论文标题
半录得
Semiseparable functors
论文作者
论文摘要
在本文中，我们介绍并研究了半播种函子的概念。它的第一个特征之一是，它可以分别从忠实和完整的功能者角度来描述可分离和自然的功能。对于任何半录制函子，我们都会连接一个不变的，由diDempotent的自然变换给出，该转换可以在函数分离时控制，并以（双重）Maschke和保守的函子来表征可分离的函子。我们证明，任何半官能函子都将典型的分解为自然函数，然后是可分离的函子。在这里，主要工具是建造相关的自然变换的相关类别类别。然后，我们将注意力转移到具有伴随的函子的半养生性上。首先，我们获得Rafael型定理。接下来，我们根据相关（CO）单子的（CO）可分离性以及相应（CO）比较函数的（CO）可分离性的（CO）可分离性来表征伴随函子的半养生性。我们还专注于与伴随三倍的一部分的函子。特别是，我们将双重反射描述为半部分（CO）反射，或等效地为Frobenius或Naturally Full（CO）反射。作为结果的应用，我们研究了传统上与圆环，煤层图，振动和双模型相关的函子的半养生性，引入了与双模数相对于cosplit corming和sugano的可分离性，相对于Bimodule的分析性相对于BiModule的分析性相对于BiModule，引入了半平原的加油和半养生的概念。
In this paper we introduce and investigate the notion of semiseparable functor. One of its first features is that it allows a novel description of separable and naturally full functors in terms of faithful and full functors, respectively. To any semiseparable functor we attach an invariant, given by an idempotent natural transformation, which controls when the functor is separable and yields a characterization of separable functors in terms of (dual) Maschke and conservative functors. We prove that any semiseparable functor admits a canonical factorization as a naturally full functor followed by a separable functor. Here the main tool is the construction of the coidentifier category attached to the associated idempotent natural transformation. Then we move our attention to the semiseparability of functors that have an adjoint. First we obtain a Rafael-type Theorem. Next we characterize the semiseparability of adjoint functors in terms of the (co)separability of the associated (co)monads and the natural fullness of the corresponding (co)comparison functor. We also focus on functors that are part of an adjoint triple. In particular, we describe bireflections as semiseparable (co)reflections, or equivalently, as either Frobenius or naturally full (co)reflections. As an application of our results, we study the semiseparability of functors traditionally attached to ring homomorphisms, coalgebra maps, corings and bimodules, introducing the notions of semicosplit coring and semiseparability relative to a bimodule which extend those of cosplit coring and Sugano's separability relative to a bimodule, respectively.
