 
Summary: Wtypes in sheaves
Benno van den Berg & Ieke Moerdijk
September 25, 2008
Abstract
In this small note we give a concrete description of Wtypes in categories
of sheaves.
It can be shown that any topos with a natural numbers object has all W
types. Although there is this general result, it can be useful to have a concrete
description of Wtypes in various toposes. For example, a concrete description of
Wtypes in the effective topos can be found in [2, 3], and a concrete description
of Wtypes in categories of presheaves was given in [5]. It was claimed in [5] that
Wtypes 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 Wtype 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 Wtypes 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
