose underlying categorical group is of the type (Π, A), i.e., a strict 2‑group with object group Π and automorphism group A (viewed as a Π‑module). The central theme is the description of such graded categories by a “factor set” (φ, f) and the subsequent cohomological classification via Γ‑operator 3‑cocycles.

First, the authors recall that a Γ‑graded categorical group is a monoidal category equipped with a grading functor to the group Γ, together with coherence data that respects the grading. For a Gr‑category of type (Π, A) the only non‑trivial data are the group Π of objects, the Π‑module A of automorphisms, and the associator, which in the strict case is trivial. To endow this Gr‑category with a Γ‑grading one must specify how Γ acts on Π and A and how the grading twists the monoidal product. This is precisely encoded by a factor set:

• For each g∈Γ a pair of automorphisms φ_g = (φ_g^Π, φ_g^A) where φ_g^Π : Π→Π is a group automorphism and φ_g^A : A→A is a Π‑module automorphism compatible with φ_g^Π.
• For each pair (g,h)∈Γ×Γ an element f(g,h)∈A measuring the deviation of the composition φ_g ∘ φ_h from φ_{gh}.

These data must satisfy three coherence equations: (i) φ_g ∘ φ_h = Inn_{f(g,h)} ∘ φ_{gh}, (ii) the 2‑cocycle condition δf(g,h,k)=0, and (iii) compatibility of φ and f with the Π‑module structure on A.

The next step is to translate the factor set into a Γ‑operator 3‑cocycle θ : Γ³ → A. The authors define
θ(g₁,g₂,g₃) = φ_{g₁}(f(g₂,g₃)) + f(g₁,g₂g₃) – f(g₁,g₂) – f(g₁g₂,g₃)
and prove that θ satisfies the operator 3‑cocycle condition
δθ(g₁,g₂,g₃,g₄) = φ_{g₁}(θ(g₂,g₃,g₄)) + θ(g₁,g₂g₃,g₄) – θ(g₁,g₂,g₃g₄) + θ(g₁g₂,g₃,g₄) = 0.
Conversely, given a Γ‑operator 3‑cocycle θ and a fixed action φ of Γ on (Π, A), one can recover a factor set by setting f(g,h)=θ(1,g,h). Two factor sets are considered equivalent if they differ by a suitable 2‑cochain; this equivalence precisely corresponds to cohomologous 3‑cocycles.

With this bijection in place, the authors prove the main classification theorem: the set of equivalence classes of Γ‑graded Gr‑categories of type (Π, A) is in natural bijection with the third cohomology group H³_Γ(Π, A) of the Γ‑module (Π, A) with respect to the operator cohomology. The proof proceeds by (1) assigning to any graded Gr‑category its factor set and then its associated 3‑cocycle, thereby defining a map to H³_Γ(Π, A); (2) showing that equivalent graded categories give cohomologous cocycles, so the map is well‑defined on equivalence classes; (3) constructing, for any class