Every even lecture will finish with one problem. Thus, there will be at most 12 of those. You need to write down a careful solution with all the details and send it to me via email. Every student who solves all the problems will successfully pass the course.
From monoids to categories, categorical monomorphisms and epimorphisms, products and equalizers through universal properties, initial and final objects.
References: [1, Section 1.2].
More on functors, representability, limits and colimits, adjoint functors.
References: [1, Sections 1.3–1.5].
Homework: Before the next lecture be sure that you are comfortable with modules over (noncommutative) rings. A refresher can be found in [2, Section 1.2].
More on adjoint functors, monoidal categories, enriched categories, (pre)additive categories and additive functors.
References: [1, Section 1.5], [3, Section VII.1], [2, Section 3.1].
Adjoint functors and limits, additive functors, categories of complexes and homotopy categories, abelian categories and exactness.
References: [2, Sections 3.1–3.2, 4.1].
Subobjects, injective and projective objects, generators, completeness and cocompleteness, Mitchell’s theorem.
References: [4, Chapters 2–4].
Injective objects and (co)generators, Grothendieck categories, essential extensions and injective envelopes, weak embedding theorem.
References: [4, Chapter 6].
Freyd–Mitchell theorem.
References: [4, Chapter 7].
Serre subcategories, quotients, localization.
References: [5, Tags 02MN, 04VB].
Categories of complexes, cones of morphisms, long exact sequence of cohomology, injective modules.
References: [6, Chapter 1].
Classical derived functors.
References: [6, Chapter 2].
Spectral sequences of double complexes, Grothendieck spectral sequence.
References: [6, Chapter 5].
Balancing Tor, Grothendieck spectral sequence, examples, presheaves.
References: [6, Chapter 5], [7, Section 2.7].
Sheaves, sheafification, triangulated categories.
References: [5, Chapter 6], [5, Sections 13.3–13.4].
Injective resolutions for sheaves of abelian groups, homotopy category is triangulated.
References: [5, Sections 13.9–13.10].
Homotopy category is triangulated, localization of triangulated categories.
References: [8, Sections IV.1–IV.2].
Localization of subcategories, triangulated quotient categories.
References: [5, Section 13.6].
Localization of functors, semiorthogonal decompositions, derived functors.
References: [2, Section 5.3], [8, Section III.6].
Deligne’s construction of derived functors.
References: [5, Section 13.14].
Composition of derived functors, derived bifunctors, t-structures.
References: [9, Section 1.3].
Properties of t-structures.
References: [9, Section 1.3].
Heart of a t-structure, flabby sheaves.
References: [9, Section 1.3], [10, Section II.2.4].
A brief overview of sheaf theory.
References: [10, Section II.2].
Derived categories of sheaves.
References: [10, Section II.2].
Local systems, constructible sheaves, perverse sheaves.
References: [9, 10].