Abelian Category

Definition

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.

5 Must Know Facts For Your Next Test

1. In an abelian category, every morphism can be factored into an epimorphism followed by a monomorphism.
2. The presence of kernels and cokernels ensures that every short exact sequence can be analyzed using the properties of the abelian category.
3. Examples of abelian categories include the category of abelian groups, vector spaces, and modules over a ring.
4. Every abelian category is also an additive category, which means that it allows for the addition of morphisms and has a zero object.
5. Abelian categories serve as a foundational structure for the development of derived functors, which extend the notion of homological algebra beyond simple modules.

Review Questions

• How do kernels and cokernels within an abelian category facilitate the understanding of exact sequences?
  • Kernels and cokernels are essential components in an abelian category that help define exact sequences. An exact sequence consists of a chain of objects connected by morphisms, where the image of one morphism equals the kernel of the next. This property allows us to analyze relationships between different objects, making it easier to work with concepts like short exact sequences, which are fundamental in homological algebra.
• Discuss the role of additivity in an abelian category and its implications for the manipulation of morphisms.
  • Additivity in an abelian category means that we can combine morphisms through addition and that there exists a zero object. This property allows us to treat morphisms similarly to vectors, making calculations and manipulations much simpler. Because every morphism can be expressed as a sum of other morphisms, this facilitates the construction and analysis of complex structures like exact sequences, which are vital in homological algebra.
• Evaluate how defining an abelian category impacts the study and computation of derived functors.
  • Defining an abelian category provides a robust framework for studying derived functors by ensuring that all necessary properties—like having kernels and cokernels—are present. This structured environment allows for the systematic computation of functors like Ext and Tor. The ability to construct exact sequences within this framework leads to powerful applications in various areas of mathematics, such as topology and algebraic geometry, ultimately enhancing our understanding of complex relationships between algebraic structures.

Related terms

Exact Sequence: A sequence of objects and morphisms in an abelian category where the image of one morphism equals the kernel of the next, indicating a precise relationship between them.

Functor: A mapping between categories that preserves the structure of categories, allowing one to translate concepts from one category to another.

Projective Object: An object in an abelian category such that any morphism from it can be lifted through any epimorphism, which plays a key role in defining exact sequences.