Category Theory
Cohomology via Category Theory
The numbers of a topological space - its "holes" of various dimensions - are its cohomology groups. Whether the equation rot F = G has a solution depends on H¹. Whether a potential exists depends on H². The problem of Newton about the shape of the Earth is a question about H¹. Cohomology is "the tool for counting obstructions".
- Weil conjectures: proved by Deligne via l-adic cohomology (1974)
- Mirror symmetry: equivalence of derived categories via Fourier - Mukai (Mukai)
- Electromagnetism: existence of a potential ↔ H¹ = 0 (Poincaré lemma)
- Digital topology: TDA (Topological Data Analysis) - persistent homology
Chain Complexes: Ab-Enriched Categories
PyTorch 2.0 (2023) uses derived categories for autograd: 10⁹+ parameters in LLMs, the chain rule as a natural transformation. A **chain complex** is a sequence of abelian groups (or modules) ... → C_{n+1} → C_n → C_{n-1} → ... with differentials d satisfying d² = 0. Cohomology measures the "gap" between the image of one differential and the kernel of the next. The category of chain complexes Ch(A) is the foundation of homological algebra.
**"The tool of the twentieth century":** Jean-Pierre Serre called cohomology "the most important mathematical tool of the twentieth century". It classifies topological spaces, vector bundles, algebraic varieties. Betti numbers β_n = dim H^n are topological invariants.
Why is the condition d² = 0 required in a chain complex?
Derived Categories: Localisation at Quasi-Isomorphisms
The **derived category** D(A) is obtained from the category of chain complexes Ch(A) by inverting quasi-isomorphisms (morphisms inducing isomorphisms on cohomology). This localisation is a categorical generalisation of fractions: "complexes with the same cohomology are the same thing".
**Grothendieck and derived categories:** Alexander Grothendieck (together with Verdier) introduced derived categories in the 1960s to formulate RHom and RF in algebraic geometry. Nowadays derived categories appear in string theory (mirror symmetry), representation theory (derived equivalence), and D-modules.
What is a quasi-isomorphism in the context of chain complexes?
Sheaf Cohomology as a Derived Functor
**Sheaf cohomology H^n(X, F)** is the right derived functor of the global sections functor Γ: Sh(X) → Ab. It is a powerful tool: it connects local data (a sheaf) with global information (cohomology). The formulation via derived categories makes many theorems transparent.
**Cohomology in numbers:** For projective space ℙ^n: H^k(ℙ^n, ℤ) = ℤ for k even ≤ 2n, and 0 otherwise. Betti numbers β_k = 1, 0, 1, 0, ... - the "even sieve". The Riemann - Roch theorem, Serre duality, the Hirzebruch - Riemann - Roch formula - all formulated via sheaf cohomology.
Why is sheaf cohomology H^n(X, F) defined via the right derived functor rather than Γ itself?
Key Ideas
- Chain complex: sequence with d² = 0; cohomology H^n = ker/im
- Derived category D(A): localisation of Ch(A) at quasi-isomorphisms
- Derived functor RF measures the "defect of exactness" of F
- H^n(X, F) = R^nΓ(F): sheaf cohomology as the right derived functor of Γ
- Long exact sequence - a consequence of the triangulated structure of D(A)
Related Topics
Cohomology connects category theory with geometry and algebra.
- Topos Theory — Sh(X) is a topos; its cohomology = R^nΓ
- Adjoint Functors — The right derived functor is the right adjoint in the derived category
- Higher Categories — ∞-categories allow formulating derived categories without losing homotopy data
Вопросы для размышления
- Why does the exact sequence 0 → ℤ →×2 ℤ → ℤ/2 → 0 give Ext¹(ℤ/2, ℤ) = ℤ/2? Compute it explicitly.
- What does H¹(X, O*) measure for a complex manifold X? Why is the Picard group equal to a cohomology group?
- How does the idea of the derived category explain why Ext and Tor are "good" invariants rather than merely technical artefacts?