We define a notion of grading of a monoid T in a monoidal category C, relative to a class of morphisms M (which provide a notion of M-subobject). We show that, under reasonable conditions (including that M forms a factorization system), there is a canonical grading of T. Our application is to graded monads and models of computational effects. We demonstrate our results by characterizing the canonical gradings of a number of monads, for which C is endofunctors with composition. We also show that we can obtain canonical grades for algebraic operations.

翻译：我们定义了幺半范畴C中幺半群T相对于态射类M（提供M-子对象的概念）的分次概念。研究表明，在合理条件下（包括M构成因子分解系统），T存在规范分次。本文的应用聚焦于分次单子与计算效应的模型。我们通过刻画若干单子的规范分次来展示结果，其中C是带有复合运算的自函子范畴。此外，我们还证明可以为代数运算获得规范分次。