26 research outputs found
Functional Kan Simplicial Sets: Non-Constructivity of Exponentiation
Functional Kan simplicial sets are simplicial sets in which the horn-fillers required by the Kan extension condition are given explicitly by functions. We show the non-constructivity of the following basic result: if B and A are functional Kan simplicial sets, then A^B is a Kan simplicial set. This strengthens a similar result for the case of non-functional Kan simplicial sets shown by Bezem, Coquand and Parmann [TLCA 2015, v. 38 of LIPIcs]. Our
result shows that-from a constructive point of view-functional
Kan simplicial sets are, as it stands, unsatisfactory as a model of even simply typed lambda calculus. Our proof is based on a rather involved Kripke countermodel which has been encoded and verified in the Coq proof assistant
Towards a constructive simplicial model of Univalent Foundations
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 classifying fibration
for small fibrations between bifibrant objects. These results allow us to
define a comprehension category supporting identity types, -types,
-types and a univalent universe, leaving only a coherence question to be
addressed.Comment: v3: changed the definition of the type Weq(U) of weak equivalences to
fix a problem with constructivity. Other Minor changes. 31 page
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
A new foundational crisis in mathematics, is it really happening?
The article reconsiders the position of the foundations of mathematics after
the discovery of HoTT. Discussion that this discovery has generated in the
community of mathematicians, philosophers and computer scientists might
indicate a new crisis in the foundation of mathematics. By examining the
mathematical facts behind HoTT and their relation with the existing
foundations, we conclude that the present crisis is not one. We reiterate a
pluralist vision of the foundations of mathematics. The article contains a
short survey of the mathematical and historical background needed to understand
the main tenets of the foundational issues.Comment: Final versio
Relative elegance and cartesian cubes with one connection
We establish a Quillen equivalence between the Kan-Quillen model structure
and a model structure, derived from a model of a cubical type theory, on the
category of cartesian cubical sets with one connection. We thereby identify a
second model structure which both constructively models homotopy type theory
and presents infinity-groupoids, the first known example being the equivariant
cartesian model of Awodey-Cavallo-Coquand-Riehl-Sattler.Comment: 60 pages. Comments welcome