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Infinity
Prose by Eric Baugh. Finalist in the 2018 Manuscripts Prose Contest
Infinity
This essay surveys the different types of infinity that occur in pure and applied mathematics, with emphasis on: 1. the contrast between potential infinity and actual infinity; 2. Cantor's distinction between transfinite sets and absolute infinity; 3. the constructivist view of infinite quantifiers and the meaning of constructive proof; 4. the concept of feasibility and the philosophical problems surrounding feasible arithmetic; 5. Zeno's paradoxes and modern paradoxes of physical infinity involving supertasks
Infinity-harmonic maps and morphisms
We propose a new notion called \emph{infinity-harmonic maps}between
Riemannain manifolds. These are natural generalizations of the well known
notion of infinity harmonic functions and are also the limiting case of %
-harmonic maps as .
Infinity harmoncity appears in many familiar contexts. For example, metric
projection onto the orbit of an isometric group action from a tubular
neighborhood is infinity harmonic.
Unfortunately, infinity-harmonicity is not preserved under composition. Those
infinity harmonic maps that always preserve infinity harmonicity under pull
back are called infinity harmonic morphisms. We show that infinity harmonic
morphisms are precisely horizontally homothetic mas.
Many example of infinity-harmonic maps are given, including some very
important and well-known classes of maps between Riemannian manifolds
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
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