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Groupoid G-spans and matrices over group rings

Joachim Kock, Jesper M. Møller

Abstract

When G is a finite abelian group, we define G-spans of groupoids and their associated matrices with entries in the group ring QG and show that composition of spans corresponds to multiplication of matrices.

Groupoid G-spans and matrices over group rings

Abstract

When G is a finite abelian group, we define G-spans of groupoids and their associated matrices with entries in the group ring QG and show that composition of spans corresponds to multiplication of matrices.
Paper Structure (5 sections, 9 theorems, 71 equations, 1 figure)

This paper contains 5 sections, 9 theorems, 71 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Homotopy pull-backs of groupoids
  3. $G$-spans of groupoids and their matrices
  4. Categorical aspects of $G$-spans
  5. Examples

Key Result

Lemma 2.6

imma_kock_tonks2018 The homotopy pull-back groupoid of Definition defn:htpypullback has Euler characteristic

Figures (1)

  • Figure 1: Three $G$-spans

Theorems & Definitions (36)

  • Definition 2.1: The homotopy pull-back groupoid BHW2010
  • Remark 2.3: Two-sided homotopy pull-backs
  • Lemma 2.6
  • proof
  • Remark 2.7: The homotopy fibre versus the full inverse image BHW2010
  • Definition 3.1
  • Lemma 3.3
  • proof
  • Remark 3.4: More on composable $G$-spans
  • Lemma 3.6
  • ...and 26 more