Category Theory
Routing Summary
This folder contains notes ingested from Basic Category Theory by Tom Leinster (Cambridge, 2014; arXiv 1612.09375v2), plus a Monads sub-topic from Riehl’s Category Theory in Context (Ch. 5). Covers categories through adjoint functor theorems and monads, with the Yoneda lemma as the central result.
- For an overview of the book and its structure → Basic Category Theory - Overview
- For foundational language (categories, functors, nat. trans.) → Foundations
- For adjunctions (free/forgetful, units/counits, universal maps) → Adjunctions
- For the Yoneda lemma and representability → Representables
- For limits, colimits, products, equalizers → Limits and Colimits
- For the synthesis (all three unified: RAPL, GAFT, CCC) → Synthesis
- For monads, algebras, Eilenberg-Moore/Kleisli, and monadicity → Monads
- For motivation (universal properties overview) → Universal Properties
Sub-folders
- Foundations — COVERS: category definition, functors, natural transformations, functor categories, size
- Adjunctions — COVERS: adjunction definition (4 forms), units/counits, initial objects / comma categories
- Representables — COVERS: hom-functors, representable functors, Yoneda lemma, Yoneda embedding
- Limits and Colimits — COVERS: products, equalizers, pullbacks, general limits, colimits, functors and limits
- Synthesis — COVERS: limits via representables, pointwise limits, right adjoints preserve limits, GAFT, SAFT, CCC
- Monads — COVERS: monad definition & laws, adjunctions induce monads, Eilenberg-Moore & Kleisli categories, Beck’s monadicity theorem (Riehl, Category Theory in Context Ch. 5)
- Universal Properties — COVERS: motivating overview of universal constructions
Cross-Cutting Concepts
| Concept | Primary Note | Also Appears In |
|---|---|---|
| Universal property | Universal Properties - Introduction | All sub-folders |
| Yoneda lemma | Yoneda Lemma | Limits via Representables, Adjoints and Limits |
| Representability | Representable Functors | Limits via Representables, Adjoint Functors |
| Right adjoints preserve limits | Adjoints and Limits | Functors and Limits, Adjoint Functor Theorems |
| Duality (op-category) | Categories | Every colimit result |
Overview Note
- Basic Category Theory - Overview — Full book map with chapter-to-folder mapping and key theorem index
Sources
- 1612.09375v2.pdf — Basic Category Theory, Tom Leinster, Cambridge University Press 2014