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Feb. 26th, 2015 03:54 pmВ пятницу Ален Конн выложил две крайне занимательных публикации на тему "поля с одним элементом", в особенности вот это: http://www.alainconnes.org/docs/gammasets.pdf. Ещё никто не разбирался?
Цитатки для острастки:
"In [11], N. Durov developed a geometry over F1 suitable for Arakelov theory applications by using monads as generalizations of classical rings. [...] The main result of this article states that all the above structures and constructions can be naturally subsumed by a theory which is well-known in homotopy theory,
namely the theory of s-algebras and Segal's Г-sets."
"We show that the basic categorical concept of an s-algebra as derived from the theory of Segal's Г-sets provides a unified description of several constructions attempting to model an algebraic geometry over the absolute point. It merges, in particular, the approaches using monoids, semirings and hyperrings as well as the development by means of monads and generalized rings in Arakelov geometry. The assembly map determines a functorial way to associate an s-algebra to a monad on pointed sets. The notion of an s-algebra is very familiar in algebraic topology where it also provides a suitable groundwork to the definition of topological cyclic homology. The main contribution of this paper is to point out its relevance and unifying role in arithmetic, in relation with the development of an algebraic geometry over symmetric closed monoidal categories."
Звучит как прорыв!
Цитатки для острастки:
"In [11], N. Durov developed a geometry over F1 suitable for Arakelov theory applications by using monads as generalizations of classical rings. [...] The main result of this article states that all the above structures and constructions can be naturally subsumed by a theory which is well-known in homotopy theory,
namely the theory of s-algebras and Segal's Г-sets."
"We show that the basic categorical concept of an s-algebra as derived from the theory of Segal's Г-sets provides a unified description of several constructions attempting to model an algebraic geometry over the absolute point. It merges, in particular, the approaches using monoids, semirings and hyperrings as well as the development by means of monads and generalized rings in Arakelov geometry. The assembly map determines a functorial way to associate an s-algebra to a monad on pointed sets. The notion of an s-algebra is very familiar in algebraic topology where it also provides a suitable groundwork to the definition of topological cyclic homology. The main contribution of this paper is to point out its relevance and unifying role in arithmetic, in relation with the development of an algebraic geometry over symmetric closed monoidal categories."
Звучит как прорыв!
