440 research outputs found
Coherent and Archimedean choice in general Banach spaces
I introduce and study a new notion of Archimedeanity for binary and
non-binary choice between options that live in an abstract Banach space,
through a very general class of choice models, called sets of desirable option
sets. In order to be able to bring an important diversity of contexts into the
fold, amongst which choice between horse lottery options, I pay special
attention to the case where these linear spaces don't include all `constant'
options.I consider the frameworks of conservative inference associated with
Archimedean (and coherent) choice models, and also pay quite a lot of attention
to representation of general (non-binary) choice models in terms of the
simpler, binary ones.The representation theorems proved here provide an
axiomatic characterisation for, amongst many other choice methods, Levi's
E-admissibility and Walley-Sen maximality.Comment: 34 pages, 7 figure
Fr\'echet Modules and Descent
We study several aspects of the study of Ind-Banach modules over Banach rings
thereby synthesizing some aspects of homological algebra and functional
analysis. This includes a study of nuclear modules and of modules which are
flat with respect to the projective tensor product. We also study metrizable
and Fr\'{e}chet Ind-Banach modules. We give explicit descriptions of projective
limits of Banach rings as ind-objects. We study exactness properties of
projective tensor product with respect to kernels and countable products. As
applications, we describe a theory of quasi-coherent modules in Banach
algebraic geometry. We prove descent theorems for quasi-coherent modules in
various analytic and arithmetic contexts.Comment: improved versio
Dagger Geometry As Banach Algebraic Geometry
In this article, we apply the approach of relative algebraic geometry towards
analytic geometry to the category of bornological and Ind-Banach spaces
(non-Archimedean or not). We are able to recast the theory of Grosse-Kl\"onne
dagger affinoid domains with their weak G-topology in this new language. We
prove an abstract recognition principle for the generators of their standard
topology (the morphisms appearing in the covers). We end with a sketch of an
emerging theory of dagger affinoid spaces over the integers, or any Banach
ring, where we can see the Archimedean and non-Archimedean worlds coming
together
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