16,816 research outputs found
Metrics for generalized persistence modules
We consider the question of defining interleaving metrics on generalized
persistence modules over arbitrary preordered sets. Our constructions are
functorial, which implies a form of stability for these metrics. We describe a
large class of examples, inverse-image persistence modules, which occur
whenever a topological space is mapped to a metric space. Several standard
theories of persistence and their stability can be described in this framework.
This includes the classical case of sublevelset persistent homology. We
introduce a distinction between `soft' and `hard' stability theorems. While our
treatment is direct and elementary, the approach can be explained abstractly in
terms of monoidal functors.Comment: Final version; no changes from previous version. Published online Oct
2014 in Foundations of Computational Mathematics. Print version to appea
Tameness in generalized metric structures
We broaden the framework of metric abstract elementary classes (mAECs) in
several essential ways, chiefly by allowing the metric to take values in a
well-behaved quantale. As a proof of concept we show that the result of Boney
and Zambrano on (metric) tameness under a large cardinal assumption holds in
this more general context. We briefly consider a further generalization to
partial metric spaces, and hint at connections to classes of fuzzy structures,
and structures on sheaves
Metric 1-spaces
A generalization of metric space is presented which is shown to admit a
theory strongly related to that of ordinary metric spaces. To avoid the
topological effects related to dropping any of the axioms of metric space,
first a new, and equivalent, axiomatization of metric space is given which is
then generalized from a fresh point of view. Naturally arising examples from
metric geometry are presented
A unified framework for generalized multicategories
Notions of generalized multicategory have been defined in numerous contexts
throughout the literature, and include such diverse examples as symmetric
multicategories, globular operads, Lawvere theories, and topological spaces. In
each case, generalized multicategories are defined as the "lax algebras" or
"Kleisli monoids" relative to a "monad" on a bicategory. However, the meanings
of these words differ from author to author, as do the specific bicategories
considered. We propose a unified framework: by working with monads on double
categories and related structures (rather than bicategories), one can define
generalized multicategories in a way that unifies all previous examples, while
at the same time simplifying and clarifying much of the theory.Comment: 76 pages; final version, to appear in TA
Homologies of Algebraic Structures via Braidings and Quantum Shuffles
In this paper we construct "structural" pre-braidings characterizing
different algebraic structures: a rack, an associative algebra, a Leibniz
algebra and their representations. Some of these pre-braidings seem original.
On the other hand, we propose a general homology theory for pre-braided vector
spaces and braided modules, based on the quantum co-shuffle comultiplication.
Applied to the structural pre-braidings above, it gives a generalization and a
unification of many known homology theories. All the constructions are
categorified, resulting in particular in their super- and co-versions. Loday's
hyper-boundaries, as well as certain homology operations are efficiently
treated using the "shuffle" tools
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