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 investigates a categorical construction that generalizes the notion of a dual monoidal category, originally introduced by S. Majid, to the richer setting of Ann‑categories. An Ann‑category is a category equipped with two monoidal structures, usually denoted ((\oplus,\mathbf 0)) (additive) and ((\otimes,\mathbf 1)) (multiplicative), together with a distributive law that mimics the interaction of addition and multiplication in a ring. A functor between Ann‑categories that respects both monoidal structures and the distributive law is called an Ann‑functor.

Majid’s dual monoidal category (\mathcal C^{\ast}) associated to a monoidal functor (F:\mathcal C\to\mathcal V) is defined as the category whose objects are pairs ((X,\gamma)) where (X) is an object of (\mathcal C) and (\gamma) is a natural family of isomorphisms (F(Y)\otimes X\cong X\otimes F(Y)) satisfying the usual coherence conditions. Morphisms are those maps in (\mathcal C) that are compatible with the families (\gamma). This construction yields a “center” of (\mathcal C) inside the ambient monoidal category (\mathcal V) and automatically carries a braiding.

The present work lifts this idea to Ann‑categories. Given an Ann‑functor (F:\mathcal B\to\mathcal A), the authors define a dual Ann‑category (\mathcal A^{\ast}) as follows. An object of (\mathcal A^{\ast}) consists of an object (a\in\mathcal A) together with two families of natural isomorphisms
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