961 research outputs found
Higher Homotopies in a Hierarchy of Univalent Universes
For Martin-Lof type theory with a hierarchy U(0): U(1): U(2): ... of
univalent universes, we show that U(n) is not an n-type. Our construction also
solves the problem of finding a type that strictly has some high truncation
level without using higher inductive types. In particular, U(n) is such a type
if we restrict it to n-types. We have fully formalized and verified our results
within the dependently typed language and proof assistant Agda.Comment: v1: 30 pages, main results and a connectedness construction; v2: 14
pages, only main results, improved presentation, final journal version,
ancillary files with electronic appendix; v3: content unchanged, different
documentclass reduced the number of pages to 1
Two-Level Type Theory and Applications
We define and develop two-level type theory (2LTT), a version of Martin-L\"of
type theory which combines two different type theories. We refer to them as the
inner and the outer type theory. In our case of interest, the inner theory is
homotopy type theory (HoTT) which may include univalent universes and higher
inductive types. The outer theory is a traditional form of type theory
validating uniqueness of identity proofs (UIP). One point of view on it is as
internalised meta-theory of the inner type theory.
There are two motivations for 2LTT. Firstly, there are certain results about
HoTT which are of meta-theoretic nature, such as the statement that
semisimplicial types up to level can be constructed in HoTT for any
externally fixed natural number . Such results cannot be expressed in HoTT
itself, but they can be formalised and proved in 2LTT, where will be a
variable in the outer theory. This point of view is inspired by observations
about conservativity of presheaf models.
Secondly, 2LTT is a framework which is suitable for formulating additional
axioms that one might want to add to HoTT. This idea is heavily inspired by
Voevodsky's Homotopy Type System (HTS), which constitutes one specific instance
of a 2LTT. HTS has an axiom ensuring that the type of natural numbers behaves
like the external natural numbers, which allows the construction of a universe
of semisimplicial types. In 2LTT, this axiom can be stated simply be asking the
inner and outer natural numbers to be isomorphic.
After defining 2LTT, we set up a collection of tools with the goal of making
2LTT a convenient language for future developments. As a first such
application, we develop the theory of Reedy fibrant diagrams in the style of
Shulman. Continuing this line of thought, we suggest a definition of
(infinity,1)-category and give some examples.Comment: 53 page
Turing-Completeness of Polymorphic Stream Equation Systems
Polymorphic stream functions operate on the structure of streams, infinite sequences of elements, without inspection of the contained data, having to work on all streams over all signatures uniformly. A natural, yet restrictive class of polymorphic stream functions comprises those definable by a system of equations using only stream constructors and destructors and recursive calls. Using methods reminiscent of prior results in the field, we first show this class consists of exactly the computable polymorphic stream functions. Using much more intricate techniques, our main result states this holds true even for unary equations free of mutual recursion, yielding an elegant model of Turing-completeness in a severely restricted environment and allowing us to recover previous complexity results in a much more restricted setting
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
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