Category theory
Prerequisites:
Fundamental Algebra and Analysis II
Course description:
This course is designed for third year or fourth year undergraduate students as well as graduate students, who intend to pursue studies in fundamental mathematics. Category theory is the theory of mathematical structures, providing an abstract framework for identifying common patterns across different fields in mathematics. The teaching objective is to enable students to master the basic language of category theory, in order to prepare them for in-depth study of many areas in modern mathematics, including but not limited to, algebraic geometry, algebraic topology, sheaf theory, and even mathematical physics. This course will employ a variety of teaching methods, including lectures, class discussions, problem-solving sessions, and independent reading.
Learning objectives:
By the end of this course, students will be able to:
1. Explain the necessity of Grothendieck universes to handle "size" issues in category theory.
Distinguish between small, large, categories. Define regular and inaccessible cardinals.
2. Formally define categories, functors, and natural transformations, and provide lots of examples. State and prove the Yoneda Lemma in its full generality.
3. Define and compute all standard (co)limits (initial, products, equalizers, pullbacks, filtered colimits) by their universal properties. State and apply key commutativity theorems (e.g., limits commute with limits, filtered colimits commute with finite limits in Set).
4. Define an adjunction using the hom-set isomorphism, unit-counit, and universal morphism formulations, and prove their equivalence. Define accessible functors and presentable categories. State, understand the hypotheses of, and apply the General Adjoint Functor Theorem and its specializations.
5. Define a monad formally and connect it to adjunctions (every adjunction gives a monad).
Construct the categories of algebras for a monad. Explain the Barr-Beck Monadicity Theorem.
Analyze limits/colimits in categories of algebras.
6. Define (symmetric) monoidal categories, monoidal functors, and monoidal natural transformations. State and MacLane's Coherence Theorem. Define algebras/monoids and modules over them within a monoidal category. Explain the concept of braiding.
7.Define abelian, additive and closed symmetric monoidal abelian categories. Perform diagram chasing in abelian categories using generalized elements. State and prove the Snake Lemma in an abelian category. Define the category of chain complexes and constructions such as truncations, the tensor product/hom complexes. Define projective and injective resolutions and explain their role in derived functors.
8. Define a Grothendieck topology (site). Define sheaves on a site and the sheafification functor. State the Giraud's Theorem axioms characterizing Grothendieck topoi.
Details topic covered：
1 Very basic set theory: including Grothendieck universe, regular cardinal, inaccessible cardinal.
2 Basic definitions: categories, functors, and natural transformations. Basic examples of categories, like the category of sets, topological spaces, groups, abelian groups, commutative rings. The category of presheaves, and Yoneda lemma.
3 Limits and colimits: initial and final objects, limits/colimits as well as their universal properties, examples: products/coproducts, pullback/pushout, equalizer/coequalizer, filtered colimits/cofiltered limits, limits and colimits in functor categories, commutative properties of limits and colimits.
4 Adjoint functors: definition of adjoint functors, unit and counit natural transformations, adjoint functors and limits/colimits, accessible categories and accessible functors, presentable categories, adjoint functor theorem.
5 Monads: definition of monads, monads and algebras, monads and adjoint functors, Barr-Beck theorem, category of algebras and limits/colimits.
6 Monoidal categories: definition of monoidal categories, monoidal functors, monoidal natural transformations, strict monoidal categories. MacLane's strictness and coherence theorem, monoids in monoidal categories, braidings, symmetric monoidal braidings.
7 Abelian categories: definition of abelian categories, enriched categories, additive categories, kernel and cokernel, closed symmetric monoidal abelian categories, categories of chain complexes, truncations of chain complexes, tensor and internal hom of chain complexes, homotopy of chain complexes, exact sequence of chain complexes, snake lemma, projective/injective resolutions.
8 Grothendieck topos: Grothendieck topology and Grothendieck site, sheaves over Grothendieck site, Grothendieck topos, sheafification functor, Giraud’s axiom.
