[Paper Review] W-types in sheaves

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

Homotopy and Cohomology in Algebraic Topology5 references3 citations

TL;DR

This paper provides a correct and concrete description of W-types in categories of sheaves, correcting an erroneous claim in prior work that W-types in sheaves could be computed identically to those in presheaves. It demonstrates that the initial object (a W-type under the identity map on the terminal object) differs between presheaves and sheaves, and offers a precise construction using Grothendieck sites and sieves, establishing a foundational tool for type theory in sheaf toposes.

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:

Motivation & Objective

• To address the lack of a correct, concrete description of W-types in categories of sheaves, despite their existence in toposes with a natural numbers object.
• To correct the incorrect claim in [5] that W-types in sheaves are computed identically to those in presheaves.
• To provide a precise, usable construction of W-types in sheaf toposes using the language of Grothendieck sites and sieves.
• To establish a foundation for type-theoretic constructions in sheaf toposes, particularly for applications in constructive and categorical logic.

Proposed method

• Uses the framework of Grothendieck sites with sieves to describe sheaf categories, focusing on the 'sifted' formulation for clarity and utility.
• Defines a sieve S on an object a as a set of arrows with codomain a, closed under precomposition, and introduces the pullback operation f∗S for sieves.
• Applies the standard inductive construction of W-types via a dependent family f: A → B, interpreting the W-type as the initial algebra for the polynomial functor associated with f.
• Demonstrates that the W-type construction in sheaves differs from that in presheaves by exhibiting a counterexample: the W-type associated to the identity map on the terminal object is not the same in sheaves as in presheaves.
• Establishes that the correct construction in sheaves requires sheafification of the presheaf-level W-type, ensuring the universal property holds in the sheaf topos.
• Relies on ZFC as the metatheory, including the Axiom of Choice, and acknowledges that weaker metatheories would require separate treatment.

Experimental results

Research questions

• RQ1 How are W-types constructed in categories of sheaves, and how do they differ from their presheaf counterparts?
• RQ2 Why is the claim in [5] that W-types in sheaves are computed as in presheaves incorrect?
• RQ3 What is the correct, concrete description of W-types in sheaf toposes using Grothendieck sites and sieves?
• RQ4 How does the initial object in the category of sheaves compare to that in the category of presheaves, and what does this imply for W-types?
• RQ5 Can a uniform construction of W-types in sheaves be given that respects the sheaf condition and the universal property of W-types?

Key findings

• The claim in [5] that W-types in sheaves are computed identically to those in presheaves is incorrect, as demonstrated by a counterexample involving the identity map on the terminal object.
• The W-type associated to the identity map f: 1 → 1 is the initial object in any topos, but this object differs between the category of presheaves and the category of sheaves.
• The construction of W-types in sheaves requires sheafification of the presheaf-level W-type, meaning the W-type in sheaves is not simply the presheaf W-type viewed as a sheaf.
• A correct and concrete description of W-types in sheaves is provided using the language of Grothendieck sites and sieves, ensuring the universal property of W-types is satisfied.
• The paper establishes that while all toposes with a natural numbers object have all W-types, a concrete description in sheaf toposes is non-trivial and requires careful handling of the sheaf condition.
• The result provides a foundational tool for type theory in sheaf toposes, particularly for constructive and predicative mathematics.