27 research outputs found
Formalizing the Ring of Witt Vectors
The ring of Witt vectors over a base ring is an important
tool in algebraic number theory and lies at the foundations of modern -adic
Hodge theory. has the interesting property that it constructs a
ring of characteristic out of a ring of characteristic , and it can
be used more specifically to construct from a finite field containing
the corresponding unramified field extension of the
-adic numbers (which is unique up to isomorphism).
We formalize the notion of a Witt vector in the Lean proof assistant, along
with the corresponding ring operations and other algebraic structure. We prove
in Lean that, for prime , the ring of Witt vectors over
is isomorphic to the ring of -adic integers
. In the process we develop idioms to cleanly handle calculations
of identities between operations on the ring of Witt vectors. These
calculations are intractable with a naive approach, and require a proof
technique that is usually skimmed over in the informal literature. Our proofs
resemble the informal arguments while being fully rigorous
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
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