10 research outputs found
Idempotents in intensional type theory
We study idempotents in intensional Martin-L\"of type theory, and in
particular the question of when and whether they split. We show that in the
presence of propositional truncation and Voevodsky's univalence axiom, there
exist idempotents that do not split; thus in plain MLTT not all idempotents can
be proven to split. On the other hand, assuming only function extensionality,
an idempotent can be split if and only if its witness of idempotency satisfies
one extra coherence condition. Both proofs are inspired by parallel results of
Lurie in higher category theory, showing that ideas from higher category theory
and homotopy theory can have applications even in ordinary MLTT.
Finally, we show that although the witness of idempotency can be recovered
from a splitting, the one extra coherence condition cannot in general; and we
construct "the type of fully coherent idempotents", by splitting an idempotent
on the type of partially coherent ones. Our results have been formally verified
in the proof assistant Coq.Comment: 24 pages. v2: final version, to appear in LMC
Predicative aspects of order theory in univalent foundations
We investigate predicative aspects of order theory in constructive univalent foundations. By predicative and constructive, we respectively mean that we do not assume Voevodsky’s propositional resizing axioms or excluded middle. Our work complements existing work on predicative mathematics by exploring what cannot be done predicatively in univalent foundations. Our first main result is that nontrivial (directed or bounded) complete posets are necessarily large. That is, if such a nontrivial poset is small, then weak propositional resizing holds. It is possible to derive full propositional resizing if we strengthen nontriviality to positivity. The distinction between nontriviality and positivity is analogous to the distinction between nonemptiness and inhabitedness. We prove our results for a general class of posets, which includes directed complete posets, bounded complete posets and sup-lattices, using a technical notion of a δ_V-complete poset. We also show that nontrivial locally small δ_V-complete posets necessarily lack decidable equality. Specifically, we derive weak excluded middle from assuming a nontrivial locally small δ_V-complete poset with decidable equality. Moreover, if we assume positivity instead of nontriviality, then we can derive full excluded middle. Secondly, we show that each of Zorn’s lemma, Tarski’s greatest fixed point theorem and Pataraia’s lemma implies propositional resizing. Hence, these principles are inherently impredicative and a predicative development of order theory must therefore do without them. Finally, we clarify, in our predicative setting, the relation between the traditional definition of sup-lattice that requires suprema for all subsets and our definition that asks for suprema of all small families
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
On Small Types in Univalent Foundations
We investigate predicative aspects of constructive univalent foundations. By
predicative and constructive, we respectively mean that we do not assume
Voevodsky's propositional resizing axioms or excluded middle. Our work
complements existing work on predicative mathematics by exploring what cannot
be done predicatively in univalent foundations. Our first main result is that
nontrivial (directed or bounded) complete posets are necessarily large. That
is, if such a nontrivial poset is small, then weak propositional resizing
holds. It is possible to derive full propositional resizing if we strengthen
nontriviality to positivity. The distinction between nontriviality and
positivity is analogous to the distinction between nonemptiness and
inhabitedness. Moreover, we prove that locally small, nontrivial (directed or
bounded) complete posets necessarily lack decidable equality. We prove our
results for a general class of posets, which includes e.g. directed complete
posets, bounded complete posets, sup-lattices and frames. Secondly, we discuss
the unavailability of Zorn's lemma, Tarski's greatest fixed point theorem and
Pataraia's lemma in our predicative setting, and prove the ordinal of ordinals
in a univalent universe to have small suprema in the presence of set quotients.
