Basic Category Theory — Overview

Summary

Tom Leinster’s Basic Category Theory (Cambridge, 2014; arXiv 1612.09375v2) is a concise introduction to category theory emphasising universal properties. The book argues that the central concept unifying all of category theory is the representable functor and its companion, the Yoneda lemma. By the end, every major construction (limits, adjoints, exponentials) is understood as an instance of representability.

Book Structure → Folder Map

Chapter	Title	Notes Location
1	Categories, functors, natural transformations	Foundations
2	Adjoints	Adjunctions
3	Interlude on sets	Foundations (size)
4	Representables	Representables
5	Limits	Limits and Colimits
6	Adjoints, representables, limits	Synthesis
App	Proof of GAFT	Adjoint Functor Theorems

The Central Thesis

The book is built around one insight: universal properties are the right way to define mathematical objects, and universal properties are precisely initial or terminal objects in appropriate categories, which are in turn precisely representations of functors. The progression:

Categories/Functors/Nat. Trans.
         ↓
    Adjoint Functors   ←→   Representable Functors
         ↓                         ↓
              Yoneda Lemma
                   ↓
              Limits & Colimits
                   ↓
           Synthesis (all three unified)

Key Theorems and Results

Result	Location
Natural transformation definition	Ch. 1.3
Adjunction definition (4 equivalent forms)	Ch. 2.1–2.2
Yoneda Lemma	Ch. 4.2
Yoneda embedding full and faithful	Ch. 4.3
General limit definition	Ch. 5.1
Cones are representable	Ch. 6.1
Adjoints preserve (co)limits	Ch. 6.3
General Adjoint Functor Theorem	Ch. 6.3

Sub-folder Index

• Foundations — Categories, functors, natural transformations, functor categories, size
• Adjunctions — Definition, units/counits, adjunctions via initial objects
• Representables — Representable functors, Yoneda lemma, Yoneda embedding
• Limits and Colimits — Products, equalizers, pullbacks, general limits, colimits, functors and limits
• Synthesis — Limits via representables, adjoints and limits, adjoint functor theorems, cartesian closed categories

See Also

• 1612.09375v2.pdf — Source PDF