🧬Homological Algebra Unit 4 – Functors and Natural Transformations

Key Concepts and Definitions

• Functors map objects and morphisms between categories while preserving composition and identity
• Natural transformations provide a way to compare functors by defining morphisms between them
• Categories consist of objects and morphisms (arrows) that satisfy composition and identity axioms
• Homological algebra studies algebraic structures (abelian categories) using functors and natural transformations
• Exact sequences are key tools in homological algebra that capture the behavior of functors
• Chain complexes are sequences of objects and morphisms with the property that the composition of consecutive morphisms is zero
• Homology measures the failure of a sequence to be exact by taking quotients of kernels and images
• Derived functors extend the notion of functors to measure the deviation from exactness

Functors: Types and Properties

• Covariant functors preserve the direction of morphisms between categories
• Contravariant functors reverse the direction of morphisms between categories
• Faithful functors inject morphisms, ensuring distinct morphisms in the domain category map to distinct morphisms in the codomain category
• Full functors surject morphisms, ensuring every morphism in the codomain category has a preimage in the domain category
• Essentially surjective functors map objects up to isomorphism, meaning every object in the codomain category is isomorphic to an object in the image of the functor
• Equivalence of categories occurs when there exist functors in both directions that are full, faithful, and essentially surjective
  • Equivalence preserves categorical properties while allowing for different representations of the same abstract structure
• Adjoint functors form a pair (left adjoint and right adjoint) that satisfy a natural isomorphism between hom-sets
  • Adjunctions capture universal properties and provide a way to move between different categorical contexts

Natural Transformations Explained

• Natural transformations are morphisms between functors that respect the structure of the categories involved
• For each object $A$ in the domain category, a natural transformation assigns a morphism $\eta_A: F(A) \to G(A)$ in the codomain category
• Naturality square commutes for every morphism $f: A \to B$ in the domain category: $G(f) \circ \eta_A = \eta_B \circ F(f)$
• Natural isomorphisms are natural transformations where each component morphism is an isomorphism
• Vertical composition of natural transformations allows for the composition of morphisms between functors
• Horizontal composition of natural transformations enables the composition of natural transformations along functors
• The functor category $\mathcal{C}^{\mathcal{D}}$ has functors from $\mathcal{D}$ to $\mathcal{C}$ as objects and natural transformations as morphisms
• The Yoneda lemma establishes a natural isomorphism between an object and its representable functor, providing a fully faithful embedding of a category into its functor category

Category Theory Connections

• Functors and natural transformations are the building blocks of category theory
• Monoidal categories have a tensor product that allows for the composition of objects and morphisms
• Monoidal functors preserve the monoidal structure between categories
• Symmetric monoidal categories have a natural isomorphism that allows for the swapping of tensor products
• Braided monoidal categories relax the symmetry condition, allowing for a braiding isomorphism
• Enriched categories replace hom-sets with objects from a monoidal category, providing a more general framework
• Kan extensions provide a way to extend functors along a given functor, generalizing limits and colimits
• Grothendieck topologies and sheaves provide a foundation for topos theory, a generalization of algebraic geometry using categorical tools

Important Theorems and Proofs

• Freyd-Mitchell embedding theorem shows that every small abelian category can be embedded into the category of modules over a ring
• Snake lemma is a powerful tool for studying the relationship between kernels and cokernels in a commutative diagram of abelian groups
• Five lemma proves the isomorphism of the middle objects in a commutative diagram of exact sequences
• Short five lemma is a special case of the five lemma for short exact sequences
• Grothendieck spectral sequence is a tool for computing derived functors in the presence of a composition of functors
• Künneth formula relates the homology of a tensor product to the homology of its factors
• Universal coefficient theorem relates homology and cohomology groups via a short exact sequence
• Dold-Kan correspondence establishes an equivalence between the category of chain complexes and the category of simplicial abelian groups

Applications in Homological Algebra

• Derived functors, such as Ext and Tor, measure the failure of a functor to be exact
  • Ext groups capture extensions of modules and provide a way to study the structure of modules
  • Tor groups measure the torsion in tensor products and are related to the flatness of modules
• Group cohomology uses homological algebra to study the algebraic properties of groups
• Lie algebra cohomology applies homological techniques to the study of Lie algebras and their representations
• Sheaf cohomology extends the notion of cohomology to sheaves on topological spaces, providing a bridge between algebra and geometry
• Hochschild homology and cohomology are important tools in the study of associative algebras and their bimodules
• Cyclic homology is a generalization of Hochschild homology that incorporates the action of the cyclic group
• Spectral sequences are powerful computational tools that arise from filtered complexes and converge to the homology of the total complex
• Derived categories provide a framework for studying chain complexes up to quasi-isomorphism, allowing for the localization of the category of chain complexes

Common Examples and Exercises

• Compute the homology groups of simple chain complexes (e.g., $0 \to \mathbb{Z} \xrightarrow{2} \mathbb{Z} \to 0$)
• Determine the effect of functors on specific categories (e.g., the forgetful functor from $\mathbf{Grp}$ to $\mathbf{Set}$)
• Construct natural transformations between familiar functors (e.g., the determinant functor and the identity functor on $\mathbf{Mat}_n(\mathbb{R})$)
• Prove that a given functor is full, faithful, or essentially surjective
• Compute derived functors in simple cases (e.g., $\mathrm{Ext}_{\mathbb{Z}}(\mathbb{Z}/n\mathbb{Z}, \mathbb{Z})$ and $\mathrm{Tor}^{\mathbb{Z}}_*(\mathbb{Z}/n\mathbb{Z}, \mathbb{Z}/m\mathbb{Z})$)
• Apply the snake lemma and five lemma to commutative diagrams of exact sequences
• Calculate the cohomology groups of simple spaces using cellular cohomology or de Rham cohomology
• Compute the Hochschild homology and cohomology of elementary algebras (e.g., polynomial algebras)

Advanced Topics and Further Reading

• Triangulated categories generalize the notion of derived categories and provide a framework for studying homological algebra in non-abelian settings
• Model categories are a general framework for homotopy theory that encompasses topological spaces, chain complexes, and simplicial sets
• Infinity categories (quasi-categories) extend the notion of categories to allow for higher-dimensional morphisms and provide a foundation for higher category theory
• Operads encode algebraic structures with multiple inputs and outputs, such as associative algebras and commutative algebras
• Homotopical algebra studies the interplay between homotopy theory and algebraic structures, leading to the development of derived algebraic geometry
• Topological quantum field theories (TQFTs) use monoidal categories and functors to describe quantum field theories in a mathematically rigorous way
• Homotopy type theory is a foundation for mathematics that combines type theory, homotopy theory, and category theory, providing a new perspective on the nature of mathematical objects and proofs
• Further reading:
  • "Algebra" by Serge Lang
  • "An Introduction to Homological Algebra" by Charles A. Weibel
  • "Categories for the Working Mathematician" by Saunders Mac Lane
  • "Sheaves in Geometry and Logic: A First Introduction to Topos Theory" by Saunders Mac Lane and Ieke Moerdijk