631 research outputs found
An Algebraic Weak Factorisation System on 01-Substitution Sets: A Constructive Proof
We will construct an algebraic weak factorisation system on the category of
01 substitution sets such that the R-algebras are precisely the Kan fibrations
together with a choice of Kan filling operation. The proof is based on Garner's
small object argument for algebraic weak factorization systems. In order to
ensure the proof is valid constructively, rather than applying the general
small object argument, we give a direct proof based on the same ideas. We use
this us to give an explanation why the J computation rule is absent from the
original cubical set model and suggest a way to fix this
CZF does not have the Existence Property
Constructive theories usually have interesting metamathematical properties
where explicit witnesses can be extracted from proofs of existential sentences.
For relational theories, probably the most natural of these is the existence
property, EP, sometimes referred to as the set existence property. This states
that whenever (\exists x)\phi(x) is provable, there is a formula \chi(x) such
that (\exists ! x)\phi(x) \wedge \chi(x) is provable. It has been known since
the 80's that EP holds for some intuitionistic set theories and yet fails for
IZF. Despite this, it has remained open until now whether EP holds for the most
well known constructive set theory, CZF. In this paper we show that EP fails
for CZF
Every metric space is separable in function realizability
We first show that in the function realizability topos every metric space is
separable, and every object with decidable equality is countable. More
generally, working with synthetic topology, every -space is separable and
every discrete space is countable. It follows that intuitionistic logic does
not show the existence of a non-separable metric space, or an uncountable set
with decidable equality, even if we assume principles that are validated by
function realizability, such as Dependent and Function choice, Markov's
principle, and Brouwer's continuity and fan principles
Evidence of the Poisson/Gaudin-Mehta phase transition for banded matrices on global scales
We prove that the Poisson/Gaudin--Mehta phase transition conjectured to occur
when the bandwidth of an symmetric banded matrix grows like is observable as a critical point in the fourth moment of the level density
for a wide class of symmetric banded matrices. A second critical point when the
bandwidth grows like leads to a new conjectured phase
transition in the eigenvalue localization, whose existence we demonstrate in
numerical experiments
Unifying Cubical Models of Univalent Type Theory
We present a new constructive model of univalent type theory based on cubical sets. Unlike prior work on cubical models, ours depends neither on diagonal cofibrations nor connections. This is made possible by weakening the notion of fibration from the cartesian cubical set model, so that it is not necessary to assume that the diagonal on the interval is a cofibration. We have formally verified in Agda that these fibrations are closed under the type formers of cubical type theory and that the model satisfies the univalence axiom. By applying the construction in the presence of diagonal cofibrations or connections and reversals, we recover the existing cartesian and De Morgan cubical set models as special cases. Generalizing earlier work of Sattler for cubical sets with connections, we also obtain a Quillen model structure
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