Six Elements, One Comparison
A teaching companion to Quotients, Refinement and Reconstruction
First published October 2026; revised November 3, 2026

Read online:
https://refacte.com/notes/six-elements-one-comparison.html
Main exposition:
https://refacte.com/notes/quotients-and-reconstruction.html
Notes on proofs, dependencies, and revisions:
https://refacte.com/notes/quotients-and-reconstruction-audit.html

Reading route
  Pages 1-8: C6, quotient maps, relation composition, the comparison
             rectangle, saturation, refinement and factor matching.
  Page 9: optional spectral calculation of the same C6 comparison.
  Page 10: seven exercises.
  Pages 11-12: worked answers.
  Pages 13-14: higher-chapter guide, with concrete C6 groupoid examples
               and prerequisites for each route.
Section numbers agree with the web edition. Print page references follow
the PDF pagination.

Editable sources
  main.tex is the main document.
  01-concrete-route.tex contains sections 1-9.
  02-exercises.tex contains sections 10-11.
  03-higher-guide.tex contains section 12.

Compile twice with pdfLaTeX:
  pdflatex -jobname=six-elements-one-comparison -interaction=nonstopmode -halt-on-error main.tex
  pdflatex -jobname=six-elements-one-comparison -interaction=nonstopmode -halt-on-error main.tex
Standard packages include amsmath, amssymb, amsthm, mathtools, booktabs,
array, microtype, tikz-cd, lmodern, geometry, xcolor, hyperref,
bookmark and fancyhdr.

The booklet proves the common-quotient mechanism for subgroup relations
and computes the C6 example. The general exact Mal'cev refinement theorem,
spectral framework and higher foundations are developed in the main
exposition. This companion's editorial update leaves its mathematical
proofs, calculations, diagrams, exercises and answers unchanged.
