论文标题
笛卡尔差异类别作为偏斜的类别
Cartesian differential categories as skew enriched categories
论文作者
论文摘要
我们展示了笛卡尔差异类别的长毛，科贝特和Seely是一种特殊的丰富类别。富集的基础是交换性单体的类别 - 或直接概括，即通勤钻机上的模块类别。但是，此类别上的张量产品不是通常的产品，而是某种单型comonad $ q $对其进行翘曲。因此，从通常的意义上讲，富集基础不是一个单体类别，而是szlachányi意义上的偏斜的单体类别。我们的第一个主要结果是，笛卡尔差异类别与在这个偏斜的单体基础上富含有限产品的类别相同。 实际上，涉及的comonad $ q $是差异方式的一个例子。差异方式是一种对称单体$ k $ - 线性类别的共生，其特征性功能是其Co-Kleisli类别是笛卡尔差异类别。使用我们的第一个主要结果，我们能够证明我们的第二个：每个小型笛卡尔差异类别都承认由差异方式诱导的完整的，结构上的嵌入到笛卡尔差异类别中（实际上，是单型封闭类别上的单差差形式 - 因此，直觉差异线性线性lineal Linal Lineal Liolic locic lineal lineal loollogic模型）。这解决了这一领域的重要开放问题。
We exhibit the cartesian differential categories of Blute, Cockett and Seely as a particular kind of enriched category. The base for the enrichment is the category of commutative monoids -- or in a straightforward generalisation, the category of modules over a commutative rig $k$. However, the tensor product on this category is not the usual one, but rather a warping of it by a certain monoidal comonad $Q$. Thus the enrichment base is not a monoidal category in the usual sense, but rather a skew monoidal category in the sense of Szlachányi. Our first main result is that cartesian differential categories are the same as categories with finite products enriched over this skew monoidal base. The comonad $Q$ involved is, in fact, an example of a differential modality. Differential modalities are a kind of comonad on a symmetric monoidal $k$-linear category with the characteristic feature that their co-Kleisli categories are cartesian differential categories. Using our first main result, we are able to prove our second one: that every small cartesian differential category admits a full, structure-preserving embedding into the cartesian differential category induced by a differential modality (in fact, a monoidal differential modality on a monoidal closed category -- thus, a model of intuitionistic differential linear logic). This resolves an important open question in this area.
