论文标题
2类语言的理解和商结构
Comprehension and quotient structures in the language of 2-categories
论文作者
论文摘要
律师在他在高多次方面的著名工作中观察到，理解的理论模式可以用分类逻辑的函数语言优雅地表达为函数上的理解结构$ p：\ mathscr {e} \ to \ to \ mathscr {b} $定义超多动物。 在本文中，我们制定并研究了给定函数$ p上三个理解结构概念的严格订购层次结构：\ mathscr {e} \ to \ mathscr {b} $，我们称之为（i）理解结构，（ii）具有图像的理解结构（ii）与图像的理解结构。 我们的方法是2类别的方法，因此我们在一般形态上的三个级别的理解结构$ p：\ mathrm {\ mathbf {e}}} \ to \ mathrm {\ mathrm {\ mathbf {b}} $中的2级别$ \ mathscr {k} $。 关于理解结构的这种概念的观点使我们能够重新审视Fumex，Ghani和Johann对给定函数上的理解结构和商结构之间的二元性的作品。 特别是，我们展示了如何将函子上的理解和商结构提升到\ Mathscr {e} \至\ Mathscr {b} $的类别中，以$ f _ {\ Mathscr {e}}}：\ Mathscr { $ f _ {\ mathscr {b}}：\ Mathscr {B} \ to \ Mathscr {B} $感兴趣的，以便通过归纳和共同诱导和共同诱导分类逻辑的传统语言来解释推理，以适当的2类别方式表达。
Lawvere observed in his celebrated work on hyperdoctrines that the set-theoretic schema of comprehension can be elegantly expressed in the functorial language of categorical logic, as a comprehension structure on the functor $p:\mathscr{E}\to\mathscr{B}$ defining the hyperdoctrine. In this paper, we formulate and study a strictly ordered hierarchy of three notions of comprehension structure on a given functor $p:\mathscr{E}\to\mathscr{B}$, which we call (i) comprehension structure, (ii) comprehension structure with section, and (iii) comprehension structure with image. Our approach is 2-categorical and we thus formulate the three levels of comprehension structure on a general morphism $p:\mathrm{\mathbf{E}}\to\mathrm{\mathbf{B}}$ in a 2-category $\mathscr{K}$. This conceptual point of view on comprehension structures enables us to revisit the work by Fumex, Ghani and Johann on the duality between comprehension structures and quotient structures on a given functor $p:\mathscr{E}\to\mathscr{B}$. In particular, we show how to lift the comprehension and quotient structures on a functor $p:\mathscr{E}\to\mathscr{B}$ to the categories of algebras or coalgebras associated to functors $F_{\mathscr{E}}:\mathscr{E}\to\mathscr{E}$ and $F_{\mathscr{B}}:\mathscr{B}\to\mathscr{B}$ of interest, in order to interpret reasoning by induction and coinduction in the traditional language of categorical logic, formulated in an appropriate 2-categorical way.
