Glossary

4.2 Additive and exact functors

2 min read•august 7, 2024

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⊕B)≅F(A)⊕F(B)F(A \oplus B) \cong F(A) \oplus F(B)
  • Examples of additive functors include:
    • Identity functor 1A:A→A1_{\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 Hom⁡(A,−):A→Ab\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 −⊗B:Ab→Ab- \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
    • 0→A→fB→gC→00 \to A \xrightarrow{f} B \xrightarrow{g} C \to 0
    • ff is a monomorphism (injective), gg is an epimorphism (surjective)
    • im⁡f=ker⁡g\operatorname{im} f = \ker g
  • Short exact sequences capture the idea of BB being an extension of AA by CC
  • Examples of short exact sequences in the category of abelian groups:
    • 0→Z→×2Z→ mod 2Z/2Z→00 \to \mathbb{Z} \xrightarrow{\times 2} \mathbb{Z} \xrightarrow{\bmod 2} \mathbb{Z}/2\mathbb{Z} \to 0
    • 0→Z/2Z→Z/4Z→Z/2Z→00 \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→Bf: A \to B is the equalizer of ff and the zero morphism
  • Cokernels are the dual notion of kernels
    • The cokernel of a morphism f:A→Bf: A \to B is the coequalizer of ff and the zero morphism
  • Examples of abelian categories include:
    • The category of abelian groups Ab\mathbf{Ab}
    • The category of modules over a ring RR-Mod\mathbf{Mod}

Key Terms to Review (15)

Abelian Category: An abelian category is a type of category in mathematics where morphisms can be added together, and every morphism has a kernel and a cokernel, allowing for the construction of exact sequences. This structure provides a framework to discuss concepts like exactness, kernels, cokernels, and homological algebra more generally, making it crucial for understanding how to work with chain complexes and derived functors.
Additive Functor: An additive functor is a type of functor between categories that preserves the structure of addition in a way that maps zero objects to zero objects and additive structures to additive structures. This means that when you apply an additive functor to a direct sum of objects, it results in the direct sum of the images of those objects. Additive functors play a crucial role in the study of exact sequences and derived functors, helping to maintain the relationships between algebraic structures while facilitating computations and applications.
Chain Complex: A chain complex is a sequence of abelian groups or modules connected by homomorphisms, where the composition of any two consecutive homomorphisms is zero. This structure allows for the study of algebraic properties and relationships in various mathematical contexts, particularly in the fields of topology and algebra. Chain complexes are fundamental in defining homology and cohomology theories, which provide powerful tools for analyzing mathematical objects.
Commutative Diagram: A commutative diagram is a visual representation in category theory where objects are represented as points and morphisms as arrows, illustrating relationships between objects in such a way that all paths with the same start and endpoints yield the same result when composed. This concept is essential for understanding how different mathematical structures interact and is crucial for analyzing concepts like exact sequences and functors.
Derived functor: A derived functor is a concept in homological algebra that extends the idea of a functor by associating it with a sequence of objects (called derived objects) which capture the 'homological information' of the original functor. This allows one to study properties of functors that are not preserved under direct application, particularly in the context of resolving modules and understanding exact sequences.
Exact Functor: An exact functor is a type of functor that preserves the exactness of sequences between categories. In simple terms, if you have a sequence of objects and morphisms that is exact in one category, applying an exact functor will produce an exact sequence in another category as well. This property is crucial for maintaining the relationships between structures when transforming them through different mathematical contexts.
Henri Cartan: Henri Cartan was a prominent French mathematician known for his significant contributions to algebraic topology and homological algebra, particularly in developing the theory of sheaves and derived functors. His work laid essential groundwork for later developments in category theory and cohomology, impacting various mathematical areas including group cohomology and spectral sequences.
Isomorphism: An isomorphism is a mapping between two mathematical structures that establishes a one-to-one correspondence, preserving the operations and relations of the structures. It shows that two structures are fundamentally the same in terms of their properties, even if they may appear different at first glance. This concept connects deeply with how we understand relationships in category theory and other mathematical frameworks.
Jean-Pierre Serre: Jean-Pierre Serre is a prominent French mathematician known for his work in algebraic geometry, topology, and number theory. His contributions have had a significant impact on various areas of mathematics, particularly through the development of cohomological methods and the theory of derived categories.
Left Exact Functor: A left exact functor is a type of functor that preserves finite limits, particularly kernels and finite products, while not necessarily preserving cokernels or coequalizers. This means that when a left exact functor is applied to a sequence of morphisms, it will maintain the structure of the kernel and product but may distort or not reflect the cokernel. Left exact functors are important in studying homological algebra as they help us understand the behavior of sequences in abelian categories.
Naturality: Naturality refers to a property of morphisms between functors that expresses a certain coherence when changing categories. It captures the idea that there are consistent ways to relate different structures, allowing transformations to be applied without losing essential relationships. This concept is crucial in ensuring that relationships between objects and their mappings remain intact when working with covariant and contravariant functors, as well as in contexts involving additive and exact functors.
Right Exact Functor: A right exact functor is a type of functor between two categories that preserves the exactness of sequences at the right end. This means that if you have an exact sequence of morphisms, applying a right exact functor will ensure that the image of the last morphism maps into the cokernel of the previous morphism, thus maintaining the structure of the sequence. Right exactness is crucial for understanding how functors interact with various algebraic structures, particularly in the context of additive categories and homological algebra.
Theorem of the Five Lemmas: The theorem of the five lemmas is a key result in homological algebra that provides a way to connect various exact sequences through a series of commutative diagrams. It establishes conditions under which the existence of certain morphisms can be guaranteed when dealing with exact functors and short exact sequences, thus allowing one to derive important conclusions about the structure of modules or objects in an abelian category.
Tor Functor: The Tor functor is a derived functor that measures the extent to which a sequence fails to be exact when applied to modules. It plays a vital role in homological algebra, connecting algebraic properties of modules with topological invariants, and it helps in understanding the relationships between different algebraic structures.
Transformation: In mathematics, a transformation is a function or mapping that takes a set of elements from one space and maps them to another space, often altering their structure or properties in the process. In the context of additive and exact functors, transformations help establish relationships between different categories by preserving certain structures, which allows for comparisons and applications of homological concepts across these categories.
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