The latter also leads us to investigate the inter-definability and interaction
of type universes of propositional truncations and set quotients, as well as a
set replacement principle. Thirdly, we clarify, in our predicative setting, the
relation between the traditional definition of sup-lattice that requires
suprema for all subsets and our definition that asks for suprema of all small
families.Comment: Extended version of arXiv:2102.08812. v2: Revised and expanded
following referee report
On Small Types in Univalent Foundations
We investigate predicative aspects of constructive univalent foundations. By
predicative and constructive, we respectively mean that we do not assume
Voevodsky's propositional resizing axioms or excluded middle. Our work
complements existing work on predicative mathematics by exploring what cannot
be done predicatively in univalent foundations. Our first main result is that
nontrivial (directed or bounded) complete posets are necessarily large. That
is, if such a nontrivial poset is small, then weak propositional resizing
holds. It is possible to derive full propositional resizing if we strengthen
nontriviality to positivity. The distinction between nontriviality and
positivity is analogous to the distinction between nonemptiness and
inhabitedness. Moreover, we prove that locally small, nontrivial (directed or
bounded) complete posets necessarily lack decidable equality. We prove our
results for a general class of posets, which includes e.g. directed complete
posets, bounded complete posets, sup-lattices and frames. Secondly, the fact
that these nontrivial posets are necessarily large has the important
consequence that Tarski's theorem (and similar results) cannot be applied in
nontrivial instances. Furthermore, we explain that generalizations of Tarski's
theorem that allow for large structures are provably false by showing that the
ordinal of ordinals in a univalent universe has small suprema in the presence
of set quotients. The latter also leads us to investigate the
inter-definability and interaction of type universes of propositional
truncations and set quotients, as well as a set replacement principle. Thirdly,
we clarify, in our predicative setting, the relation between the traditional
definition of sup-lattice that requires suprema for all subsets and our
definition that asks for suprema of all small families
Domain Theory in Constructive and Predicative Univalent Foundations
We develop domain theory in constructive and predicative univalent
foundations (also known as homotopy type theory). That we work predicatively
means that we do not assume Voevodsky's propositional resizing axioms. Our work
is constructive in the sense that we do not rely on excluded middle or the
axiom of (countable) choice. Domain theory studies so-called directed complete
posets (dcpos) and Scott continuous maps between them and has applications in
programming language semantics, higher-type computability and topology. A
common approach to deal with size issues in a predicative foundation is to work
with information systems, abstract bases or formal topologies rather than
dcpos, and approximable relations rather than Scott continuous functions. In
our type-theoretic approach, we instead accept that dcpos may be large and work
with type universes to account for this. A priori one might expect that complex
constructions of dcpos result in a need for ever-increasing universes and are
predicatively impossible. We show that such constructions can be carried out in
a predicative setting. We illustrate the development with applications in the
semantics of programming languages: the soundness and computational adequacy of
the Scott model of PCF and Scott's model of the untyped
-calculus. We also give a predicative account of continuous and
algebraic dcpos, and of the related notions of a small basis and its rounded
ideal completion. The fact that nontrivial dcpos have large carriers is in fact
unavoidable and characteristic of our predicative setting, as we explain in a
complementary chapter on the constructive and predicative limitations of
univalent foundations. Our account of domain theory in univalent foundations is
fully formalised with only a few minor exceptions. The ability of the proof
assistant Agda to infer universe levels has been invaluable for our purposes.Comment: PhD thesis, extended abstract in the pdf. v5: Fixed minor typos in
6.2.18, 6.2.19 and 6.4.
Idempotents in intensional type theory
We study idempotents in intensional Martin-L\"of type theory, and inparticular the question of when and whether they split. We show that in thepresence of propositional truncation and Voevodsky's univalence axiom, thereexist idempotents that do not split; thus in plain MLTT not all idempotents canbe proven to split. On the other hand, assuming only function extensionality,an idempotent can be split if and only if its witness of idempotency satisfiesone extra coherence condition. Both proofs are inspired by parallel results ofLurie in higher category theory, showing that ideas from higher category theoryand homotopy theory can have applications even in ordinary MLTT. Finally, we show that although the witness of idempotency can be recoveredfrom a splitting, the one extra coherence condition cannot in general; and weconstruct "the type of fully coherent idempotents", by splitting an idempotenton the type of partially coherent ones. Our results have been formally verifiedin the proof assistant Coq.Comment: 24 pages. v2: final version, to appear in LMC
Idempotents in intensional type theory
We study idempotents in intensional Martin-L\"of type theory, and in
particular the question of when and whether they split. We show that in the
presence of propositional truncation and Voevodsky's univalence axiom, there
exist idempotents that do not split; thus in plain MLTT not all idempotents can
be proven to split. On the other hand, assuming only function extensionality,
an idempotent can be split if and only if its witness of idempotency satisfies
one extra coherence condition. Both proofs are inspired by parallel results of
Lurie in higher category theory, showing that ideas from higher category theory
and homotopy theory can have applications even in ordinary MLTT.
Finally, we show that although the witness of idempotency can be recovered
from a splitting, the one extra coherence condition cannot in general; and we
construct "the type of fully coherent idempotents", by splitting an idempotent
on the type of partially coherent ones. Our results have been formally verified
in the proof assistant Coq