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

We set out to study the consequences of the assumption of types of wellfounded trees in dependent type theories. We do so by investigating the categorical notion of wellfounded tree introduced in [16]. Our main result shows that wellfounded trees allow us to define initial algebras for a wide class of endofunctors on locally cartesian closed categories.

Image factorizations in regular categories are stable under pullbacks, so they model a natural modal operator in dependent type theory. This unary type constructor [A] has turned up previously in a syntactic form as a way of erasing computational content, and formalizing a notion of proof irrelevance. Indeed, semantically, the notion of a support is sometimes used as surrogate proposition asserting inhabitation of an indexed family. We give rules for bracket types in dependent type theory and provide complete semantics using regular categories. We show that dependent type theory with the unit type, strong extensional equality types, strong dependent sums, and bracket types is the internal type theory of regular categories, in the same way that the usual dependent type theory with dependent sums and products is the internal type theory of locally cartesian closed categories. We also show how to interpret rst-order logic in type theory with brackets, and we make use of the translation to compare type theory with logic. Specically, we show that the propositions-as-types interpretation is complete with respect to a certain fragment of intuitionistic rst-order logic. As a consequence, a modied double-negation translation into type theory (without bracket types) is complete for all of classical rst-order logic.

Abstract. We provide a monadic view on implicit and noncomputational arguments. This allows us to treat Berger’s non-computational quantifiers in the Coq-system. We use Tait’s normalization proof and the concatenation of vectors as case studies for the extraction of programs. With little effort one can eliminate noncomputational arguments from extracted programs. One thus obtains extracted code that is not only closer to the intended one, but also decreases both the running time and the memory usage dramatically. We also study the connection between Harrop formulas, lax modal logic and the Coq type theory.