4.2 Additive and exact functors

Functors are key players in category theory, bridging different mathematical structures. Additive functors preserve the addition of morphisms, while exact functors maintain the exactness of sequences. These concepts are crucial for understanding how information flows between categories.

Left exact functors preserve kernels, while right exact functors preserve cokernels. This distinction helps us analyze how functors interact with short exact sequences, shedding light on the relationships between different mathematical objects and structures.

Additive and Exact Functors

Additive Functors and Their Properties

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• Additive functors preserve the additive structure of categories
  • Map objects to objects and morphisms to morphisms
  • Respect addition of morphisms and the zero morphism
• Additive functors commute with finite direct sums
  • $F(A \oplus B) \cong F(A) \oplus F(B)$
• Examples of additive functors include:
  • Identity functor $1_{\mathcal{A}}: \mathcal{A} \to \mathcal{A}$
  • Forgetful functor from the category of abelian groups to the category of sets

Exact Functors and Sequences

• Exact functors preserve exact sequences
  • Map short exact sequences to short exact sequences
  • Preserve kernels and cokernels
• Left exact functors preserve exactness at the beginning of a sequence
  • Preserve kernels and monomorphisms
  • Example: Hom functor $\operatorname{Hom}(A, -): \mathcal{A} \to \mathbf{Ab}$
• Right exact functors preserve exactness at the end of a sequence
  • Preserve cokernels and epimorphisms
  • Example: Tensor product functor $- \otimes B: \mathbf{Ab} \to \mathbf{Ab}$

Short Exact Sequences and Abelian Categories

Short Exact Sequences

• A short exact sequence is a sequence of objects and morphisms in an abelian category
  • $0 \to A \xrightarrow{f} B \xrightarrow{g} C \to 0$
  • $f$ is a monomorphism (injective), $g$ is an epimorphism (surjective)
  • $\operatorname{im} f = \ker g$
• Short exact sequences capture the idea of $B$ being an extension of $A$ by $C$
• Examples of short exact sequences in the category of abelian groups:
  • $0 \to \mathbb{Z} \xrightarrow{\times 2} \mathbb{Z} \xrightarrow{\bmod 2} \mathbb{Z}/2\mathbb{Z} \to 0$
  • $0 \to \mathbb{Z}/2\mathbb{Z} \to \mathbb{Z}/4\mathbb{Z} \to \mathbb{Z}/2\mathbb{Z} \to 0$

Abelian Categories and Their Properties

• Abelian categories are categories with additional structure
  • Have a zero object, binary products and coproducts, and kernels and cokernels
  • Satisfy certain axioms related to the behavior of morphisms and exact sequences
• Kernels are the categorical generalization of the kernel of a group homomorphism
  • The kernel of a morphism $f: A \to B$ is the equalizer of $f$ and the zero morphism
• Cokernels are the dual notion of kernels
  • The cokernel of a morphism $f: A \to B$ is the coequalizer of $f$ and the zero morphism
• Examples of abelian categories include:
  • The category of abelian groups $\mathbf{Ab}$
  • The category of modules over a ring $R$-$\mathbf{Mod}$