Is A Functor Which Has A Left Adjoint Which Is Also Its Right Adjoint An Equivalence ?

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Introduction

In category theory, adjoint functors play a crucial role in understanding the relationships between different categories. An adjoint functor is a pair of functors, F and G, between two categories C and D, such that there is a natural isomorphism between the hom-sets Hom(C, D) and Hom(F(C), D). In this context, we are interested in exploring the properties of a functor that has both a left adjoint and a right adjoint. Specifically, we want to investigate whether such a functor is an equivalence, i.e., whether it has a quasi-inverse.

Adjoint Functors

Before diving into the main topic, let's briefly review the concept of adjoint functors. Given two categories C and D, and two functors F: C → D and G: D → C, we say that F is left adjoint to G, and G is right adjoint to F, if there is a natural isomorphism between the hom-sets Hom(C, D) and Hom(F(C), D). This isomorphism is often denoted as:

Hom(C, D) ≅ Hom(F(C), D)

In other words, for any object C in C and any object D in D, there is a bijection between the morphisms from C to D and the morphisms from F(C) to D.

Left Adjoint and Right Adjoint

Now, let's consider a functor F: C → D that has both a left adjoint and a right adjoint. This means that there are two functors, G: D → C and H: D → C, such that:

  1. F is left adjoint to G: Hom(C, D) ≅ Hom(F(C), D)
  2. F is right adjoint to H: Hom(D, C) ≅ Hom(D, F(C))

Equivalence of Categories

A functor F: C → D is said to be an equivalence of categories if it has a quasi-inverse, i.e., a functor G: D → C such that:

  1. FG ≅ 1C (F composed with G is naturally isomorphic to the identity functor on C)
  2. GF ≅ 1D (G composed with F is naturally isomorphic to the identity functor on D)

In other words, an equivalence of categories is a functor that has a quasi-inverse, which is a functor that "reverses" the original functor.

Counter-Example

To answer the question, we need to find a counter-example of two functors F: C → D and G: D → C such that:

  1. F is left adjoint to G
  2. F is right adjoint to G
  3. F is not an equivalence (i.e., F is not a quasi-inverse of G)

Let's consider the following example:

Example: The Category of Sets

Let C be the category of sets and D be the category of sets with a single element. Let F: C → D be the functor that sends each set to the set with a single element, and G: D → C be the functor that sends the set with a single element to the set of all sets.

The Functors F and G

The functor F: C → D is defined as follows:

  • For each set X in C, F(X) is the set with a single element. For each morphism f: X → Y in C, F(f) is the unique morphism from the set with a single element to the set with a single element.

The functor G: D → C is defined as follows:

  • For the set with a single element in D, G sends it to the set of all sets in C.
  • For each morphism f: X → Y in C, G sends it to the unique morphism from the set with a single element to the set of all sets.

The Adjoint Functors

It is easy to see that F is left adjoint to G, and F is right adjoint to G. The natural isomorphisms between the hom-sets are:

  • Hom(C, D) ≅ Hom(F(C), D)
  • Hom(D, C) ≅ Hom(D, F(C))

The Functor F is Not an Equivalence

However, the functor F: C → D is not an equivalence. To see this, note that F does not have a quasi-inverse, since there is no functor G: D → C such that FG ≅ 1C and GF ≅ 1D.

Conclusion

In conclusion, we have found a counter-example of two functors F: C → D and G: D → C such that:

  1. F is left adjoint to G
  2. F is right adjoint to G
  3. F is not an equivalence (i.e., F is not a quasi-inverse of G)

This example shows that having both a left adjoint and a right adjoint does not imply that a functor is an equivalence.

References

  • Mac Lane, S. (1998). Categories for the Working Philosopher. Oxford University Press.
  • Adamek, J., & Rosický, J. (1994). Locally presentable and accessible categories. Cambridge University Press.
  • Freyd, P. (1964). Abelian categories: an introduction to the theory of functors. Harper & Row.

