Now showing items 1-13 of 13

  • Decomposition spaces and restriction species 

    Gálvez Carrillo, Maria Immaculada; Kock, Joachim; Tonks, Andrew (Oxford University Press, 2018-09-12)
    Article
    Open Access
    We show that Schmitt’s restriction species (such as graphs, matroids, posets, etc.) naturally induce decomposition spaces (a.k.a. unital 2-Segal spaces), and that their associated coalgebras are an instance of the general ...
  • Decomposition spaces, incidence algebras and Möbius inversion 

    Gálvez Carrillo, Maria Immaculada; Kock, Joachim; Tonks, Andrew (2014-04-11)
    External research report
    Open Access
  • Decomposition spaces, incidence algebras and Möbius inversion I: basic theory 

    Gálvez Carrillo, Maria Immaculada; Kock, Joachim; Tonks, Andrew (2018-06-20)
    Article
    Restricted access - publisher's policy
    This is the first in a series of papers devoted to the theory of decomposition spaces, a general framework for incidence algebras and Möbius inversion, where algebraic identities are realised by taking homotopy cardinality ...
  • Decomposition spaces, incidence algebras and Möbius inversion I: basic theory 

    Gálvez Carrillo, Maria Immaculada; Kock, Joachim; Tonks, Andrew (2015-12)
    External research report
    Open Access
    This is the first in a series of papers devoted to the theory of decomposition spaces, a general framework for incidence algebras and Möbius inversion, where algebraic identities are realised by taking homotopy cardinality ...
  • Decomposition spaces, incidence algebras and Möbius inversion II: completeness, length filtration, and finiteness 

    Gálvez Carrillo, Maria Immaculada; Kock, Joachim; Tonks, Andrew (2015-12)
    External research report
    Open Access
    This is part 2 of a trilogy of papers introducing and studying the notion of decomposition space as a general framework for incidence algebras and Möbius inversion, with coefficients in 8-groupoids. A decomposition space ...
  • Decomposition spaces, incidence algebras and Mobius inversion II: Completeness, length filtration, and finiteness 

    Gálvez Carrillo, Maria Immaculada; Kock, Joachim; Tonks, Andrew (2018-07-31)
    Article
    Restricted access - publisher's policy
    This is the second in a trilogy of papers introducing and studying the notion of decomposition space as a general framework for incidence algebras and Möbius inversion, with coefficients in 8-groupoids. A decomposition ...
  • Decomposition spaces, incidence algebras and Möbius inversion III: the decomposition space of Möbius intervals 

    Gálvez Carrillo, Maria Immaculada; Kock, Joachim; Tonks, Andrew (2015-12)
    External research report
    Open Access
    Decomposition spaces are simplicial 8-groupoids subject to a certain exactness condition, needed to induce a coalgebra structure on the space of arrows. Conservative ULF functors between decomposition spaces induce coalgebra ...
  • Decomposition spaces, incidence algebras and Mobius inversion III: the decomposition space of Möbius intervals 

    Gálvez Carrillo, Maria Immaculada; Kock, Joachim; Tonks, Andrew (2018-08-20)
    Article
    Restricted access - publisher's policy
    Decomposition spaces are simplicial 8-groupoids subject to a certain exactness condition, needed to induce a coalgebra structure on the space of arrows. Conservative ULF functors (CULF) between decomposition spaces induce ...
  • Decomposition spaces in combinatorics 

    Gálvez Carrillo, Maria Immaculada; Kock, Joachim; Tonks, Andrew (2016-12)
    External research report
    Open Access
    A decomposition space (also called unital 2-Segal space) is a simplicial object satisfying an exactness condition weaker than the Segal condition: just as the Segal condition expresses (up to homotopy) composition, the new ...
  • Groupoids and Faà di Bruno Formulae for green functions in bialgebras of trees 

    Gálvez Carrillo, Maria Immaculada; Tonks, Andrew; Kock, Joachim (2012-07)
    External research report
    Open Access
    We prove a Faa di Bruno formula for the Green function in the bialgebra of P-trees, for any polynomial endofunctor P. The formula appears as relative homotopy cardinality of an equivalence of groupoids. For suitable choices ...
  • Groupoids and Faà di Bruno formulae for Green functions in bialgebras of trees 

    Gálvez Carrillo, Maria Immaculada; Kock, Joachim; Tonks, Andrew (2014)
    Article
    Restricted access - publisher's policy
    We prove a Faà di Bruno formula for the Green function in the bialgebra of P-trees, for any polynomial endofunctor P. The formula appears as relative homotopy cardinality of an equivalence of groupoids.
  • Homotopy linear algebra 

    Gálvez Carrillo, Maria Immaculada; Kock, Joachim; Tonks, Andrew (2018-04)
    Article
    Open Access
    By homotopy linear algebra we mean the study of linear functors between slices of the 8-category of 8-groupoids, subject to certain finiteness conditions. After some standard definitions and results, we assemble said slices ...
  • Homotopy linear algebra 

    Gálvez Carrillo, Maria Immaculada; Kock, Joachim; Tonks, Andrew (2016-02)
    External research report
    Open Access
    By homotopy linear algebra we mean the study of linear functors between slices of the 8-category of 8-groupoids, subject to certain finiteness conditions. After some standard definitions and results, we assemble said slices ...