Distinguished Triangle

Definition

A distinguished triangle is a fundamental concept in derived categories, representing a sequence of morphisms between objects that encodes information about the relationships between complexes. This triangle consists of three objects and three morphisms arranged in a specific way, often illustrating how one complex can be viewed as an extension of others. The properties of distinguished triangles help to study derived functors and provide a way to discuss exact sequences in a homotopical context.

5 Must Know Facts For Your Next Test

1. Distinguished triangles can be thought of as a generalized notion of exact sequences, helping to manage relationships between objects in derived categories.
2. They are used to define and understand various properties like cone constructions and morphisms, which lead to important results in homological algebra.
3. In a derived category, every distinguished triangle has associated long exact sequences in cohomology, enabling deeper insights into the structure of complexes.
4. The notion of distinguished triangles leads to the concept of equivalences in derived categories, making them critical for establishing categorical results and transformations.
5. Any distinguished triangle can be represented up to isomorphism, meaning that the exact position of the objects can vary without changing the essential information conveyed by the triangle.

Review Questions

• How does the concept of distinguished triangles relate to the study of exact sequences in derived categories?
  • Distinguished triangles generalize the notion of exact sequences by providing a framework for understanding relationships between complexes through morphisms. In derived categories, these triangles illustrate how one object can be viewed as an extension or modification of others, capturing both algebraic and topological properties. Thus, they allow mathematicians to express and analyze ideas about cohomology and exactness in a more flexible way than traditional exact sequences.
• Discuss the role of distinguished triangles in defining derived functors and their significance in homological algebra.
  • Distinguished triangles play a crucial role in defining derived functors because they help manage resolutions and capture essential relationships between different complexes. This structure allows for the derivation process to be applied effectively, leading to significant results in homological algebra such as spectral sequences and vanishing results. As such, they are foundational elements that enable mathematicians to understand deeper relationships within derived categories and their associated functors.
• Evaluate how distinguished triangles contribute to the understanding of equivalences between different derived categories.
  • Distinguished triangles contribute significantly to understanding equivalences between different derived categories by providing a categorical framework where morphisms can be analyzed up to isomorphism. This enables mathematicians to establish equivalences between categories by demonstrating that certain morphisms behave identically under transformations. The ability to work with these triangles facilitates comparisons between seemingly different derived categories, allowing insights into their structural properties and enhancing the broader study of homotopical algebra.

Related terms

Derived Functor: A derived functor is an extension of a classical functor that accounts for resolutions, allowing one to analyze and study derived categories more effectively.

Exact Triangle: An exact triangle is a specific case of a distinguished triangle where the mapping cones are utilized to illustrate exactness in the derived category.

Homotopy Category: The homotopy category is a category that arises from the process of formally inverting homotopy equivalences, providing a framework to study topological properties using homological methods.