Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The diagrammatic presentation of equations in categories

Kevin Arlin, James Fairbanks, Tim Hosgood, Evan Patterson

Abstract

Lifts of categorical diagrams $D\colon\mathsf{J}\to\mathsf{X}$ against discrete opfibrations $π\colon\mathsf{E}\to\mathsf{X}$ can be interpreted as presenting solutions to systems of equations. With this interpretation in mind, it is natural to ask if there is a notion of equivalence of diagrams $D\simeq D'$ that precisely captures the idea of the two diagrams "having the same solutions''. We give such a definition, and then show how the localisation of the category of diagrams in $\mathsf{X}$ along such equivalences is isomorphic to the localisation of the slice category $\mathsf{Cat}/\mathsf{X}$ along the class of initial functors. Finally, we extend this result to the 2-categorical setting, proving the analogous statement for any locally presentable 2-category in place of $\mathsf{Cat}$.

The diagrammatic presentation of equations in categories

Abstract

Lifts of categorical diagrams against discrete opfibrations can be interpreted as presenting solutions to systems of equations. With this interpretation in mind, it is natural to ask if there is a notion of equivalence of diagrams that precisely captures the idea of the two diagrams "having the same solutions''. We give such a definition, and then show how the localisation of the category of diagrams in along such equivalences is isomorphic to the localisation of the slice category along the class of initial functors. Finally, we extend this result to the 2-categorical setting, proving the analogous statement for any locally presentable 2-category in place of .
Paper Structure (7 sections, 31 theorems, 43 equations, 2 figures)

This paper contains 7 sections, 31 theorems, 43 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Conventions
  3. Acknowledgments
  4. Diagram categories
  5. Initial and relatively initial functors
  6. Localisations of categories of diagrams
  7. The general 2-categorical story

Key Result

Lemma 2.4

Let $\mathsf{X}$ be a complete category. Then giving a choice of limit for every diagram in $\mathsf{X}$ it is equivalent to giving a functor The dual statement for cocomplete categories also holds, giving a functor $\mathop{\mathrm{colim}}\limits\colon\operatorname{\mathsf{Diag}}_{\rightarrow}(\mathsf{X})\longrightarrow\mathsf{X}$.

Figures (2)

  • Figure 1: Left: a morphism in $\operatorname{\mathsf{Diag}}_{\rightarrow}(\mathsf{X})$. Right: a morphism in $\operatorname{\mathsf{Diag}}_{\leftarrow}(\mathsf{X})$.
  • Figure 2: A discrete opfibration $\pi$ at $f\colon x\longrightarrow y$. As the picture suggests, the codomain of the lift $\overline{f}$ is not given, but is instead part of the existence and uniqueness statement.

Theorems & Definitions (89)

  • Definition 2.1
  • Remark 2.2
  • Definition 2.3
  • Lemma 2.4
  • proof
  • Definition 2.5
  • Definition 2.6
  • Remark 2.7
  • Remark 2.8
  • Definition 2.9
  • ...and 79 more