论文标题
部分评估和条形结构的组成结构
Partial Evaluations and the Compositional Structure of the Bar Construction
论文作者
论文摘要
代数表达式$ 3 + 2 + 6 $可以评估为$ 11 $，但也可以部分评估为$ 5 + 6 $。在分类代数中，可以根据单元代数的bar构造的$ 1 $骨骼来定义这种部分评估。我们表明，这种部分评估关系可以看作是通过将正式表达与其总评估相关联的代数类别的内部关系。这种关系是许多单月的传递性，这些单子描述了通常遇到的代数结构，更普遍地是在$ \ mathsf {set} $上的卑诗省monads（这是基础函数和乘法较弱的那些单子是弱的，这是弱的卡特西亚人）。我们发现，对于所有单调，这并不是如此：我们在$ \ mathsf {set} $上描述了终端代数的部分评估关系的限制单元。考虑到更高的代数重写的观点，我们随后研究了所有维度的条形结构的组成结构。我们表明，对于卑诗省元的代数，棒构建具有$δ^n $的所有定向无环构型的填充物，但通常并非全部内角。
The algebraic expression $3 + 2 + 6$ can be evaluated to $11$, but it can also be partially evaluated to $5 + 6$. In categorical algebra, such partial evaluations can be defined in terms of the $1$-skeleton of the bar construction for algebras of a monad. We show that this partial evaluation relation can be seen as the relation internal to the category of algebras generated by relating a formal expression to its total evaluation. The relation is transitive for many monads which describe commonly encountered algebraic structures, and more generally for BC monads on $\mathsf{Set}$ (which are those monads for which the underlying functor and the multiplication are weakly cartesian). We find that this is not true for all monads: we describe a finitary monad on $\mathsf{Set}$ for which the partial evaluation relation on the terminal algebra is not transitive. With the perspective of higher algebraic rewriting in mind, we then investigate the compositional structure of the bar construction in all dimensions. We show that for algebras of BC monads, the bar construction has fillers for all directed acyclic configurations in $Δ^n$, but generally not all inner horns.
