Related papers: Fibred sets within a predicative and constructive …
Here, we present a subcategory pEff of Hyland's Effective Topos Eff which can be considered a predicative variant of Eff itself. The construction of pEff is motivated by the desire of providing a "predicative" categorical universe of…
We show that Martin Hyland's effective topos can be exhibited as the homotopy category of a path category $\mathbb{EFF}$. Path categories are categories of fibrant objects in the sense of Brown satisfying two additional properties and as…
In the context of relative topos theory via stacks, we introduce the notion of existential fibred site and of existential topos of such a site. These notions allow us to develop relative topos theory in a way which naturally generalizes the…
In Feferman's work, explicit mathematics and theories of generalized inductive definitions play a central role. One objective of this article is to describe the connections with Martin-Lof type theory and constructive Zermelo-Fraenkel set…
This is the author's PhD thesis. It is a contribution to categorical logic, in particular to the theory of realizability toposes. While the tools of categorical logic have proven very successful in analyzing and organizing proof theoretic…
We present a way of constructing a Quillen model structure on a full subcategory of an elementary topos, starting with an interval object with connections and a certain dominance. The advantage of this method is that it does not require the…
Using the Feferman-Vaught Theorem, we prove that a definable subset of a product structure must be a Boolean combination of open sets, in the product topology induced by giving each factor structure the discrete topology. We prove a…
We study fibrations in the category of cubespaces/nilspaces. We show that a fibration of finite degree $f \colon X\rightarrow Y$ between compact ergodic gluing cubespaces (in particular nilspaces) factors as a (possibly countable) tower of…
We introduce the notion of an effective Kan fibration, a new mathematical structure that can be used to study simplicial homotopy theory. Our main motivation is to make simplicial homotopy theory suitable for homotopy type theory. Effective…
In this paper, we identify some categorical structures in which one can model predicative formal systems: in other words, predicative analogues of the notion of a topos, with the aim of using sheaf models to interprete predicative formal…
In this work we shall introduce a new model structure on the category of pro-simplicial sheaves, which is very convenient for the study of \'etale homotopy. Using this model structure we define a pro-space associated to a topos, as a result…
Definability is a key notion in the theory of Grothendieck fibrations that characterises when an external property of objects can be accessed from within the internal logic of the base of a fibration. In this paper we consider a…
We study a new proof principle in the context of constructive Zermelo-Fraenkel set theory based on what we will call "non-deterministic inductive definitions". We give applications to formal topology as well as a predicative justification…
In this survey, we remind some fibrations structure theorems (also called Milnor's fibrations) recently proved in the real and complex case, in the local and global settings. We give several Poincar\'e-Hopf type formulae which relates the…
We develop a categorical framework for reasoning about abstract properties of differentiation, based on the theory of fibrations. Our work encompasses the first-order fragments of several existing categorical structures for differentiation,…
We reconcile the two different category-theoretic semantics of regular theories in predicate logic. A 2-category of `regular fibrations' is constructed, as well as a 2-category of `regular proarrow equipments', and it is shown that the two…
Choice and independence of premise principles play an important role in characterizing Kreisel's modified realizability and G\"odel's Dialectica interpretation. In this paper we show that a great many intuitionistic set theories are closed…
A special final coalgebra theorem, in the style of Aczel's, is proved within standard Zermelo-Fraenkel set theory. Aczel's Anti-Foundation Axiom is replaced by a variant definition of function that admits non-well-founded constructions.…
We show how one may establish proof-theoretic results for constructive Zermelo-Fraenkel set theory, such as the compactness rule for Cantor space and the Bar Induction rule for Baire space, by constructing sheaf models and using their…
This is the third installment in a series of papers on algebraic set theory. In it, we develop a uniform approach to sheaf models of constructive set theories based on ideas from categorical logic. The key notion is that of a "predicative…