Algebra i Analiz
RUS  ENG    JOURNALS   PEOPLE   ORGANISATIONS   CONFERENCES   SEMINARS   VIDEO LIBRARY   PACKAGE AMSBIB  
General information
Latest issue
Archive
Impact factor

Search papers
Search references

RSS
Latest issue
Current issues
Archive issues
What is RSS



Algebra i Analiz:
Year:
Volume:
Issue:
Page:
Find






Personal entry:
Login:
Password:
Save password
Enter
Forgotten password?
Register


Algebra i Analiz, 2017, Volume 29, Issue 6, Pages 35–98 (Mi aa1562)  

This article is cited in 1 scientific paper (total in 1 paper)

Research Papers

Homotopy theory of normed sets I. Basic constructions

N. V. Durov

St. Petersburg Department of the Steklov Mathematical Institute, 27 Fontanka emb., 191023, St. Petersburg, Russia
Full-text PDF (527 kB) Citations (1)
References:
Abstract: We would like to present an extension of the theory of $\mathbb R_{\ge0}$-graded (or "$\mathbb R_{\ge0}$-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 $\mathbb R_{\ge0}$ can be replaced with more or less arbitrary (partially) ordered commutative monoid $\Delta$, yielding a symmetric monoidal category $\mathcal N_\Delta$ of $\Delta$-normed sets. However, this category fails to be closed under some important categorical constructions. We deal with this problems by embedding $\mathcal N_\Delta$ into a larger category $\mathrm{Sets}^\Delta$ of $\Delta$-graded sets.
Next, we show that most constructions make sense with $\Delta$ replaced by a small symmetric monoidal category $\mathcal I$. In particular, we have a symmetric monoidal category $\mathrm{Sets}^\mathcal I$ of $\mathcal I$-graded sets.
We use these foundations for two further developments: a homotopy theory for normed and graded sets, essentially consisting of a well-behaved combinatorial model structure on simplicial $\mathcal I$-graded sets and a theory of $\Delta$-graded monads. This material will be exposed elsewhere.
Keywords: normed sets, normed groups, norms, normed algebraic structures, graded algebraic structures, filtered algebraic structures, fuzzy sets, linear logic, presheaf categories, finitary monads, generalized rings, metric spaces, model categories, homotopy categories, higher categories.
Received: 09.09.2017
English version:
St. Petersburg Mathematical Journal, 2018, Volume 29, Issue 6, Pages 887–934
DOI: https://doi.org/10.1090/spmj/1520
Bibliographic databases:
Document Type: Article
MSC: 06D72
Language: English
Citation: N. V. Durov, “Homotopy theory of normed sets I. Basic constructions”, Algebra i Analiz, 29:6 (2017), 35–98; St. Petersburg Math. J., 29:6 (2018), 887–934
Citation in format AMSBIB
\Bibitem{Dur17}
\by N.~V.~Durov
\paper Homotopy theory of normed sets~I. Basic constructions
\jour Algebra i Analiz
\yr 2017
\vol 29
\issue 6
\pages 35--98
\mathnet{http://mi.mathnet.ru/aa1562}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=3723811}
\elib{https://elibrary.ru/item.asp?id=30381767}
\transl
\jour St. Petersburg Math. J.
\yr 2018
\vol 29
\issue 6
\pages 887--934
\crossref{https://doi.org/10.1090/spmj/1520}
\isi{https://gateway.webofknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=Publons&SrcAuth=Publons_CEL&DestLinkType=FullRecord&DestApp=WOS_CPL&KeyUT=000444495400002}
Linking options:
  • https://www.mathnet.ru/eng/aa1562
  • https://www.mathnet.ru/eng/aa/v29/i6/p35
    Cycle of papers
    This publication is cited in the following 1 articles:
    Citing articles in Google Scholar: Russian citations, English citations
    Related articles in Google Scholar: Russian articles, English articles
    Алгебра и анализ St. Petersburg Mathematical Journal
     
      Contact us:
     Terms of Use  Registration to the website  Logotypes © Steklov Mathematical Institute RAS, 2024