ar X iv : 0 81 0 . 23 98 v 1 [ m at h . C T ] 1 4 O ct 2 00 8 W - types in sheaves

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 W-types. 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 metathe-ory 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 f g). We call the set M a 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 : f g ∈ S} on b. In case f belongs to S, we have f * S = M b. A (Grothendieck) topology Cov …