862 research outputs found
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
Principal infinity-bundles - Presentations
We discuss two aspects of the presentation of the theory of principal
infinity-bundles in an infinity-topos, introduced in [NSSa], in terms of
categories of simplicial (pre)sheaves.
First we show that over a cohesive site C and for G a presheaf of simplicial
groups which is C-acyclic, G-principal infinity-bundles over any object in the
infinity-topos over C are classified by hyper-Cech-cohomology with coefficients
in G. Then we show that over a site C with enough points, principal
infinity-bundles in the infinity-topos are presented by ordinary simplicial
bundles in the sheaf topos that satisfy principality by stalkwise weak
equivalences. Finally we discuss explicit details of these presentations for
the discrete site (in discrete infinity-groupoids) and the smooth site (in
smooth infinity-groupoids, generalizing Lie groupoids and differentiable
stacks).
In the companion article [NSSc] we use these presentations for constructing
classes of examples of (twisted) principal infinity-bundles and for the
discussion of various applications.Comment: 55 page
The Sierpinski Object in the Scott Realizability Topos
We study the Sierpinski object in the realizability topos based on
Scott's graph model of the -calculus. Our starting observation is that
the object of realizers in this topos is the exponential , where
is the natural numbers object. We define order-discrete objects by
orthogonality to . We show that the order-discrete objects form a
reflective subcategory of the topos, and that many fundamental objects in
higher-type arithmetic are order-discrete. Building on work by Lietz, we give
some new results regarding the internal logic of the topos. Then we consider
as a dominance; we explicitly construct the lift functor and
characterize -subobjects. Contrary to our expectations the dominance
is not closed under unions. In the last section we build a model for
homotopy theory, where the order-discrete objects are exactly those objects
which only have constant paths
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 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
Orbifolds as Groupoids: an Introduction
This is a survey paper based on my talk at the Workshop on Orbifolds and
String Theory, the goal of which was to explain the role of groupoids and their
classifying spaces as a foundation for the theory of orbifolds
Homotopical Algebraic Geometry I: Topos theory
This is the first of a series of papers devoted to lay the foundations of
Algebraic Geometry in homotopical and higher categorical contexts (for part II,
see math.AG/0404373). In this first part we investigate a notion of higher
topos. For this, we use S-categories (i.e. simplicially enriched categories) as
models for certain kind of \infty-categories, and we develop the notions of
S-topologies, S-sites and stacks over them. We prove in particular, that for an
S-category T endowed with an S-topology, there exists a model category of
stacks over T, generalizing the model category structure on simplicial
presheaves over a Grothendieck site of A. Joyal and R. Jardine. We also prove
some analogs of the relations between topologies and localizing subcategories
of the categories of presheaves, by proving that there exists a one-to-one
correspondence between S-topologies on an S-category T, and certain left exact
Bousfield localizations of the model category of pre-stacks on T. Based on the
above results, we study the notion of model topos introduced by C. Rezk, and we
relate it to our model categories of stacks over S-sites. In the second part of
the paper, we present a parallel theory where S-categories, S-topologies and
S-sites are replaced by model categories, model topologies and model sites. We
prove that a canonical way to pass from the theory of stacks over model sites
to the theory of stacks over S-sites is provided by the simplicial localization
construction of Dwyer and Kan. We also prove a Giraud's style theorem
characterizing model topoi internally. As an example of application, we propose
a definition of etale K-theory of ring spectra, extending the etale K-theory of
commutative rings.Comment: 71 pages. Final version to appear in Adv. Mat
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