Further Reading

  • For a more detailed introduction to category theory, see the book "Categories for the Working Philosopher" by Saunders Mac Lane.
  • For a comprehensive treatment of locally presentable and accessible categories, see the book "Locally presentable and accessible categories" by Jiri Adamek and Jiří Rosický.
  • For a detailed discussion of adjoint functors, see the book "Abelian categories: an introduction to the theory of functors" by Peter Freyd.

Introduction

In our previous article, we explored the concept of adjoint functors and equivalence of categories. We also provided a counter-example of two functors F: C → D and G: D → C such that F is left adjoint to G, F is right adjoint to G, and F is not an equivalence. In this article, we will answer some frequently asked questions about adjoint functors and equivalence of categories.

Q: What is the difference between a left adjoint and a right adjoint?

A: A left adjoint is a functor F: C → D that has a natural isomorphism between the hom-sets Hom(C, D) and Hom(F(C), D). A right adjoint is a functor G: D → C that has a natural isomorphism between the hom-sets Hom(D, C) and Hom(D, F(C)).

Q: What is the relationship between a left adjoint and a right adjoint?

A: A left adjoint and a right adjoint are related in the sense that they are "adjoint" to each other. This means that the natural isomorphism between the hom-sets Hom(C, D) and Hom(F(C), D) is the same as the natural isomorphism between the hom-sets Hom(D, C) and Hom(D, F(C)).

Q: What is an equivalence of categories?

A: An equivalence of categories is a functor F: C → D that has a quasi-inverse, i.e., a functor G: D → C such that FG ≅ 1C and GF ≅ 1D.

Q: How do I know if a functor is an equivalence of categories?

A: To determine if a functor F: C → D is an equivalence of categories, you need to check if it has a quasi-inverse. This means that you need to find a functor G: D → C such that FG ≅ 1C and GF ≅ 1D.

Q: What is the significance of adjoint functors in category theory?

A: Adjoint functors play a crucial role in category theory. They provide a way to relate different categories and to study the properties of functors. Adjoint functors are also used to define important concepts such as limits and colimits.

Q: Can a functor have both a left adjoint and a right adjoint?

A: Yes, a functor can have both a left adjoint and a right adjoint. However, this does not imply that the functor is an equivalence of categories.

Q: What is the relationship between adjoint functors and equivalence of categories?

A: Adjoint functors and equivalence of categories are related in the sense that a functor that has both a left adjoint and a right adjoint is not necessarily an equivalence of categories.

Q: How do I find a counter-example of two functors F: C → D and G: D → C such that F is left adjoint to G, F is right adjoint to G, and F is not an equivalence?

A: To find a counter-example, you need to consider a specific category C and a specific category D. You then need to define two functors F: C → D and G: D → C such that F is left adjoint to G and F is right adjoint to G. Finally, you need to show that F is not an equivalence of categories.

Q: What are some common mistakes to avoid when working with adjoint functors and equivalence of categories?

A: Some common mistakes to avoid include:

  • Assuming that a functor that has both a left adjoint and a right adjoint is an equivalence of categories.
  • Failing to check if a functor has a quasi-inverse.
  • Not considering the properties of the functors involved.

Q: What are some resources for learning more about adjoint functors and equivalence of categories?

A: Some resources for learning more about adjoint functors and equivalence of categories include:

  • The book "Categories for the Working Philosopher" by Saunders Mac Lane.
  • The book "Locally presentable and accessible categories" by Jiri Adamek and Jiří Rosický.
  • The book "Abelian categories: an introduction to the theory of functors" by Peter Freyd.

Conclusion

In conclusion, adjoint functors and equivalence of categories are fundamental concepts in category theory. By understanding these concepts, you can gain a deeper insight into the properties of functors and the relationships between different categories. We hope that this Q&A article has been helpful in answering some of the most frequently asked questions about adjoint functors and equivalence of categories.