We show that for any uncountable cardinal λ, the category of sets of cardinality at least λ and monomorphisms between them cannot appear as the category of point of a topos, in particular is not the category of models of a L (∞,ω) -theory. More generally we show that for any regular cardinal κ < λ it is neither the category of κ-points of a κ-topos, in particular, not the category of models of a L (∞,κ) -theory.The proof relies on the construction of a categorified version of the Scott topology, which constitute a left adjoint to the functor sending any topos to its category of points and the computation of this left adjoint evaluated on the category of sets of cardinality at least λ and monomorphisms between them. The same techniques also applies to a few other categories. At least to the category of vector spaces of with bounded below dimension and the category of algebraic closed fields of fixed characteristic with bounded below transcendence degree. This is also mentioned in section 5 of [2]. And we will prove some more general claims along the same lines ( 2.9, 3.3).

In this paper we prove, as conjectured by B.Banachewski and C.J.Mulvey, that the constructive Gelfand duality can be extended into a duality between compact regular locales and unital abelian localic C * -algebras. In order to do so we develop a constructive theory of localic metric spaces and localic Banach spaces, we study the notion of localic completion of such objects and the behaviour of these constructions with respect to pull-back along geometric morphisms.

We provide a partial solution to the problem of defining a constructive version of Voevodsky's simplicial model of Univalent Foundations. For this, we prove constructive counterparts of the necessary results of simplicial homotopy theory, building on the constructive version of the Kan‐Quillen model structure established by the second‐named author. In particular, we show that dependent products along fibrations with cofibrant domains preserve fibrations, establish the weak equivalence extension property for weak equivalences between fibrations with cofibrant domain and define a univalent fibration that classifies small fibrations between bifibrant objects. These results allow us to define a comprehension category supporting identity types, Σ$\Sigma$‐types, Π$\Pi$‐types and a univalent universe, leaving only a coherence question to be addressed.

In this paper we develop a notion of measure theory over boolean toposes which is analogous to noncommutative measure theory, i.e. to the theory of von Neumann algebras. This is part of a larger project to study relations between topos theory and noncommutative geometry. The main result is a topos theoretic version of the modular time evolution of von Neumann algebra which take the form of a canonical R >0 -principal bundle over any integrable locally separated boolean topos.

For a category $\mathcal {E}$ with finite limits and well-behaved countable coproducts, we construct a model structure, called the effective model structure, on the category of simplicial objects in $\mathcal {E}$ , generalising the Kan–Quillen model structure on simplicial sets. We then prove that the effective model structure is left and right proper and satisfies descent in the sense of Rezk. As a consequence, we obtain that the associated $\infty $ -category has finite limits, colimits satisfying descent, and is locally Cartesian closed when $\mathcal {E}$ is but is not a higher topos in general. We also characterise the $\infty $ -category presented by the effective model structure, showing that it is the full sub-category of presheaves on $\mathcal {E}$ spanned by Kan complexes in $\mathcal {E}$ , a result that suggests a close analogy with the theory of exact completions.