论文标题
双定位对象和双代表并没有什么不同
Bi-initial objects and bi-representations are not so different
论文作者
论文摘要
我们介绍了一个函数$ \ MATHCAL V \ COLON \ MATHRM {dBLCAT} _ {H，NPS} \ to \ Mathrm {2cat} _ {H，NPS} $从双重类别的A $ 2 $ - 股票中提取的对象和孔子是Vermicts and vertical Morphisms and Squaress and Squaress和Squaress和Squares。我们给出了正常伪函数$ f \ colon \ mathbf c^{\ propatatOrname {op}} \ to \ m athrm {cat} $的表征$ 2 $ -STAGERY $ \ MATHCAL V \ MATHBB {e} l（f）$ f $的形态。尽管总体上并非不正确，但在特殊情况下，$ 2 $ -2CATEGORY $ \ MATHBF C $具有按类别按$ \ Mathbf {2} = \ {0 \ {0 \ to 1 \} $和$ f $保留的张量的张量$ f $的元素的$ \ mathbf {e} l（f）$。我们将该理论应用于双重功能和加权双限度。
We introduce a functor $\mathcal V\colon \mathrm{DblCat}_{h,nps}\to \mathrm{2Cat}_{h,nps}$ extracting from a double category a $2$-category whose objects and morphisms are the vertical morphisms and squares. We give a characterisation of bi-representations of a normal pseudo-functor $F\colon \mathbf C^{\operatorname{op}}\to \mathrm{Cat}$ in terms of double bi-initial objects in the double category $\mathbb{E}l(F)$ of elements of $F$, or equivalently as bi-initial objects of a special form in the $2$-category $\mathcal V\mathbb{E}l(F)$ of morphisms of $F$. Although not true in general, in the special case where the $2$-category $\mathbf C$ has tensors by the category $\mathbf{2}=\{0\to 1\}$ and $F$ preserves those tensors, we show that a bi-representation of $F$ is then precisely a bi-initial object in the $2$-category $\mathbf{E}l(F)$ of elements of $F$. We give applications of this theory to bi-adjunctions and weighted bi-limits.
