11 research outputs found
The Category of Iterative Sets in Homotopy Type Theory and Univalent Foundations
When working in Homotopy Type Theory and Univalent Foundations, the
traditional role of the category of sets, Set, is replaced by the category hSet
of homotopy sets (h-sets); types with h-propositional identity types. Many of
the properties of Set hold for hSet ((co)completeness, exactness, local
cartesian closure, etc.). Notably, however, the univalence axiom implies that
Ob(hSet) is not itself an h-set, but an h-groupoid. This is expected in
univalent foundations, but it is sometimes useful to also have a stricter
universe of sets, for example when constructing internal models of type theory.
In this work, we equip the type of iterative sets V0, due to Gylterud (2018) as
a refinement of the pioneering work of Aczel (1978) on universes of sets in
type theory, with the structure of a Tarski universe and show that it satisfies
many of the good properties of h-sets. In particular, we organize V0 into a
(non-univalent strict) category and prove that it is locally cartesian closed.
This enables us to organize it into a category with families with the structure
necessary to model extensional type theory internally in HoTT/UF. We do this in
a rather minimal univalent type theory with W-types, in particular we do not
rely on any HITs, or other complex extensions of type theory. Furthermore, the
construction of V0 and the model is fully constructive and predicative, while
still being very convenient to work with as the decoding from V0 into h-sets
commutes definitionally for all type constructors. Almost all of the paper has
been formalized in Agda using the agda-unimath library of univalent
mathematics
Bisimulation as path type for guarded recursive types
In type theory, coinductive types are used to represent processes, and are
thus crucial for the formal verification of non-terminating reactive programs
in proof assistants based on type theory, such as Coq and Agda. Currently,
programming and reasoning about coinductive types is difficult for two reasons:
The need for recursive definitions to be productive, and the lack of
coincidence of the built-in identity types and the important notion of
bisimilarity.
Guarded recursion in the sense of Nakano has recently been suggested as a
possible approach to dealing with the problem of productivity, allowing this to
be encoded in types. Indeed, coinductive types can be encoded using a
combination of guarded recursion and universal quantification over clocks. This
paper studies the notion of bisimilarity for guarded recursive types in Ticked
Cubical Type Theory, an extension of Cubical Type Theory with guarded
recursion. We prove that, for any functor, an abstract, category theoretic
notion of bisimilarity for the final guarded coalgebra is equivalent (in the
sense of homotopy type theory) to path equality (the primitive notion of
equality in cubical type theory). As a worked example we study a guarded notion
of labelled transition systems, and show that, as a special case of the general
theorem, path equality coincides with an adaptation of the usual notion of
bisimulation for processes. In particular, this implies that guarded recursion
can be used to give simple equational reasoning proofs of bisimilarity. This
work should be seen as a step towards obtaining bisimilarity as path equality
for coinductive types using the encodings mentioned above
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
Constructing Higher Inductive Types as Groupoid Quotients
In this paper, we study finitary 1-truncated higher inductive types (HITs) in
homotopy type theory. We start by showing that all these types can be
constructed from the groupoid quotient. We define an internal notion of
signatures for HITs, and for each signature, we construct a bicategory of
algebras in 1-types and in groupoids. We continue by proving initial algebra
semantics for our signatures. After that, we show that the groupoid quotient
induces a biadjunction between the bicategories of algebras in 1-types and in
groupoids. Then we construct a biinitial object in the bicategory of algebras
in groupoids, which gives the desired algebra. From all this, we conclude that
all finitary 1-truncated HITs can be constructed from the groupoid quotient.
We present several examples of HITs which are definable using our notion of
signature. In particular, we show that each signature gives rise to a HIT
corresponding to the freely generated algebraic structure over it. We also
start the development of universal algebra in 1-types. We show that the
bicategory of algebras has PIE limits, i.e. products, inserters and equifiers,
and we prove a version of the first isomorphism theorem for 1-types. Finally,
we give an alternative characterization of the foundamental groups of some
HITs, exploiting our construction of HITs via the groupoid quotient. All the
results are formalized over the UniMath library of univalent mathematics in
Coq
Greatest HITs: Higher Inductive Types in Coinductive Definitions via Induction under Clocks
Guarded recursion is a powerful modal approach to recursion that can be seen
as an abstract form of step-indexing. It is currently used extensively in
separation logic to model programming languages with advanced features by
solving domain equations also with negative occurrences. In its multi-clocked
version, guarded recursion can also be used to program with and reason about
coinductive types, encoding the productivity condition required for recursive
definitions in types. This paper presents the first type theory combining
multi-clocked guarded recursion with the features of Cubical Type Theory, as
well as a denotational semantics. Using the combination of Higher Inductive
Types (HITs) and guarded recursion allows for simple programming and reasoning
about coinductive types that are traditionally hard to represent in type
theory, such as the type of finitely branching labelled transition systems. For
example, our results imply that bisimilarity for these imply path equality, and
so proofs can be transported along bisimilarity proofs. Among our technical
contributions is a new principle of induction under clocks. This allows
universal quantification over clocks to commute with HITs up to equivalence of
types, and is crucial for the encoding of coinductive types. Such commutativity
requirements have been formulated for inductive types as axioms in previous
type theories with multi-clocked guarded recursion, but our present formulation
as an induction principle allows for the formulation of general computation
rules.Comment: 29 page
Finite sets in homotopy type theory
Contains fulltext :
184426.pdf (publisher's version ) (Closed access)CPP 2018: 7th ACM SIGPLAN International Conference on Certified Programs and Proofs, Los Angeles, CA, USA — January 08 - 09, 201
Kuratowski finite sets in the UniMath Library
This paper focuses on implementing and verifying the proofs presented in ``Finite Sets in Homotopy Type Theory" within the UniMath library. The UniMath library currently lacks support for higher inductive types, which are crucial for reasoning about finite sets in Homotopy Type Theory. This paper addresses that issue and introduces higher inductive types to UniMath. This is used to develop a computer-checked implementation of the proofs within "Finite Sets in Homotopy Type Theory." This implementation enables future research on finite sets in HoTT by providing accessible and reliable proofs. This paper defines finite sets as Kuratowski-finite. This is in contrast with the most common notion of finiteness, e.g. Bishop-finite and enumerated types. I argue that Kuratowski-finiteness is the most general finite for which the usual operations of finite types and sub-objects can be operated upon.CSE3000 Research ProjectComputer Science and Engineerin