6,831 research outputs found
Homotopy Theoretic Models of Type Theory
We introduce the notion of a logical model category which is a Quillen model
category satisfying some additional conditions. Those conditions provide enough
expressive power that one can soundly interpret dependent products and sums in
it. On the other hand, those conditions are easy to check and provide a wide
class of models some of which are listed in the paper.Comment: Corrected version of the published articl
Fibred Fibration Categories
We introduce fibred type-theoretic fibration categories which are fibred
categories between categorical models of Martin-L\"{o}f type theory. Fibred
type-theoretic fibration categories give a categorical description of logical
predicates for identity types. As an application, we show a relational
parametricity result for homotopy type theory. As a corollary, it follows that
every closed term of type of polymorphic endofunctions on a loop space is
homotopic to some iterated concatenation of a loop
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
Models of Martin-L\"of type theory from algebraic weak factorisation systems
We introduce type-theoretic algebraic weak factorisation systems and show how
they give rise to homotopy-theoretic models of Martin-L\"of type theory. This
is done by showing that the comprehension category associated to a
type-theoretic algebraic weak factorisation system satisfies the assumptions
necessary to apply a right adjoint method for splitting comprehension
categories. We then provide methods for constructing several examples of
type-theoretic algebraic weak factorisation systems, encompassing the existing
groupoid model and cubical sets models, as well as some models based on normal
fibrationsComment: Changed title (it used to be "Type-theoretic algebraic weak
factorisation systems"); rewritten introduction; fixed typos; fixed
inaccuracy in Lemma 2.3 spotted by Paige North; added references. 37 page
Models of Martin-Löf Type Theory from Algebraic Weak Factorisation Systems
We introduce type-theoretic algebraic weak factorisation systems and show how they give rise to homotopy-theoretic models of Martin-Löf type theory. This is done by showing that the comprehension category associated to a type-theoretic algebraic weak factorisation system satisfies the assumptions necessary to apply a right adjoint method for splitting comprehension categories. We then provide methods for constructing several examples of type-theoretic algebraic weak factorisation systems, encompassing the existing groupoid and cubical sets models, as well as new models based on normal fibrations
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
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