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Simplification for Graph-like Objects

Will Grilliette

Abstract

The simplification of a multigraph into a simple graph can be abstracted to a more general comma category under some common conditions. When using the identity functor, the category of simple objects in a comma category generalizes the functor-structured category. Seated in categorical terms, simplification can be dualized to "antisimplification", which manifests as removal of isolated vertices and loose edges.

Simplification for Graph-like Objects

Abstract

The simplification of a multigraph into a simple graph can be abstracted to a more general comma category under some common conditions. When using the identity functor, the category of simple objects in a comma category generalizes the functor-structured category. Seated in categorical terms, simplification can be dualized to "antisimplification", which manifests as removal of isolated vertices and loose edges.

Paper Structure

This paper contains 11 sections, 16 theorems, 26 equations.

Table of Contents

  1. Introduction
  2. Simplification
  3. Subobject-structured Categories
  4. Examples of Simplification
  5. Set-System Hypergraphs
  6. Quivers
  7. Incidence Hypergraphs
  8. Slice Category
  9. Examples of Antisimplification
  10. Incidence Hypergraphs
  11. Quivers

Key Result

Proposition 2.2

Let $H\in\mathop{\mathrm{Ob}}\nolimits(\mathfrak{G})$.

Theorems & Definitions (36)

  • Definition 2.1: Generalized complete & simple
  • Proposition 2.2: Simple & unit map
  • proof
  • Proposition 2.3: Maps into simple
  • proof
  • Definition 2.4: Simplification
  • Theorem 2.5: Universal property
  • proof
  • Corollary 2.6: Reflective subcategory
  • Proposition 3.1: Unit map
  • ...and 26 more