Triangulated categories

Definition

Triangulated categories are a type of category equipped with a distinguished class of morphisms called 'triangles,' which generalize the notion of exact sequences in homological algebra. They provide a framework for studying various cohomological theories, allowing for the development of tools like the derived category and providing insight into the relationships between objects through their morphisms and the behavior of these triangles.

5 Must Know Facts For Your Next Test

1. A triangulated category consists of a class of morphisms that form triangles satisfying certain axioms, including the rotation axiom and the octahedral axiom.
2. The concept of a triangulated category helps formalize the notion of derived functors, allowing for the study of cohomological invariants in a more generalized setting.
3. Triangulated categories have applications beyond homological algebra, including algebraic geometry and representation theory, helping to connect various mathematical fields.
4. Every derived category has an associated triangulated structure, which facilitates understanding the relationships between different derived functors and their associated cohomology theories.
5. Key examples of triangulated categories include the derived category of modules over a ring and the homotopy category of chain complexes.

Review Questions

• How do triangulated categories generalize the concept of exact sequences, and what implications does this have for cohomological theories?
  • Triangulated categories extend the idea of exact sequences by introducing a framework where morphisms can be organized into triangles that satisfy specific axioms. This generalization allows mathematicians to analyze complex relationships between objects while maintaining a coherent structure similar to exactness. By doing so, triangulated categories provide powerful tools for developing cohomological theories and understanding derived functors more effectively.
• Discuss how triangulated categories facilitate connections between different areas of mathematics such as algebraic geometry and representation theory.
  • Triangulated categories serve as a unifying framework that allows for insights from one mathematical area to be applied to another. In algebraic geometry, they help analyze derived categories of coherent sheaves, while in representation theory, they provide a means to study modular representations through their homotopy categories. By leveraging the properties of triangulated categories, mathematicians can draw parallels and uncover deep relationships across these diverse fields.
• Evaluate the role of triangulated categories in current research trends within homological algebra, highlighting their significance in developing new theories.
  • In contemporary research trends in homological algebra, triangulated categories play a pivotal role by offering a structured approach to explore various cohomological concepts and derived functors. Their significance is seen in advancements such as stable homotopy theory and the development of new invariants that characterize topological spaces. By continually refining our understanding of these structures, researchers are expanding the frontiers of homological algebra and establishing new connections with other mathematical disciplines.

Related terms

derived categories: Derived categories are a construction in homological algebra that allows one to systematically study complexes of objects by formalizing the idea of 'homotopy' between them, leading to the definition of functors that preserve homological information.

exact sequences: Exact sequences are sequences of objects and morphisms in a category that capture the notion of 'exactness,' indicating how image and kernel relate to one another, which is fundamental in studying algebraic structures.

stable homotopy theory: Stable homotopy theory is an area of algebraic topology dealing with stable phenomena of topological spaces and spectra, often utilizing triangulated categories to analyze the relationships between homotopy types.