Module 3 Readings: Functors and Natural Transformations

Required

• Textbook — Chapter
  Category Theory in Context — §1.3: Functors and §1.4: Naturality
  Emily Riehl — Complete §1.3 and §1.4
  Focus on: Example 1.3.2 (functors between posets), Example 1.3.4 (forgetful functors), the definition of natural transformation in §1.4, and Examples 1.4.3–1.4.5. Work through the proof that natural transformations compose to form a category.
• Textbook — Section
  Seven Sketches in Compositionality — §3.3: Functors and Data Migration
  Fong and Spivak — Read §3.3
  Focus on: how functors between database schemas correspond to data migration operations. This directly connects Module 3 to Module 7.

Recommended

• Video
  Category Theory for Programmers — Functors
  Bartosz Milewski — 45 min — Strong programmer-native intuition for functors as "containers" and as structure-preserving maps
• Paper — Section
  Geometric Deep Learning — §2.2: Symmetry and Equivariance
  Bronstein et al. — 2021 — Read §2.2 only — How equivariance (the condition that F(g·x) = g·F(x)) is a naturality condition in disguise
• Classic
  Category Theory: The Language of Mathematics
  Eilenberg and Mac Lane wrote the first paper on category theory in 1945 motivated by natural transformations — the "naturality" in algebraic topology. This short expository article gives historical context.

Reading Notes

The naturality square is the hardest concept in this module for most students. To build fluency:

• For each natural transformation example in Riehl §1.4, draw the full naturality square explicitly with all four vertices and four arrows labeled.
• For each natural transformation, identify: What are F and G? What is C? What is D? What is η_c for a specific choice of c?
• Find one example from your own CS background (a polymorphic function, an API adapter, a data transformation) and check whether it is a natural transformation. Draw its naturality square.