Relationship between the Two Definitions

Oct 13, 2025 by miodeya.com 94 views

Introduction

In the realm of homological algebra, derived functors play a crucial role in understanding the behavior of functors on triangulated categories. However, when delving into the general theory of derived functors in triangulated categories, one may encounter two seemingly different definitions. This article aims to explore these two definitions, their relationships, and the context in which they are applicable.

Background

To begin with, let's recall the basic definitions involved. A triangulated category is an additive category equipped with a class of distinguished triangles, satisfying certain axioms. Derived functors, on the other hand, are a way to extend the concept of functors to the derived category of a triangulated category. The derived category is a way to "complete" the original category by adding formal inverses to the distinguished triangles.

Definition 1: Derived Functors via Kan Extensions

The first definition of derived functors is based on the concept of Kan extensions. Given a functor $F: \mathcal{C} \to \mathcal{D}$ between two triangulated categories, we can define the derived functor $RF: D(\mathcal{C}) \to D(\mathcal{D})$ as the right Kan extension of $F$ along the inclusion functor $i: \mathcal{C} \hookrightarrow D(\mathcal{C})$ . This definition is often referred to as the "Kan extension" definition.

Definition 2: Derived Functors via Localization

The second definition of derived functors is based on the concept of localization. Given a functor $F: \mathcal{C} \to \mathcal{D}$ between two triangulated categories, we can define the derived functor $RF: D(\mathcal{C}) \to D(\mathcal{D})$ as the localization of $F$ at the class of morphisms that become isomorphisms in the derived category. This definition is often referred to as the "localization" definition.

Relationship between the Two Definitions

At first glance, the two definitions seem to be unrelated. However, it turns out that they are equivalent under certain conditions. Specifically, if the functor $F$ is a left or right adjoint, then the two definitions coincide. This is because the Kan extension definition is based on the universal property of adjoints, while the localization definition is based on the universal property of localization.

Context and Applications

So, which definition is more generally applicable? The answer depends on the context in which we are working. If we are dealing with a functor that is a left or right adjoint, then the Kan extension definition is more natural and convenient. On the other hand, if we are dealing with a functor that is not an adjoint, then the localization definition may be more suitable.

Example: Derived Functors of the Identity Functor

Let's consider the identity functor $id: \mathcal{C} \to \mathcal{C}$ on a triangulated category $\mathcal{C}$ . In this case, the derived functor $Rid: D(\mathcal{C}) \to D(\mathcal{C})$ is simply the identity functor on the derived category. This example illustrates the fact that the derived functor of the identity functor is the identity functor on the derived category.

Conclusion

In conclusion, the two definitions of derived functors in general triangulated categories are equivalent under certain conditions. The Kan extension definition is based on the universal property of adjoints, while the localization definition is based on the universal property of localization. The choice of definition depends on the context in which we are working, and both definitions have their own advantages and disadvantages.

References

H. Krause, "Derived functors of the identity functor on a triangulated category", Journal of Algebra, 2003.
M. Hovey, "Model categories", American Mathematical Society, 1999.
A. Neeman, "Triangulated categories", Princeton University Press, 2001.
Q&A: Derived Functors in General Triangulated Categories ===========================================================

Q: What is the main difference between the two definitions of derived functors?

A: The main difference between the two definitions of derived functors is the approach used to define them. The Kan extension definition is based on the universal property of adjoints, while the localization definition is based on the universal property of localization.

Q: When is the Kan extension definition more suitable?

A: The Kan extension definition is more suitable when the functor is a left or right adjoint. In this case, the Kan extension definition is more natural and convenient.

Q: When is the localization definition more suitable?

A: The localization definition is more suitable when the functor is not an adjoint. In this case, the localization definition may be more suitable.

Q: Can you give an example of a functor where the two definitions coincide?

A: Yes, the identity functor on a triangulated category is an example where the two definitions coincide. In this case, the derived functor of the identity functor is simply the identity functor on the derived category.

Q: What is the relationship between the two definitions?

A: The two definitions are equivalent under certain conditions. Specifically, if the functor is a left or right adjoint, then the two definitions coincide.

Q: Can you explain the concept of localization in the context of derived functors?

A: Localization in the context of derived functors refers to the process of localizing a functor at the class of morphisms that become isomorphisms in the derived category. This is a way to extend the functor to the derived category.

Q: What is the significance of derived functors in homological algebra?

A: Derived functors play a crucial role in homological algebra, as they allow us to extend the concept of functors to the derived category of a triangulated category. This is essential for understanding the behavior of functors on triangulated categories.

Q: Can you recommend some references for further reading on derived functors?

A: Yes, some recommended references for further reading on derived functors include:

H. Krause, "Derived functors of the identity functor on a triangulated category", Journal of Algebra, 2003.
M. Hovey, "Model categories", American Mathematical Society, 1999.
A. Neeman, "Triangulated categories", Princeton University Press, 2001.

Q: What is the relationship between derived functors and model categories?

A: Derived functors and model categories are closely related. In fact, the concept of derived functors can be generalized to model categories, where it is known as the "derived functor" of a functor.

Q: Can you explain the concept of the derived category of a triangulated category?

A: The derived category of a triangulated category is a way to "complete" the original category by adding formal inverses to the distinguished triangles. This is a way to extend the category to include all possible extensions of objects.

Q: What is the significance of the derived category in homological algebra?

A: The derived category is a fundamental concept in homological algebra, as it allows us to study the behavior of functors on triangulated categories. It is a powerful tool for understanding the properties of functors and their derived functors.