AI 中文总结
本文将Gabi-代数推广到偏闭范畴上的Gabi-单子，通过重构定理建立其与Eilenberg–Moore范畴偏闭结构的联系，刻画其与Kock闭单子、Hopf单子的关系并给出实例。
AI 中文摘要
我们研究偏闭范畴上的Gabi-单子，将Berger、第二作者与Vercruysse提出的Gabi-代数从线性情形推广。我们的主要重构定理确定了：在一个单子上的Gabi-单子结构，对应其Eilenberg–Moore范畴上的偏闭结构，使得典范忘却函子为严格闭函子。我们将该概念与Kock意义下的闭单子作比较，表明在表示论情形下这些概念差异显著。在闭幺半范畴上，每个左Hopf单子都是正规Gabi-单子，但反之一般不成立。我们通过对应参数伙伴的可逆性刻画Gabi-单子何时为Hopf单子，这推广了环论结果：交换基环上的正规Gabi-代数是Hopf代数。Gabi-单子理论有多个自然例子，如无挠模、自反有向图、单纯复形，我们将详细探讨；还研究了带点集作为拟例子。
英文摘要
We study gabi-monads on skew-closed categories, extending the gabi-algebras of Berger, the second author, and Vercruysse beyond the linear case. Our main reconstruction theorem identifies gabi-monad structures on a monad with skew-closed structures on its Eilenberg--Moore category for which the canonical forgetful functor is strict closed. We compare this notion with closed monads in the sense of Kock, showing that in representation-theoretic cases these notions are quite different. On closed monoidal categories, every left Hopf monad is a normal gabi-monad, but the converse fails in general. We characterise when a gabi-monad is Hopf by the invertibility of the corresponding parametric mates, which recovers the ring-theoretic result that normal gabi-algebras over a commutative base ring are Hopf algebras. The theory of gabi-monads admits several natural examples, such as torsion-free modules, reflexive digraphs, and simplicial complexes, that we will explore in detail; we also study pointed sets as a quasi-example.
Comments61 pages, lots of figures; comments very welcome!
