Hopf comonads on naturally Frobenius map-monoidales

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• 作者：Gabriella Bö ; hm^a ; ^{bohm.gabriella@wigner.mta.hu" class="auth_mail" title="E-mail the corresponding author} ; Stephen Lack^b ; ^{steve.lack@mq.edu.au" class="auth_mail" title="E-mail the corresponding author}
• 刊名：Journal of Pure and Applied Algebra
• 出版年：2016
• 出版时间：June 2016
• 年：2016
• 卷：220
• 期：6
• 页码：2177-2213
• 全文大小：780 K

文摘

We study monoidal comonads on a naturally Frobenius map-monoidale M   in a monoidal bicategory M$\mathcal{M}$. We regard them as bimonoids in the duoidal hom-category M(M,M)$\mathcal{M}(M,M)$, and generalize to that setting various conditions distinguishing classical Hopf algebras among bialgebras; in particular, we define a notion of antipode in that context. Assuming the existence of certain conservative functors and the splitting of idempotent 2-cells in M$\mathcal{M}$, we show all these Hopf-like conditions to be equivalent. Our results imply in particular several equivalent characterizations of Hopf monoids in braided monoidal categories, of small groupoids, of Hopf algebroids over commutative base algebras, of weak Hopf algebras, and of Hopf monads in the sense of Bruguières and Virelizier.