6.3 Universal properties of derived functors

Derived functors are powerful tools in homological algebra, measuring how far a functor is from being exact. They help us understand complex structures by breaking them down into simpler parts we can analyze.

Universal properties of derived functors give us a way to characterize these tools uniquely. This approach lets us work with derived functors abstractly, without worrying about the specific constructions used to define them.

Universal Properties and Functors

Universal Properties and Natural Transformations

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• A universal property defines an object in terms of its relationships with other objects, rather than by its internal structure
• Universal properties are used to characterize objects and morphisms in category theory
• Natural transformations provide a way to compare functors between categories
  • A natural transformation $\eta: F \to G$ is a family of morphisms $\eta_X: F(X) \to G(X)$ for each object $X$ in the source category, such that for every morphism $f: X \to Y$, the diagram commutes: $G(f) \circ \eta_X = \eta_Y \circ F(f)$
• Universal properties and natural transformations play a crucial role in the study of derived functors and homological algebra

Delta and Cohomological Functors

• A delta functor is a sequence of functors $T^n: \mathcal{A} \to \mathcal{B}$ between abelian categories, along with connecting homomorphisms $\delta^n: T^n(A') \to T^{n+1}(A)$ for each short exact sequence $0 \to A \to A' \to A'' \to 0$ in $\mathcal{A}$, satisfying certain axioms
  • The connecting homomorphisms form a long exact sequence: $\cdots \to T^n(A) \to T^n(A') \to T^n(A'') \xrightarrow{\delta^n} T^{n+1}(A) \to \cdots$
• A cohomological functor is a contravariant delta functor, i.e., a delta functor with $T^n(f): T^n(B) \to T^n(A)$ for each morphism $f: A \to B$ in $\mathcal{A}$
• Examples of cohomological functors include the Ext functors $\text{Ext}^n_R(A, -)$ and the cohomology functors $H^n(X; -)$ in algebraic topology

Derived Functors and Exact Sequences

Effacement and Derived Functors

• The effacement theorem states that for any additive functor $F: \mathcal{A} \to \mathcal{B}$ between abelian categories and any object $A$ in $\mathcal{A}$, there exists an epimorphism $P \to A$ with $P$ projective such that $F(P) \to F(A)$ is an epimorphism
• The effacement theorem is used to construct derived functors by taking projective resolutions of objects
• Given a left exact functor $F: \mathcal{A} \to \mathcal{B}$ and an object $A$ in $\mathcal{A}$, the right derived functors $R^nF(A)$ are defined as the cohomology of the complex $F(P_\bullet)$, where $P_\bullet \to A$ is a projective resolution of $A$
  • The derived functors measure the failure of $F$ to be exact

Long Exact Sequences and Connecting Homomorphisms

• A long exact sequence is a sequence of homomorphisms between abelian groups or modules $\cdots \to A_{n-1} \xrightarrow{f_{n-1}} A_n \xrightarrow{f_n} A_{n+1} \to \cdots$ such that the image of each homomorphism is equal to the kernel of the next: $\text{Im}(f_{n-1}) = \text{Ker}(f_n)$
• Long exact sequences often arise from short exact sequences of chain complexes by taking homology or cohomology
• The connecting homomorphism in a long exact sequence is a homomorphism $\delta_n: H_n(C'') \to H_{n-1}(C)$ induced by the short exact sequence of chain complexes $0 \to C \to C' \to C'' \to 0$
  • The connecting homomorphism measures the obstruction to lifting cycles in $C''$ to cycles in $C'$
• Examples of long exact sequences include the long exact sequence in homology associated with a short exact sequence of chain complexes and the long exact sequence of a pair in singular homology