Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Closed symmetric monoidal structures on the category of graphs

Chris Kapulkin, Nathan Kershaw

TL;DR

The paper classifies closed symmetric monoidal structures on the category of (reflexive) graphs. Using Day convolution in the setting of a reflective subcategory of presheaves, it reduces the problem to analyzing promonoidal data on a small indexing category. By a sequence of finiteness and labeling arguments on the image of the edge object, it is shown that only two viable Kan-extended products exist, corresponding to the box product and the categorical product. Consequently, Graph admits exactly two closed symmetric monoidal structures, both realized as Day convolution from two canonical promonoidal functors.

Abstract

We show that the category of (reflexive) graphs and graph maps carries exactly two closed symmetric monoidal products: the box product and the categorical product.

Closed symmetric monoidal structures on the category of graphs

TL;DR

The paper classifies closed symmetric monoidal structures on the category of (reflexive) graphs. Using Day convolution in the setting of a reflective subcategory of presheaves, it reduces the problem to analyzing promonoidal data on a small indexing category. By a sequence of finiteness and labeling arguments on the image of the edge object, it is shown that only two viable Kan-extended products exist, corresponding to the box product and the categorical product. Consequently, Graph admits exactly two closed symmetric monoidal structures, both realized as Day convolution from two canonical promonoidal functors.

Abstract

We show that the category of (reflexive) graphs and graph maps carries exactly two closed symmetric monoidal products: the box product and the categorical product.
Paper Structure (4 sections, 19 theorems, 25 equations, 2 figures)

This paper contains 4 sections, 19 theorems, 25 equations, 2 figures.

Table of Contents

  1. Monoidal categories
  2. The category of graphs
  3. Products via the left Kan extension
  4. Main Theorem

Key Result

Theorem 1

The category of (reflexive) graphs carries precisely two closed symmetric monoidal structures: the box product and the categorical product.

Figures (2)

  • Figure 1: The graph $I_4$
  • Figure 2: The graphs $I_0$ and $I_1$ as the image of the Yoneda embedding

Theorems & Definitions (46)

  • Theorem : \ref{['main th']}
  • definition 1
  • definition 2
  • example 3
  • definition 4
  • example 5
  • definition 6
  • proposition 7
  • proof
  • theorem 8: day
  • ...and 36 more