Abstract

this paper, we give an abstract 2 categorical characterization of W-types. We calculate these W-types explicitly in some categories of presheaves and sheaves on a site, and in the gluing category or Freyd cover. (We also have an explicit description in the case of Hyland's realizability topos, which will be presented in [17].) These explicit calculations can be formalized in a weak predicative metatheory, and lead to the result that if E is any suitably filtered pretopos with dependent products and W-types, then so is the category of internal sheaves on a site in E (Remark 5.9). Our paper is organized as follows. In Section 2 we review some standard definitions concerning pretoposes and dependent products. In Section 3 we present the categorical definition of the W-construction, and in Section 4 we prove some of its basic functoriality properties; e.g., that it turns coequalizers into equalizers. In Section 5, a construction is presented which to each map between (pre)sheaves of sets associates a sheaf of wellfounded trees, and it is proved that this is in fact the W-type in the category (pre)sheaves of sets (Theorem 5.6). In Section 6, we discuss the W-construction for the Freyd cover. Finally, in Section 7 it is shown how these categorical constructions are not only analogous to but explicitly related to Martin-Lof type theory. 2 Pretoposes and dependent products

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