Roadmap

The vision

Mathematical Structures is intended as a coherent multi-book series, not a collection of independent texts. The first volume — Mathematical Structures I — provides the structural foundation the rest of the series assumes. Three further volumes, on linear algebra and geometric structure, on calculus and analysis, and on probability, are written from that foundation rather than from scratch, so that subjects share their underlying machinery instead of each rebuilding it. Beyond the core four, the longer horizon includes further mathematics, and texts in physics and engineering that use the same apparatus.

Progress and plans

Updated May 25, 2026.

Done

  • May 2026 — first draft of Categories, Maps, and Diagrams (the first chapter of the Categorical Foundations block). First draft of this site.

Next

Within 1 month (through June 2026)

  • ~50–75% drafted of Categories, Maps, and Diagrams
  • ~50–75% drafted of Functors and Profunctors

Within 3 months (through August 2026)

  • ~50–75% drafted across the intro block (five chapters; common number systems is revisited three times — once qualitatively, once tightening up language and notation, and once as a horizon chapter using representability and related categorical tools)
  • ~50–75% drafted across the Categorical Foundations block overall

Within 6 months (through November 2026)

  • Intro and Categorical Foundations blocks with main material in place — structure, exposition, and worked examples drafted (exercises and final polish are layered in later, so “in place” doesn’t mean “done”)
  • ~50–75% drafted on the next block — an in-depth treatment of the category of Set, doubling as an introduction to set theory for readers without prior exposure

For reference, the Categorical Foundations block is planned to cover, in order:

  1. Categories, Maps, and Diagrams
  2. Functors and Profunctors
  3. Natural Transformations and Functor Categories
  4. Representability, Universal Elements, and Yoneda
  5. Comma Categories, Slices, and Universal Arrows
  6. Limits and Colimits
  7. Adjunctions as Two-Sided Representability
  8. Enriched and Additive Structures
  9. Monoidal Categories
  10. The Two-Categorical Horizon