9,138 research outputs found
Constructing Infinitary Quotient-Inductive Types
This paper introduces an expressive class of quotient-inductive types, called
QW-types. We show that in dependent type theory with uniqueness of identity
proofs, even the infinitary case of QW-types can be encoded using the
combination of inductive-inductive definitions involving strictly positive
occurrences of Hofmann-style quotient types, and Abel's size types. The latter,
which provide a convenient constructive abstraction of what classically would
be accomplished with transfinite ordinals, are used to prove termination of the
recursive definitions of the elimination and computation properties of our
encoding of QW-types. The development is formalized using the Agda theorem
prover
Constructing Infinitary Quotient-Inductive Types
This paper introduces an expressive class of quotient-inductive types, called
QW-types. We show that in dependent type theory with uniqueness of identity
proofs, even the infinitary case of QW-types can be encoded using the
combination of inductive-inductive definitions involving strictly positive
occurrences of Hofmann-style quotient types, and Abel's size types. The latter,
which provide a convenient constructive abstraction of what classically would
be accomplished with transfinite ordinals, are used to prove termination of the
recursive definitions of the elimination and computation properties of our
encoding of QW-types. The development is formalized using the Agda theorem
prover.Comment: The accompanying Agda code can be found at
https://doi.org/10.17863/CAM.4818
W-types in setoids
W-types and their categorical analogue, initial algebras for polynomial
endofunctors, are an important tool in predicative systems to replace
transfinite recursion on well-orderings. Current arguments to obtain W-types in
quotient completions rely on assumptions, like Uniqueness of Identity Proofs,
or on constructions that involve recursion into a universe, that limit their
applicability to a specific setting. We present an argument, verified in Coq,
that instead uses dependent W-types in the underlying type theory to construct
W-types in the setoid model. The immediate advantage is to have a proof more
type-theoretic in flavour, which directly uses recursion on the underlying
W-type to prove initiality. Furthermore, taking place in intensional type
theory and not requiring any recursion into a universe, it may be generalised
to various categorical quotient completions, with the aim of finding a uniform
construction of extensional W-types.Comment: 17 pages, formalised in Coq; v2: added reference to formalisatio
Homotopy Type Theory in Lean
We discuss the homotopy type theory library in the Lean proof assistant. The
library is especially geared toward synthetic homotopy theory. Of particular
interest is the use of just a few primitive notions of higher inductive types,
namely quotients and truncations, and the use of cubical methods.Comment: 17 pages, accepted for ITP 201
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
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