1.4 Category-theoretic approach to homological algebra

Homological algebra blends category theory and algebra, offering powerful tools for studying algebraic structures. It introduces abelian categories, which generalize abelian groups and provide a framework for working with kernels, cokernels, and exact sequences.

This approach allows us to analyze complex algebraic systems using categorical concepts. We'll explore chain complexes, homology, and derived categories, seeing how these ideas connect and apply to various mathematical fields.

Categories in Homological Algebra

Abelian Categories and Their Properties

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• Abelian categories generalize the notion of abelian groups to a broader setting
  • Objects in an abelian category have a group structure compatible with morphisms
  • Morphisms between objects in an abelian category form abelian groups under addition
• Abelian categories have kernels and cokernels for every morphism
  • Kernel of a morphism $f: A \rightarrow B$ is an object $K$ with a morphism $k: K \rightarrow A$ such that for any object $X$ and morphism $x: X \rightarrow A$ with $f \circ x = 0$, there exists a unique morphism $x': X \rightarrow K$ with $x = k \circ x'$
  • Cokernel of a morphism $f: A \rightarrow B$ is an object $C$ with a morphism $c: B \rightarrow C$ such that for any object $Y$ and morphism $y: B \rightarrow Y$ with $y \circ f = 0$, there exists a unique morphism $y': C \rightarrow Y$ with $y = y' \circ c$
• Every monomorphism in an abelian category is the kernel of its cokernel
• Every epimorphism in an abelian category is the cokernel of its kernel

Additive Categories and Universal Properties

• Additive categories are categories in which the morphism sets are abelian groups and composition of morphisms is bilinear
  • Objects in an additive category have a zero object and biproducts (direct sums)
  • Morphisms between objects in an additive category can be added and have additive inverses
• Universal properties characterize objects and morphisms in terms of their relationships with other objects and morphisms
  • Examples of universal properties include the universal property of the product (uniqueness of morphisms into the product) and the universal property of the coproduct (uniqueness of morphisms out of the coproduct)
• Universal properties allow for the construction of objects and morphisms satisfying certain conditions without explicitly specifying their internal structure

Derived Categories and Localization

• Derived categories are obtained from categories of chain complexes by formally inverting quasi-isomorphisms (morphisms inducing isomorphisms on homology)
  • Derived categories provide a way to study homological invariants (e.g., Ext and Tor functors) in a more intrinsic setting
  • Objects in a derived category are chain complexes, and morphisms are equivalence classes of chain maps modulo homotopy
• Localization is a process of adding formal inverses to a category with respect to a collection of morphisms
  • Derived categories can be viewed as the localization of the category of chain complexes with respect to quasi-isomorphisms
  • Localization allows for the construction of new categories with desired properties (e.g., the homotopy category of a model category)

Chain Complexes and Homology

Chain Complexes and Their Properties

• A chain complex is a sequence of abelian groups (or modules) connected by homomorphisms (differentials) such that the composition of any two consecutive differentials is zero
  • A chain complex $C_*$ is denoted as $\cdots \rightarrow C_{n+1} \xrightarrow{d_{n+1}} C_n \xrightarrow{d_n} C_{n-1} \rightarrow \cdots$, where $d_n \circ d_{n+1} = 0$ for all $n$
  • The condition $d_n \circ d_{n+1} = 0$ ensures that the image of each differential is contained in the kernel of the next differential
• Chain complexes can be used to model various algebraic and topological objects, such as simplicial complexes, CW complexes, and resolutions of modules
• Morphisms between chain complexes are chain maps, which are collections of homomorphisms between the corresponding abelian groups that commute with the differentials

Homology and Cohomology of Chain Complexes

• The homology of a chain complex $C_*$ at degree $n$ is defined as the quotient group $H_n(C_*) = \ker(d_n) / \operatorname{im}(d_{n+1})$
  • Elements of $\ker(d_n)$ are called $n$-cycles, and elements of $\operatorname{im}(d_{n+1})$ are called $n$-boundaries
  • Homology measures the extent to which the sequence of differentials fails to be exact
• The cohomology of a chain complex $C_*$ at degree $n$ is defined as the homology of the dual cochain complex $C^*$, where $C^n = \operatorname{Hom}(C_n, A)$ for some fixed abelian group $A$
  • The differentials in the cochain complex are the duals of the differentials in the original chain complex
  • Cohomology can be interpreted as a contravariant functor from the category of chain complexes to the category of abelian groups
• Homology and cohomology are functorial, meaning that chain maps between chain complexes induce homomorphisms between their homology and cohomology groups

Applications and Examples of Homology

• Singular homology is a homology theory for topological spaces, constructed using the chain complex of singular simplices
  • Singular homology groups are topological invariants that capture information about the "holes" in a space
  • The singular homology of a sphere $S^n$ is $H_k(S^n) = \mathbb{Z}$ for $k = 0, n$ and $H_k(S^n) = 0$ otherwise
• Group homology is a homology theory for groups, constructed using the chain complex of the group ring with coefficients in a module
  • Group homology measures the failure of a module to be projective over the group ring
  • The group homology of a cyclic group $C_n$ with coefficients in $\mathbb{Z}$ is $H_0(C_n; \mathbb{Z}) = \mathbb{Z}$, $H_1(C_n; \mathbb{Z}) = \mathbb{Z}/n\mathbb{Z}$, and $H_k(C_n; \mathbb{Z}) = 0$ for $k \geq 2$