1,191 research outputs found
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
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
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
Copredication in homotopy type theory
This paper applies homotopy type theory to formal semantics of natural languages and proposes a new model for the linguistic phenomenon of copredication. Copredication refers to sentences where two predicates which assume different requirements for their arguments are asserted for one single entity, e.g., "the lunch was delicious but took forever". This paper is particularly concerned with copredication sentences with quantification, i.e., cases where the two predicates impose distinct criteria of quantification and individuation, e.g., "Fred picked up and mastered three books." In our solution developed in homotopy type theory and using the rule of existential closure following Heim analysis of indefinites, common nouns are modeled as identifications of their aspects using HoTT identity types, e.g., the common noun book is modeled as identifications of its physical and informational aspects. The previous treatments of copredication in systems of semantics which are based on simple type theory and dependent type theories make the correct predictions but at the expense of ad hoc extensions (e.g., partial functions, dot types and coercive subtyping). The model proposed here, also predicts the correct results but using a conceptually simpler foundation and no ad hoc extensions
Towards a directed homotopy type theory
In this paper, we present a directed homotopy type theory for reasoning
synthetically about (higher) categories, directed homotopy theory, and its
applications to concurrency. We specify a new `homomorphism' type former for
Martin-L\"of type theory which is roughly analogous to the identity type former
originally introduced by Martin-L\"of. The homomorphism type former is meant to
capture the notions of morphism (from the theory of categories) and directed
path (from directed homotopy theory) just as the identity type former is known
to capture the notions of isomorphism (from the theory of groupoids) and path
(from homotopy theory). Our main result is an interpretation of these
homomorphism types into Cat, the category of small categories. There, the
interpretation of each homomorphism type hom(a,b) is indeed the set of
morphisms between the objects a and b of a category C. We end the paper with an
analysis of the interpretation in Cat with which we argue that our homomorphism
types are indeed the directed version of Martin-L\"of's identity types
Concrete Categories in Homotopy Type Theory
We introduce some classes of genuine higher categories in homotopy type
theory, defined as well-behaved subcategories of the category of types. We give
several examples, and some techniques for showing other things are not
examples. While only a small part of what is needed, it is a natural
construction, and may be instructive for people seeking to provide a fully
general construction.Comment: 18 page
Cellular Cohomology in Homotopy Type Theory
We present a development of cellular cohomology in homotopy type theory.
Cohomology associates to each space a sequence of abelian groups capturing part
of its structure, and has the advantage over homotopy groups in that these
abelian groups of many common spaces are easier to compute. Cellular cohomology
is a special kind of cohomology designed for cell complexes: these are built in
stages by attaching spheres of progressively higher dimension, and cellular
cohomology defines the groups out of the combinatorial description of how
spheres are attached. Our main result is that for finite cell complexes, a wide
class of cohomology theories (including the ones defined through
Eilenberg-MacLane spaces) can be calculated via cellular cohomology. This
result was formalized in the Agda proof assistant
W-types in Homotopy Type Theory
We will give a detailed account of why the simplicial sets model of the
univalence axiom due to Voevodsky also models W-types. In addition, we will
discuss W-types in categories of simplicial presheaves and an application to
models of set theory.Comment: We have corrected the statement of Theorem 3.4. We thank Christian
Sattler for alerting us to the error in the original versio
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