141 research outputs found
Partial Univalence in n-truncated Type Theory
It is well known that univalence is incompatible with uniqueness of identity
proofs (UIP), the axiom that all types are h-sets. This is due to finite h-sets
having non-trivial automorphisms as soon as they are not h-propositions.
A natural question is then whether univalence restricted to h-propositions is
compatible with UIP. We answer this affirmatively by constructing a model where
types are elements of a closed universe defined as a higher inductive type in
homotopy type theory. This universe has a path constructor for simultaneous
"partial" univalent completion, i.e., restricted to h-propositions.
More generally, we show that univalence restricted to -types is
consistent with the assumption that all types are -truncated. Moreover we
parametrize our construction by a suitably well-behaved container, to abstract
from a concrete choice of type formers for the universe.Comment: 21 pages, long version of paper accepted at LICS 202
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
Signatures and Induction Principles for Higher Inductive-Inductive Types
Higher inductive-inductive types (HIITs) generalize inductive types of
dependent type theories in two ways. On the one hand they allow the
simultaneous definition of multiple sorts that can be indexed over each other.
On the other hand they support equality constructors, thus generalizing higher
inductive types of homotopy type theory. Examples that make use of both
features are the Cauchy real numbers and the well-typed syntax of type theory
where conversion rules are given as equality constructors. In this paper we
propose a general definition of HIITs using a small type theory, named the
theory of signatures. A context in this theory encodes a HIIT by listing the
constructors. We also compute notions of induction and recursion for HIITs, by
using variants of syntactic logical relation translations. Building full
categorical semantics and constructing initial algebras is left for future
work. The theory of HIIT signatures was formalised in Agda together with the
syntactic translations. We also provide a Haskell implementation, which takes
signatures as input and outputs translation results as valid Agda code
Modalities in homotopy type theory
Univalent homotopy type theory (HoTT) may be seen as a language for the
category of -groupoids. It is being developed as a new foundation for
mathematics and as an internal language for (elementary) higher toposes. We
develop the theory of factorization systems, reflective subuniverses, and
modalities in homotopy type theory, including their construction using a
"localization" higher inductive type. This produces in particular the
(-connected, -truncated) factorization system as well as internal
presentations of subtoposes, through lex modalities. We also develop the
semantics of these constructions
Injective types in univalent mathematics
We investigate the injective types and the algebraically injective types in
univalent mathematics, both in the absence and in the presence of propositional
resizing. Injectivity is defined by the surjectivity of the restriction map
along any embedding, and algebraic injectivity is defined by a given section of
the restriction map along any embedding. Under propositional resizing axioms,
the main results are easy to state: (1) Injectivity is equivalent to the
propositional truncation of algebraic injectivity. (2) The algebraically
injective types are precisely the retracts of exponential powers of universes.
(2a) The algebraically injective sets are precisely the retracts of powersets.
(2b) The algebraically injective -types are precisely the retracts of
exponential powers of universes of -types. (3) The algebraically injective
types are also precisely the retracts of algebras of the partial-map
classifier. From (2) it follows that any universe is embedded as a retract of
any larger universe. In the absence of propositional resizing, we have similar
results which have subtler statements that need to keep track of universe
levels rather explicitly, and are applied to get the results that require
resizing.Comment: Includes revisions after review proces
The Compatibility of the Minimalist Foundation with Homotopy Type Theory
The Minimalist Foundation, for short MF, is a two-level foundation for
constructive mathematics ideated by Maietti and Sambin in 2005 and then fully
formalized by Maietti in 2009. MF serves as a common core among the most
relevant foundations for mathematics in the literature by choosing for each of
them the appropriate level of MF to be translated in a compatible way, namely
by preserving the meaning of logical and set-theoretical constructors. The
two-level structure consists of an intensional level, an extensional one, and
an interpretation of the latter in the former in order to extract intensional
computational contents from mathematical proofs involving extensional
constructions used in everyday mathematical practice. In 2013 a completely new
foundation for constructive mathematics appeared in the literature, called
Homotopy Type Theory, for short HoTT, which is an example of Voevodsky's
Univalent Foundations with a computational nature. So far no level of MF has
been proved to be compatible with any of the Univalent Foundations in the
literature. Here we show that both levels of MF are compatible with HoTT. This
result is made possible thanks to the peculiarities of HoTT which combines
intensional features of type theory with extensional ones by assuming
Voevodsky's Univalence Axiom and higher inductive quotient types. As a relevant
consequence, MF inherits entirely new computable models
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