Re: A new preprint "Higher Orbifolds and Deligne-Mumford Stacks as Structured Infinity Topoi"

Re: A new preprint "Higher Orbifolds and Deligne-Mumford Stacks as Structured Infinity Topoi"

[David Carchedi]

Dear All,

The following preprint is now available on the arXiv at:

http://arxiv.org/abs/1312.2204.

D. Carchedi, "Higher Orbifolds and Deligne-Mumford Stacks as Structured
Infinity Topoi"

Abstract:


We develop a framework for modeling higher orbifolds and Deligne-Mumford
stacks as infinity topoi equipped with a structure sheaf. This framework is
general enough to apply to differential topology, classical algebraic
geometry, and also derived and spectral variants of the latter. The general
set up is to start with a class of geometric objects, which are to be
thought of as local models, and then to use them as building blocks in much
the same way one builds manifolds out of Euclidean spaces, or schemes out
of affine schemes. When starting with Euclidean smooth manifolds, this
results in a theory of higher orbifolds and higher \'etale differentiable
stacks. When applied to affine schemes, it produces Lurie's higher
Deligne-Mumford stacks, and more generally, when applied to derived affine
schemes or spectral affine schemes, it produces Lurie's derived
Deligne-Mumford stacks and spectral Deligne-Mumford stacks. We give
categorical characterizations of the resulting geometric objects in the
general setting, as well as their functors of points, which are far
reaching generalizations of previous results of the author about \'etale
stacks. These specialize to a new characterization of classical
Deligne-Mumford stacks which extends to the derived and spectral setting as
well. In the differentiable setting, this characterization shows that there
is a natural correspondence between n-dimensional higher \'etale
differentiable stacks (generalized higher orbifolds), and classical fields
for n-dimensional field theories, in the sense of Freed and Teleman.

The underpinning idea behind this work is the following phenomenon, which
seems to really be a phenomenon in higher topos theory:

If one starts with a collection of local models (modeled as structured
infinity topoi), then the higher category of higher sheaves over the site
of local models and local homeomorphisms is equivalent to the higher
category of "higher orbifolds" constructed out of these local models and
their respective local homeomorphisms.

Kind Regards,

David Carchedi


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