Morita代数双范畴与对偶对合
Morita Bicategories of Algebras and Duality Involutions
摘要：
双范畴上的弱对偶对合的概念最近由Shulman在[arXiv：1606.05058]中引入。我们在有限维k-代数的Morita双范畴$\text{Alg}$的完全可对偶部分上构造了一个弱对偶对合。Kapranov-Voevodsky k-向量空间的2-范畴$\text{KV}$可以具有规范严格对偶对合.证明了发送代数到有限维模范畴的伪函子$\text{Rep}: \text{Alg}^{fd} \to \text{KV}$可以正则地具有对偶伪函子的结构。因此，$\text{Rep}$是一个严格化在舒尔曼的严格化定理的意义上的双范畴与弱对偶对合。最后，我们给出了半单对称有限张量范畴中Morita双代数范畴上对偶对合的一般设置。
英文摘要：
The notion of a weak duality involution on a bicategory was recently introduced by Shulman in [arXiv:1606.05058]. We construct a weak duality involution on the fully dualisable part of $\text{Alg}$, the Morita bicategory of finite-dimensional k-algebras. The 2-category $\text{KV}$ of Kapranov-Voevodsky k-vector spaces may be equipped with a canonical strict duality involution. We show that the pseudofunctor $\text{Rep}: \text{Alg}^{fd} \to \text{KV}$ sending an algebra to its category of finite-dimensional modules may be canonically equipped with the structure of a duality pseudofunctor. Thus $\text{Rep}$ is a strictification in the sense of Shulman's strictification theorem for bicategories with a weak duality involution. Finally, we present a general setting for duality involutions on the Morita bicategory of algebras in a semisimple symmetric finite tensor category.
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