The category of double categories and double functors is equipped with a symmetric closed monoidal structure. For any double category A, the corresponding internal hom functor ⟦A,−⟧ sends a double category B to the double category whose 0-cells are the double functors A → B, whose horizontal and vertical 1-cells are the horizontal and vertical pseudotransformations, respectively, and whose 2cells are the modifications. Some well-known functors of practical significance are checked to be compatible with this monoidal structure. Introduction The category 2-Cat of 2-categories and 2-functors carries different monoidal structures. The simplest one is given by the Cartesian product. It is symmetric and closed. For any 2-category A, the internal hom functor ⟨A,−⟩ sends a 2-category B to the 2-category of 2-functors A → B, 2-natural transformations, and modifications. This is, however, often too restrictive. For example, important examples of 2-categories which are intuitively monoidal, fail to be monoids for that [1, 14, 15]. A well established generalization is the so-called Gray monoidal product [13]. It is also symmetric and closed and for any 2-category A the corresponding internal hom functor [A,−] sends a 2-category B to the 2-category of 2-functors A → B, pseudonatural transformations, and modifications. The Cartesian monoidal structure is more restrictive than the Gray one in the sense that the identity functor on 2-Cat is a monoidal functor from the former to the latter one. The category DblCat of double categories and double functors is also symmetric closed monoidal via the Cartesian product ×. For any double category A, the corresponding internal hom functor jA,−o sends a double category B to the double category whose 0-cells are the double functors A→ B, whose horizontal and vertical 1-cells are the horizontal and vertical transformations, respectively, and whose 2-cells are the modifications; see [9]. The analogue of the Gray monoidal product on DblCat, however, has apparently not yet been discussed in the literature. The current paper addresses this question. For any double categories A and B, there is a bigger double category ⟦A,B⟧ in which the 0-cells are still the double functors A→ B. The horizontal and vertical 1-cells are, however, the horizontal and vertical pseudo (or strong) transformations of [9]. The 2cells are their modifications. In Section 1 we prove that for any double categoriesA and B, there is a representing object B⊗A of the functor DblCat(B, ⟦A,−⟧) ∶ DblCat→ Set. Constructing the associativity and unit constraints, as well as the symmetry, in Section 2 we show that ⊗ equips DblCat with a symmetric monoidal structure. In order to support this choice of monoidal structure on DblCat, in Section 3 monoidality of the following functors is checked. Date: Jan 2019. 1

中文翻译：

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双范畴的灰色幺半群

双范畴和双函子的范畴具有对称的封闭幺半群结构。对于任何双类A，对应的内部hom函子⟦A,−⟧发送一个双类B给双类，其0-cells是双函子A→B，其水平和垂直1-cells是水平和垂直的伪变换，分别为 2cells 是修改。检查了一些具有实际意义的知名函子是否与这种幺半群结构兼容。介绍 2-范畴和2-函子的范畴2-Cat 带有不同的幺半群结构。最简单的一种是笛卡尔积。它是对称的和封闭的。对于任何2-范畴A，内部hom函子⟨A,−⟩发送一个2-范畴B到2-函子A→B的2-范畴，2-自然变换，和修改。然而，这通常过于严格。例如，直觉上是幺半群的 2 类的重要例子，就不能成为幺半群 [1, 14, 15]。一个完善的概括是所谓的灰色幺半群产品 [13]。它也是对称和封闭的，对于任何 2 类 A，相应的内部 hom 函子 [A,−] 将 2 类 B 发送到 2 类 A → B、伪自然变换和修改的 2 类。笛卡尔幺半群结构比格雷结构更严格，因为 2-Cat 上的恒等函子是从前者到后者的幺半群函子。双范畴和双函子的范畴 DblCat 通过笛卡尔积 × 也是对称闭幺半群。对于任何双类别A，对应的内部hom函子jA，-o 发送一个双类别 B 到双类别，其 0-cells 是双函子 A→B，其水平和垂直 1-cells 分别是水平和垂直变换，其 2-cells 是修改；见[9]。然而，文献中显然还没有讨论过 DblCat 上灰色幺半群的类似物。目前的论文解决了这个问题。对于任何双类别 A 和 B，都有一个更大的双类别 ⟦A,B⟧，其中 0-细胞仍然是双函子 A→B。然而，水平和垂直 1-细胞是水平和垂直伪[9] 的（或强）转换。2cell 是他们的修改。在第 1 节中，我们证明对于任何双类别 A 和 B，都有一个函子 DblCat(B, ⟦A,−⟧) ∶ DblCat→ Set 的表示对象 B⊗A。在第 2 节中构建结合性和单位约束以及对称性，我们表明 ⊗ 为 DblCat 配备了对称幺半群结构。为了在 DblCat 上支持这种幺半群结构的选择，在第 3 节中检查了以下函子的幺半性。日期：2019 年 1 月 1