Triangulated categories

Definition

Triangulated categories are a type of category in mathematics equipped with an additional structure that allows for the study of homological properties. This structure includes distinguished triangles, which provide a way to relate objects in the category and their morphisms, making them useful in areas like algebraic K-Theory and motivic cohomology.

5 Must Know Facts For Your Next Test

1. Triangulated categories are defined by a class of morphisms called distinguished triangles, which can be thought of as a way to encapsulate exact sequences in a categorical setting.
2. They play a crucial role in modern algebraic K-Theory, as they allow mathematicians to study stable homotopy types and link various cohomological theories.
3. Every triangulated category has a suspension functor that shifts objects in the category, which is essential for defining long exact sequences.
4. The axioms governing triangulated categories ensure that many results from classical homological algebra carry over to this more abstract setting.
5. Triangulated categories provide a framework for understanding motivic cohomology, allowing mathematicians to connect algebraic geometry with stable homotopy theory.

Review Questions

• How do distinguished triangles function within triangulated categories and why are they significant?
  • Distinguished triangles serve as the core building blocks of triangulated categories, defining relationships between objects and morphisms. They allow for the encoding of exact sequences and provide the means to study homological properties in a more abstract way. The significance lies in their ability to bridge various mathematical fields, including algebraic K-Theory and motivic cohomology, facilitating connections among different areas of research.
• Discuss the role of triangulated categories in connecting algebraic K-Theory with other cohomological theories.
  • Triangulated categories play an integral role in algebraic K-Theory by providing a categorical framework that supports the study of stable homotopy types. They help in relating algebraic K-Theory with motivic cohomology through distinguished triangles, enabling mathematicians to draw parallels between these two seemingly different areas. This connection allows for greater insights into the properties of algebraic varieties and their invariants.
• Evaluate how the concepts from triangulated categories can transform our understanding of derived categories and their applications.
  • The concepts from triangulated categories significantly enhance our understanding of derived categories by providing a structured approach to handle complexes of objects. By integrating the notions of distinguished triangles and suspension functors, mathematicians can analyze derived functors more effectively. This transformation leads to new results in homological algebra, influencing various applications ranging from algebraic geometry to representation theory, thus broadening the scope of research within these fields.

Related terms

derived category: A derived category is a construction that allows for the extension of concepts from homological algebra, focusing on complexes of objects and their morphisms while ignoring certain equivalences.

distinguished triangle: A distinguished triangle is a specific type of triangle in a triangulated category that satisfies certain axioms, serving as the foundational building blocks for homological theories.

functoriality: Functoriality refers to the property that allows for the systematic transformation of mathematical structures through functors, preserving the relationships between objects and morphisms.