80 research outputs found
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
The Herbrand Topos
We define a new topos, the Herbrand topos, inspired by the modified
realizability topos and our earlier work on Herbrand realizability. We also
introduce the category of Herbrand assemblies and characterise these as the
double-negation-separated objects in the Herbrand topos. In addition, we show
that the category of sets is included as the category of
double-negation-sheaves and prove that the inclusion functor preserves and
reflects validity of first-order formulas
Types are weak omega-groupoids
We define a notion of weak omega-category internal to a model of Martin-L\"of
type theory, and prove that each type bears a canonical weak omega-category
structure obtained from the tower of iterated identity types over that type. We
show that the omega-categories arising in this way are in fact omega-groupoids.Comment: 28 pages; v2: final journal versio
Noncommutativity as a colimit
Every partial algebra is the colimit of its total subalgebras. We prove this
result for partial Boolean algebras (including orthomodular lattices) and the
new notion of partial C*-algebras (including noncommutative C*-algebras), and
variations such as partial complete Boolean algebras and partial AW*-algebras.
The first two results are related by taking projections. As corollaries we find
extensions of Stone duality and Gelfand duality. Finally, we investigate the
extent to which the Bohrification construction, that works on partial
C*-algebras, is functorial.Comment: 22 pages; updated theorem 15, added propoisition 3
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