[论文解读] W-types in sheaves

Benno van den Berg, Ieke Moerdijk|arXiv (Cornell University)|Oct 14, 2008

Homotopy and Cohomology in Algebraic Topology参考文献 5被引用 3

一句话总结

本文對层范畴中的W-类型提供了正确且具体的描述，纠正了先前工作中一个错误的断言：即层中的W-类型可与预层中的W-类型以相同方式计算。本文表明，预层与层中初始对象（即终端对象上恒等映射下的W-类型）是不同的，并利用格罗滕迪克范畴与筛子给出了精确的构造，为层拓扑中的类型理论奠定了基础。

ABSTRACT

In this small note we give a concrete description of W-types in categories of sheaves. It can be shown that any topos with a natural numbers object has all Wtypes. Although there is this general result, it can be useful to have a concrete description of W-types in various toposes. For example, a concrete description of W-types in the effective topos can be found in [2, 3], and a concrete description of W-types in categories of presheaves was given in [5]. It was claimed in [5] that W-types in categories of sheaves are computed as in presheaves (Proposition 5.7 in loc.cit.) and can therefore be described in the same way. Unfortunately, this claim is incorrect, as the following (easy) counterexample shows. Let f : 1 → 1 be the identity map on the terminal object. The W-type associated to f is the initial object, which, in general, is different in categories of presheaves and sheaves. This means that we still lack a concrete description of W-types in categories of sheaves. This note aims to fill this gap. We would like to warn readers who are sensitive to such issues that our metatheory is ZFC. In particular, we freely use the axiom of choice. We leave the issue of how to describe W-types in categories of sheaves when the metatheory is more demanding (i.e. weaker) to another occasion. Categories of sheaves are described using (Grothendieck) sites. There are different formulations of the notion of a site, all essentially equivalent ([4] provides an excellent discussion of this point), but for our purposes we find the following (“sifted”) formulation the most useful. Definition 0.1 Let C be a category. A sieve S on an object a ∈ C consists of a set of arrows in C all having codomain a and closed under precomposition (i.e., if f : b → a and g: c → b are arrows in C and f belongs to S, then so does fg). We call the set Ma of all arrows into a the maximal sieve on a. If S is a sieve on a and f : b → a is any map in C, we write f∗S for the sieve {g: c → b : fg ∈ S} on b. In case f belongs to S, we have f∗S = Mb. A (Grothendieck) topology Cov on C is given by assigning to every object a ∈ C a collection of sieves Cov(a) such that the following axioms are satisfied:

研究动机与目标

• 为层范畴中W-类型的正确且具体描述提供支持，尽管在具有自然数对象的拓扑中它们存在。
• 纠正[5]中关于层中W-类型与预层中W-类型计算方式相同的错误断言。
• 使用格罗滕迪克范畴与筛子的语言，为层拓扑中的W-类型提供精确且可操作的构造。
• 为层拓扑中的类型论构造建立基础，尤其适用于构造性与范畴逻辑的应用。

提出的方法

• 使用格罗滕迪克范畴与筛子的框架来描述层范畴，重点采用‘筛子化’表述以增强清晰度与实用性。
• 将对象a上的筛子S定义为以a为余域的箭头集合，且在前复合下封闭，并引入筛子的拉回运算f∗S。
• 通过依赖族f: A → B应用W-类型的标准归纳构造，将W-类型解释为与f相关的多项式函子的初始代数。
• 通过反例表明，层中的W-类型构造与预层中的不同：与终端对象上恒等映射相关的W-类型在层中与在预层中并不相同。
• 证明在层中正确的构造需要对预层级别的W-类型进行层化，以确保在层拓扑中满足普遍性质。
• 依赖ZFC作为元理论，包含选择公理，并承认若使用较弱的元理论则需单独处理。

实验结果

研究问题

• RQ1 W-类型在层范畴中如何构造，它们与预层中的对应物有何不同？
• RQ2 为何[5]中关于层中W-类型以与预层相同方式计算的断言是错误的？
• RQ3 使用格罗滕迪克范畴与筛子，W-类型在层拓扑中的正确且具体描述是什么？
• RQ4 层范畴中的初始对象与预层范畴中的初始对象相比如何？这对W-类型有何含义？
• RQ5 能否给出一个统一的层中W-类型构造，使其满足层条件与W-类型的普遍性质？

主要发现

• [5]中关于层中W-类型与预层中W-类型计算方式相同的断言是错误的，这一结论通过涉及终端对象上恒等映射的反例得到证实。
• 与恒等映射f: 1 → 1相关的W-类型在任何拓扑中都是初始对象，但该对象在预层范畴与层范畴中是不同的。
• 层中W-类型的构造需要对预层级别的W-类型进行层化，这意味着层中的W-类型并非简单地将预层W-类型视为层。
• 本文使用格罗滕迪克范畴与筛子的语言，为层中的W-类型提供了正确且具体的描述，确保满足W-类型的普遍性质。
• 本文确立了：尽管所有具有自然数对象的拓扑都具有所有W-类型，但在层拓扑中给出具体描述是非平凡的，需要仔细处理层条件。
• 该结果为层拓扑中的类型理论提供了基础工具，尤其适用于构造性与 predicative 数学。