1,603 research outputs found
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
Predicative toposes
We explain the motivation for looking for a predicative analogue of the
notion of a topos and propose two definitions. For both notions of a
predicative topos we will present the basic results, providing the groundwork
for future work in this area
On Constructive Axiomatic Method
In this last version of the paper one may find a critical overview of some
recent philosophical literature on Axiomatic Method and Genetic Method.Comment: 25 pages, no figure
The weak choice principle WISC may fail in the category of sets
The set-theoretic axiom WISC states that for every set there is a set of
surjections to it cofinal in all such surjections. By constructing an unbounded
topos over the category of sets and using an extension of the internal logic of
a topos due to Shulman, we show that WISC is independent of the rest of the
axioms of the set theory given by a well-pointed topos. This also gives an
example of a topos that is not a predicative topos as defined by van den Berg.Comment: v2 Change of title and abstract; v3 Almost completely rewritten after
referee pointed out critical mistake. v4 Final version. Will be published in
Studia Logica. License is CC-B
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