论文标题
内部高等类别理论中的呈现性和拓扑
Presentability and topoi in internal higher category theory
论文作者
论文摘要
本文的目的是开发出现的类别理论，并在任意$ \ intcos $ -TOPOS $ \ MATHCAL {B} $的内部内部进行托普伊。我们的主要结果是Lurie's和Lurie-Simpson的内部类似物，其特征是$ \ infty $ - 类别和$ \ infty $ -topoi。在此过程中，我们介绍了一种内部过滤性和可访问的内部类别的理论，并建立了许多关于可呈现的$ \ MATHCAL {B} $类别的结构性结果，例如伴随函数定理以及Lurie Tensor产品的内部类似物的存在。我们还将这些内部概念与外部变体进行比较。我们表明，$ \ MATHCAL {B} $ - 模块完全忠实地嵌入了$ \ Mathcal {B} $ - 类别中，并证明topoi内部到$ \ MATHCAL {B} $和$ \ iffty $ -topoi of $ \ mathcal over \ nathcal of \ nathcal {b} $ {b} $ {b} $。我们还包括许多结果的应用，例如$ \ infty $ -topoi的dioconescu定理的一般版本，以及在平稳性方面对本地合约几何形态的表征。
The goal of this article is to develop the theory of presentable categories and topoi internal to an arbitrary $\infty$-topos $\mathcal{B}$. Our main results are internal analogues of Lurie's and Lurie-Simpson's characterisations of presentable $\infty$-categories and $\infty$-topoi. In the process, we introduce a theory of internal filteredness and accessible internal categories and establish a number of structural results about presentable $\mathcal{B}$-categories such as adjoint functor theorems and the existence of an internal analogue of the Lurie tensor product. We also compare these internal notions with external variants. We show that $\mathcal{B}$-modules embed fully faithfully into presentable $\mathcal{B}$-categories and prove that there is an equivalence between topoi internal to $\mathcal{B}$ and $\infty$-topoi over $\mathcal{B}$. We also include a number of applications of our results, such as a general version of Diaconescu's theorem for $\infty$-topoi and a characterisation of locally contractible geometric morphisms in terms of smoothness.
