Elementary remarks on units in monoidal categories
Elementary remarks on units in monoidal categories
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关于幺半群范畴中单位的基本评论
DOI：
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发表时间：
2005
影响因子：
0.8
通讯作者：
Joachim Kock
中科院分区：
文献类型：
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作者：
Joachim Kock
摘要：我们探索了最初由 Saavedra 提出的幺半群范畴中单位的另一种定义：Saavedra 单位是可取消的幂等（在 1-分类意义上）。就左右约束而言，这个概念比通常的概念更经济，并且受到更高范畴论的启发。首先，我们描述给定半幺半范畴上所有可能的单元结构的半幺半范畴，并观察它是可收缩的（如果非空）。然后我们证明这两个单位概念在强函子意义上是等价的。接下来，证明（强）幺半群函子的单位相容条件正是函子提升到单位范畴的条件，并解释了 Saavedra 单位的概念如何自然地导出公平幺半群范畴的等价非代数概念，其中可收缩的多个单位被视为一个整体，而不是选择一个单位。最后，考虑单位比较的宽松版本。该论文是独立的。所有的论点都是基本的，其中一些论点具有一定的美感。
Abstract We explore an alternative definition of unit in a monoidal category originally due to Saavedra: a Saavedra unit is a cancellable idempotent (in a 1-categorical sense). This notion is more economical than the usual notion in terms of left-right constraints, and is motivated by higher category theory. To start, we describe the semi-monoidal category of all possible unit structures on a given semi-monoidal category and observe that it is contractible (if non-empty). Then we prove that the two notions of units are equivalent in a strong functorial sense. Next, it is shown that the unit compatibility condition for a (strong) monoidal functor is precisely the condition for the functor to lift to the categories of units, and it is explained how the notion of Saavedra unit naturally leads to the equivalent non-algebraic notion of fair monoidal category, where the contractible multitude of units is considered as a whole instead of choosing one unit. To finish, the lax version of the unit comparison is considered. The paper is self-contained. All arguments are elementary, some of them of a certain beauty.