We prove that the category of trees Ω is a test category in the sense of Grothendieck. This implies that the category of dendroidal sets is endowed with the structure of a model category Quillen-equivalent to spaces. We show that this model category structure, up to a change of cofibrations, can be obtained as an explicit left Bousfield localisation of the operadic model category structure.

In the article below, Theorem 3.4 requires the additional assumption that A is Kan as well. Indeed, the inductive proof as given only shows that if W(f)<α is a Kan complex, then W(f)<α+1 → A is a Kan fibration.

We will give a detailed account of why the simplicial sets model of the univalence axiom due to Voevodsky also models W-types. In addition, we will discuss W-types in categories of simplicial presheaves and an application to models of set theory.

For a site ${\mathcal S}$ (with enough points), we construct a topological space X$_($)$ and a full embedding φ$^*$ of the category of sheaves on ${\mathcal S}$ into those on X$_($) (i.e., a morphism of toposes φ:Sh (X$_($) →Sh(${\mathcal S}$)). The embedding will be shown to induce a full embedding of derived categories, hence isomorphisms coh$^*$(${\mathcal S}$,A) =coh$^**$(X$_($), φ$^*$A) for any Abelian sheaf A on ${\mathcal S}$. As a particular case, this will give for any scheme Y a topological space X and a functorial isomorphism between the étale cohomology H$^**$(et{Y},A) and the ordinary sheaf cohomology H$^**$(X,φ$^**$A), for any sheaf A for the étale topology on Y.

In this paper, we give a constructive nonstandard model of intuitionistic arithmetic (Heyting arithmetic). We present two axiomatisations of the model: one finitary and one infinitary variant. Using the model these axiomatisations are proven to be conservative over ordinary intuitionistic arithmetic. The definition of the model along with the proofs of its properties may be carried out within a constructive and predicative metatheory (such as Martin-Löf's type theory). This paper gives an illustration of the use of sheaf semantics to obtain effective proof-theoretic results.

The axiomatisations of nonstandard intuitionistic arithmetic (to be called HAI and HAIω respectively) as well as their model are based on the construction in [5] of a sheaf model for arithmetic using a site of filters. In this paper we present a “minimal” version of this model, built instead on a suitable site of provable filter bases. The construction of this site can be viewed as an extension of the well-known construction of the classifying topos for a geometric theory which uses “syntactic sites”. (Such sites can in fact be used to prove semantical completeness of first order logic in a strictly constructive framework, see [6].)

We should mention that for classical nonstandard arithmetics there are several nonconstructive methods of proving conservativity over arithmetic, e.g. the compactness theorem, Mac Dowell–Specker's theorem [3].