24 research outputs found
Univalent Foundations and the UniMath Library. The Architecture of Mathematics.
We give a concise presentation of the Univalent Foundations of mathematics outlining the main ideas, followed by a discussion of the UniMath library of formalized mathematics implementing the ideas of the Univalent Foundations (section 1), and the challenges one faces in attempting to design a large-scale library of formalized mathematics (section 2). This leads us to a general discussion about the links between architecture and mathematics where a meeting of minds is revealed between architects and mathematicians (section 3). On the way our odyssey from the foundations to the "horizon" of mathematics will lead us to meet the mathematicians David Hilbert and Nicolas Bourbaki as well as the architect Christopher Alexander
Univalent Foundations and the UniMath Library. The Architecture of Mathematics.
We give a concise presentation of the Univalent Foundations of mathematics outlining the main ideas, followed by a discussion of the UniMath library of formalized mathematics implementing the ideas of the Univalent Foundations (section 1), and the challenges one faces in attempting to design a large-scale library of formalized mathematics (section 2). This leads us to a general discussion about the links between architecture and mathematics where a meeting of minds is revealed between architects and mathematicians (section 3). On the way our odyssey from the foundations to the "horizon" of mathematics will lead us to meet the mathematicians David Hilbert and Nicolas Bourbaki as well as the architect Christopher Alexander
Univalent Foundations and the UniMath Library. The Architecture of Mathematics.
We give a concise presentation of the Univalent Foundations of mathematics outlining the main ideas, followed by a discussion of the UniMath library of formalized mathematics implementing the ideas of the Univalent Foundations (section 1), and the challenges one faces in attempting to design a large-scale library of formalized mathematics (section 2). This leads us to a general discussion about the links between architecture and mathematics where a meeting of minds is revealed between architects and mathematicians (section 3). On the way our odyssey from the foundations to the "horizon" of mathematics will lead us to meet the mathematicians David Hilbert and Nicolas Bourbaki as well as the architect Christopher Alexander
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
Constructive sheaf models of type theory
We generalise sheaf models of intuitionistic logic to univalent type theory
over a small category with a Grothendieck topology. We use in a crucial way
that we have constructive models of univalence, that can then be relativized to
any presheaf models, and these sheaf models are obtained by localisation for a
left exact modality. We provide first an abstract notion of descent data which
can be thought of as a higher version of the notion of prenucleus on frames,
from which can be generated a nucleus (left exact modality) by transfinite
iteration. We then provide several examples.Comment: Simplified the definition of lex operation, simplified the encoding
of the homotopy limit and remark that the homotopy descent data is a lex
modality without using higher inductive type