On monoidal functors between (braided) Gr-categories

In this paper, we state and prove precise theorems on the classification of the category of (braided) categorical groups and their (braided) monoidal functors, and some applications obtained from the basic studies on monoidal functors between categorical groups.

💡 Research Summary

The paper provides a complete classification of (braided) Gr‑categories—also known as categorical groups—and of the (braided) monoidal functors between them. A Gr‑category is a 2‑category whose objects form a group π₀, whose automorphism groups of the unit object form an abelian group π₁, and whose associativity constraints are encoded by a 3‑cocycle α∈H³(π₀, π₁). The authors first recall that the triple (π₀, π₁, α) determines a Gr‑category up to equivalence, and they give a precise statement of this well‑known result, together with a proof that emphasizes coherence and Mac Lane’s pentagon.

The core of the work is the analysis of monoidal functors F: 𝔾→ℍ between two Gr‑categories 𝔾 and ℍ. Assuming F is normalized (preserves the unit object), the authors show that F decomposes into two components: a group homomorphism f₀:π₀(𝔾)→π₀(ℍ) and a module homomorphism f₁:π₁(𝔾)→π₁(ℍ) which is f₀‑linear. The monoidal structure of F imposes a compatibility condition between the 3‑cocycles α (of 𝔾) and β (of ℍ). Explicitly, \