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, 2008

"... 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 co ..."

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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.