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Статья «HOMOTOPY THEORY OF NORMED SETS I. BASIC CONSTRUCTIONS, "Алгебра и анализ"»

Авторы:
  • DUROV N. V.1
стр. 35-98
Платно
1 St. Petersburg Department of the Steklov Mathematical Institute
  • SDI: 007.001.0234-0852.2017.029.006.2
Ключевые слова:
  • normed sets
  • normed groups
  • norms
  • normed algebrai strutures
  • graded algebrai strutures
  • filtered algebrai strutures
  • fuzzy sets
  • linear logi
  • presheaf categories
  • finitary monads
  • generalized rings
  • metric spaces
  • model categories
  • homotopy categories
  • higher categories
Аннотация:
We would like to present an extension of the theory of R^o-graded (or "R^o-normed") sets and monads over them as defined in recent paper by Frederic Paugam. We extend the theory of graded sets in three directions. First of all, we show that R^o can be replaced with more or less arbitrary (partially) ordered commutative monoid Д, yielding a symmetric monoidal category Na of A-normed sets. However, this category fails to be closed under some important categorical constructions. We deal with this problems by embedding Na into a larger category Sets of A-graded sets. Next, we show that most constructions make sense with Д replaced by a small symmetric monoidal category I. In particular, we have a symmetric monoidal category Sets of I-graded sets. We use these foundations for two further developments: a homotopy theory for normed and graded sets, essentially consisting of a well-behaved I A-graded monads. This material will be exposed elsewhere.

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