Preadditive category

Definition

A preadditive category is a type of category where the hom-sets between any two objects are abelian groups, and the composition of morphisms is bilinear. This structure allows for the definition of concepts like kernels and cokernels, which are essential in the study of homological algebra. Preadditive categories serve as a foundational framework for many constructions in both category theory and algebra.

5 Must Know Facts For Your Next Test

1. In a preadditive category, for any two objects A and B, the set of morphisms from A to B, denoted as Hom(A, B), forms an abelian group under addition.
2. Composition of morphisms in a preadditive category respects the structure of the abelian groups, making it bilinear: (f + g) ∘ h = f ∘ h + g ∘ h for any morphisms f, g, and h.
3. Every additive category is also a preadditive category, but not every preadditive category has zero morphisms or direct sums.
4. Preadditive categories are crucial in defining limits and colimits because they allow for the use of kernel and cokernel concepts from linear algebra.
5. Common examples of preadditive categories include the category of abelian groups and the category of modules over a ring.

Review Questions

• How does the structure of hom-sets in a preadditive category facilitate operations like addition and composition of morphisms?
  • In a preadditive category, each hom-set between two objects is not just a collection of morphisms but an abelian group. This means that you can add morphisms together to form new morphisms while also maintaining the ability to compose them. The bilinearity of composition ensures that addition distributes over composition, which is key to performing various operations such as finding kernels and cokernels.
• Discuss the implications of having a preadditive structure when analyzing morphisms in categories related to homological algebra.
  • The preadditive structure allows us to utilize tools from linear algebra, such as kernels and cokernels, when analyzing morphisms. This is vital in homological algebra because it enables us to define exact sequences and understand properties like exactness in terms of how morphisms relate to each other. Moreover, it provides a framework to explore derived functors and other concepts that rely on additive structures.
• Evaluate how the concept of a preadditive category influences the development of more complex categorical structures like triangulated categories.
  • The notion of preadditivity lays the groundwork for understanding more complex categorical structures such as triangulated categories. Triangulated categories extend ideas from preadditive categories by introducing cones and distinguished triangles, which depend on the properties of kernels and cokernels found in preadditive settings. This connection allows mathematicians to analyze derived categories and their relationships with homological properties, ultimately enriching our understanding of categorical frameworks in algebra.

Related terms

abelian group: An abelian group is a set equipped with an operation that combines two elements to form a third element, satisfying properties such as associativity, identity, inverses, and commutativity.

morphism: A morphism is a structure-preserving map between two objects in a category, which can be thought of as a generalization of functions.

additive category: An additive category is a preadditive category with additional structure that includes zero morphisms and allows for the direct sum of objects.