853 research outputs found
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
Univalent Foundations and the UniMath Library
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
The real projective spaces in homotopy type theory
Homotopy type theory is a version of Martin-L\"of type theory taking
advantage of its homotopical models. In particular, we can use and construct
objects of homotopy theory and reason about them using higher inductive types.
In this article, we construct the real projective spaces, key players in
homotopy theory, as certain higher inductive types in homotopy type theory. The
classical definition of RP(n), as the quotient space identifying antipodal
points of the n-sphere, does not translate directly to homotopy type theory.
Instead, we define RP(n) by induction on n simultaneously with its tautological
bundle of 2-element sets. As the base case, we take RP(-1) to be the empty
type. In the inductive step, we take RP(n+1) to be the mapping cone of the
projection map of the tautological bundle of RP(n), and we use its universal
property and the univalence axiom to define the tautological bundle on RP(n+1).
By showing that the total space of the tautological bundle of RP(n) is the
n-sphere, we retrieve the classical description of RP(n+1) as RP(n) with an
(n+1)-cell attached to it. The infinite dimensional real projective space,
defined as the sequential colimit of the RP(n) with the canonical inclusion
maps, is equivalent to the Eilenberg-MacLane space K(Z/2Z,1), which here arises
as the subtype of the universe consisting of 2-element types. Indeed, the
infinite dimensional projective space classifies the 0-sphere bundles, which
one can think of as synthetic line bundles.
These constructions in homotopy type theory further illustrate the utility of
homotopy type theory, including the interplay of type theoretic and homotopy
theoretic ideas.Comment: 8 pages, to appear in proceedings of LICS 201
Semi-simplicial Types in Logic-enriched Homotopy Type Theory
The problem of defining Semi-Simplicial Types (SSTs) in Homotopy Type Theory
(HoTT) has been recognized as important during the Year of Univalent
Foundations at the Institute of Advanced Study. According to the interpretation
of HoTT in Quillen model categories, SSTs are type-theoretic versions of Reedy
fibrant semi-simplicial objects in a model category and simplicial and
semi-simplicial objects play a crucial role in many constructions in homotopy
theory and higher category theory. Attempts to define SSTs in HoTT lead to some
difficulties such as the need of infinitary assumptions which are beyond HoTT
with only non-strict equality types.
Voevodsky proposed a definition of SSTs in Homotopy Type System (HTS), an
extension of HoTT with non-fibrant types, including an extensional strict
equality type. However, HTS does not have the desirable computational
properties such as decidability of type checking and strong normalization. In
this paper, we study a logic-enriched homotopy type theory, an alternative
extension of HoTT with equational logic based on the idea of logic-enriched
type theories. In contrast to Voevodskys HTS, all types in our system are
fibrant and it can be implemented in existing proof assistants. We show how
SSTs can be defined in our system and outline an implementation in the proof
assistant Plastic
Sets in homotopy type theory
Homotopy Type Theory may be seen as an internal language for the
-category of weak -groupoids which in particular models the
univalence axiom. Voevodsky proposes this language for weak -groupoids
as a new foundation for mathematics called the Univalent Foundations of
Mathematics. It includes the sets as weak -groupoids with contractible
connected components, and thereby it includes (much of) the traditional set
theoretical foundations as a special case. We thus wonder whether those
`discrete' groupoids do in fact form a (predicative) topos. More generally,
homotopy type theory is conjectured to be the internal language of `elementary'
-toposes. We prove that sets in homotopy type theory form a -pretopos. This is similar to the fact that the -truncation of an
-topos is a topos. We show that both a subobject classifier and a
-object classifier are available for the type theoretical universe of sets.
However, both of these are large and moreover, the -object classifier for
sets is a function between -types (i.e. groupoids) rather than between sets.
Assuming an impredicative propositional resizing rule we may render the
subobject classifier small and then we actually obtain a topos of sets
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