标量、单子和范畴

摘要：

这一章描述了以下三个方面之间的相互关系：（1）标量集合上的代数结构，（2）与这些标量集合相关的单子的性质，（3）与这些单子相关的范畴结构（特别是Lawvere理论）。这些相互关系将用“三角形的重叠”来表示，例如涉及各种幺半群（非交换的，交换的，对合的）和半环作为标量。它将被证明是哪种单子和范畴，这些代数结构对应通过排除。

英文摘要：

This chapter describes interrelations between: (1) algebraic structure on sets of scalars, (2) properties of monads associated with such sets of scalars, and (3) structure in categories (esp. Lawvere theories) associated with these monads. These interrelations will be expressed in terms of "triangles of adjunctions", involving for instance various kinds of monoids (non-commutative, commutative, involutive) and semirings as scalars. It will be shown to which kind of monads and categories these algebraic structures correspond via adjunctions.

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