449 research outputs found
Cyclic theories
We describe a geometric theory classified by Connes-Consani's epicylic topos
and two related theories respectively classified by the cyclic topos and by the
topos .Comment: 25 page
Ionads
The notion of Grothendieck topos may be considered as a generalisation of
that of topological space, one in which the points of the space may have
non-trivial automorphisms. However, the analogy is not precise, since in a
topological space, it is the points which have conceptual priority over the
open sets, whereas in a topos it is the other way around. Hence a topos is more
correctly regarded as a generalised locale, than as a generalised space. In
this article we introduce the notion of ionad, which stands in the same
relationship to a topological space as a (Grothendieck) topos does to a locale.
We develop basic aspects of their theory and discuss their relationship with
toposes.Comment: 24 pages; v2: diverse revisions; v3: chopped about in face of
trenchant and insightful referee feedbac
Localic Metric spaces and the localic Gelfand duality
In this paper we prove, as conjectured by B.Banachewski and C.J.Mulvey, that
the constructive Gelfand duality can be extended into a duality between compact
regular locales and unital abelian localic C*-algebras. In order to do so we
develop a constructive theory of localic metric spaces and localic Banach
spaces, we study the notion of localic completion of such objects and the
behaviour of these constructions with respect to pull-back along geometric
morphisms.Comment: 57 page
Principal infinity-bundles - General theory
The theory of principal bundles makes sense in any infinity-topos, such as
that of topological, of smooth, or of otherwise geometric
infinity-groupoids/infinity-stacks, and more generally in slices of these. It
provides a natural geometric model for structured higher nonabelian cohomology
and controls general fiber bundles in terms of associated bundles. For suitable
choices of structure infinity-group G these G-principal infinity-bundles
reproduce the theories of ordinary principal bundles, of bundle
gerbes/principal 2-bundles and of bundle 2-gerbes and generalize these to their
further higher and equivariant analogs. The induced associated infinity-bundles
subsume the notions of gerbes and higher gerbes in the literature.
We discuss here this general theory of principal infinity-bundles, intimately
related to the axioms of Giraud, Toen-Vezzosi, Rezk and Lurie that characterize
infinity-toposes. We show a natural equivalence between principal
infinity-bundles and intrinsic nonabelian cocycles, implying the classification
of principal infinity-bundles by nonabelian sheaf hyper-cohomology. We observe
that the theory of geometric fiber infinity-bundles associated to principal
infinity-bundles subsumes a theory of infinity-gerbes and of twisted
infinity-bundles, with twists deriving from local coefficient infinity-bundles,
which we define, relate to extensions of principal infinity-bundles and show to
be classified by a corresponding notion of twisted cohomology, identified with
the cohomology of a corresponding slice infinity-topos.
In a companion article [NSSb] we discuss explicit presentations of this
theory in categories of simplicial (pre)sheaves by hyper-Cech cohomology and by
simplicial weakly-principal bundles; and in [NSSc] we discuss various examples
and applications of the theory.Comment: 46 pages, published versio
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
Topological Representation of Geometric Theories
Using Butz and Moerdijk's topological groupoid representation of a topos with
enough points, a `syntax-semantics' duality for geometric theories is
constructed. The emphasis is on a logical presentation, starting with a
description of the semantical topological groupoid of models and isomorphisms
of a theory and a direct proof that this groupoid represents its classifying
topos. Using this representation, a contravariant adjunction is constructed
between theories and topological groupoids. The restriction of this adjunction
yields a contravariant equivalence between theories with enough models and
semantical groupoids. Technically a variant of the syntax-semantics duality
constructed in [Awodey and Forssell, arXiv:1008.3145v1] for first-order logic,
the construction here works for arbitrary geometric theories and uses a slice
construction on the side of groupoids---reflecting the use of `indexed' models
in the representation theorem---which in several respects simplifies the
construction and allows for an intrinsic characterization of the semantic side.Comment: 32 pages. This is the first pre-print version, the final revised
version can be found at
http://onlinelibrary.wiley.com/doi/10.1002/malq.201100080/abstract (posting
of which is not allowed by Wiley). Changes in v2: updated comment
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