Lecture notes

Algebraic Topology

A fairytale

Chapter I.1: Manifolds and bundles

Chapter I.2: Covering spaces and the fundamental group

Chapter II.1: Simplicial homology

Chapter II.2: Singular homology

Chapter II.3: Classical applications of homology

Chapter II.4: Cellular homology

Chapter II.5: The unicity theorem for homology

Outlook

Higher Categories I

A fairytale

Chapter I: Categories

Chapter II: Simplicial sets

Chapter III: Quasicategories

Chapter IV: Simplicially enriched categories

Chapter V: Simplicial homotopy theory

Chapter VI: Categorical homotopy theory

Chapter VII: The homotopy hypothesis

Higher Categories II

A fairytale

Chapter VIII: Localisations and model categories

Digression I: Fat and thin slices

Chapter IX: Cartesian fibrations

Chapter X: Straightening and Unstraightening, Gijs' notes

Digression II: Homotopy colimits

Digression III: Simplicial model categories

Chapter XI: Yoneda's lemma, adjunctions and colimits

Algebraic K-Theory

Chapter I: Preliminaries

Chapter II: Stable and monoidal categories

Chapter III: Group-completion and the K-Theory of finite fields

Chapter IV: K-Theory of stable categories

A TeXed (and in many ways improved) version of these notes was produced by Ferdinand Wagner, you can find it here.