Duals of Ann-categories

Dual monoidal category $\mathcal C^\ast$ of a monoidal functor $F:\mathcal C\to \mathcal V$ has been constructed by S. Majid. In this paper, we extend the construction of dual structures for an Ann-functor $F:\mathcal B\to \mathcal A$. In particular, when $F=id_{\mathcal A}$, then the dual category $\mathcal A^{\ast}$ is indeed the center of $\mathcal A$ and this is a braided Ann-category.

💡 Research Summary

The paper extends the construction of dual monoidal categories, originally introduced by S. Majid, to the setting of Ann‑categories, which are categorical analogues of rings equipped with two compatible binary operations: a symmetric categorical group structure ((\mathcal A,\oplus)) and a monoidal structure ((\mathcal A,\otimes)) together with distributivity constraints.

The authors begin by recalling Majid’s “full dual category” ((\mathcal C,F)^{*}) for a monoidal functor (F:\mathcal C\to\mathcal V). Objects of the dual are pairs ((V,u_{V})) where (u_{V,X}:V\otimes FX\to FX\otimes V) is a natural family satisfying coherence with the monoidal structure of (F).

To adapt this to Ann‑categories, the paper introduces the notion of a right ((\mathcal B,F))-module. Given an Ann‑functor (F:\mathcal B\to\mathcal A), a right module consists of an object (A\in\mathcal A) together with a natural transformation (u_{A,X}:A\otimes FX\to FX\otimes A) that respects both the additive and multiplicative structures of (\mathcal A). The required compatibility is expressed by two diagrams (2) and (3) in the paper, which encode the distributivity of (u) over (\oplus) and (\otimes), respectively, and the condition (u_{A,I}=id).

The collection of all right ((\mathcal B,F))-modules, together with morphisms that are maps in (\mathcal A) commuting with the (u)-structures, forms a new category denoted (\mathcal B^{}=(\mathcal B,F)^{}). The authors then endow (\mathcal B^{*}) with two binary operations:

1. Addition (\oplus): For modules ((A,u_{A})) and ((B,u_{B})), their sum is ((A\oplus B, u_{A\oplus B})) where
   \