Monoidal Functor And The Units II

Introduction

In the realm of category theory, monoidal functors play a crucial role in establishing connections between different monoidal categories. These functors not only preserve the monoidal structure but also provide a way to relate the units of the categories involved. In this article, we will delve into the concept of monoidal functors and explore the significance of the units in this context.

Monoidal Categories

Before we dive into the world of monoidal functors, let's briefly recall the definition of a monoidal category. A monoidal category is a category equipped with a bifunctor $\otimes: \mathcal{C} \times \mathcal{C} \to \mathcal{C}$, called the tensor product, and two objects $I$ and $J$ in $\mathcal{C}$, called the units, satisfying certain properties. Specifically, the tensor product must be associative up to a natural isomorphism, and the units must act as left and right identities for the tensor product.

Monoidal Functors

A monoidal functor between two monoidal categories $\mathcal{C}$ and $\mathcal{D}$ is a functor $F: \mathcal{C} \to \mathcal{D}$ that preserves the monoidal structure. This means that $F$ must commute with the tensor product and the units of the categories involved. In other words, $F$ must satisfy the following properties:

• $F(a \otimes b) \cong F(a) \otimes F(b)$ for all objects $a$ and $b$ in $\mathcal{C}$.
• $F(I) \cong I$ for the unit object $I$ in $\mathcal{C}$.

Units and Monoidal Functors

The units of a monoidal category play a crucial role in the definition of a monoidal functor. In particular, the isomorphism $F(I) \cong I$ is a key component of the definition. However, as mentioned in the introduction, Etingof, Gelaki, Nikshych, and Ostrik provide a different definition of a monoidal functor in their book Tensor Categories. In this definition, the isomorphism $F(I) \cong I$ is not required.

Strong Monoidal Functors

A strong monoidal functor is a monoidal functor that preserves the associativity of the tensor product. In other words, a strong monoidal functor $F: \mathcal{C} \to \mathcal{D}$ must satisfy the following property:

• $F(a \otimes b) \otimes F(c) \cong F(a) \otimes (F(b) \otimes F(c))$ for all objects $a$, $b$, and $c$ in $\mathcal{C}$.

Weak Monoidal Functors

A weak monoidal functor is a monoidal functor that does not preserve the associativity of the tensor product. In other words, a weak monoidal functor $F: \mathcal{C} \to \mathcal{D}$ must satisfy the following property:

• $F(a \otimes b) \otimes F(c) \cong F(a) \otimes (F(b) \otimes F(c))$ up to a natural isomorphism.

Comparison of Definitions

The definition of a monoidal functor provided by Etingof, Gelaki, Nikshych, and Ostrik is different from the traditional definition. In particular, the isomorphism $F(I) \cong I$ is not required. This difference has significant implications for the study of monoidal categories and functors.

Implications for Monoidal Categories

The definition of a monoidal functor provided by Etingof, Gelaki, Nikshych, and Ostrik has implications for the study of monoidal categories. In particular, it allows for the existence of monoidal functors that do not preserve the associativity of the tensor product. This has significant implications for the study of monoidal categories and their properties.

Conclusion

In conclusion, the concept of monoidal functors and the units plays a crucial role in the study of monoidal categories. The definition of a monoidal functor provided by Etingof, Gelaki, Nikshych, and Ostrik is different from the traditional definition, and has significant implications for the study of monoidal categories and functors. Further research is needed to fully understand the implications of this definition and its impact on the study of monoidal categories.

References

• Etingof, P., Gelaki, S., Nikshych, D., & Ostrik, V. (2015). Tensor categories. American Mathematical Society.
• Mac Lane, S. (1998). Categories for the working mathematician. Springer-Verlag.

Future Research Directions

Further research is needed to fully understand the implications of the definition of a monoidal functor provided by Etingof, Gelaki, Nikshych, and Ostrik. Some potential research directions include:

• Investigating the properties of monoidal functors that do not preserve the associativity of the tensor product.
• Studying the implications of this definition for the study of monoidal categories and their properties.
• Developing new techniques for constructing monoidal functors that do not preserve the associativity of the tensor product.

Open Problems

There are several open problems related to the study of monoidal functors and the units. Some of these problems include:

• Can we develop a theory of monoidal functors that do not preserve the associativity of the tensor product?
• What are the implications of this definition for the study of monoidal categories and their properties?
• Can we develop new techniques for constructing monoidal functors that do not preserve the associativity of the tensor product?
  Monoidal Functor and the Units II: Q&A =====================================

Q: What is a monoidal functor?

A: A monoidal functor is a functor between two monoidal categories that preserves the monoidal structure. This means that it must commute with the tensor product and the units of the categories involved.

Q: What is the difference between a strong monoidal functor and a weak monoidal functor?

A: A strong monoidal functor is a monoidal functor that preserves the associativity of the tensor product, while a weak monoidal functor does not preserve the associativity of the tensor product.

Q: Why is the isomorphism $F(I) \cong I$ important in the definition of a monoidal functor?

A: The isomorphism $F(I) \cong I$ is important because it ensures that the functor preserves the units of the categories involved. This is a key component of the definition of a monoidal functor.

Q: What is the significance of the units in a monoidal category?

A: The units in a monoidal category play a crucial role in the definition of a monoidal functor. They act as left and right identities for the tensor product and must be preserved by the functor.

Q: Can a monoidal functor be both strong and weak?

A: No, a monoidal functor cannot be both strong and weak. If it preserves the associativity of the tensor product, it is a strong monoidal functor, and if it does not preserve the associativity of the tensor product, it is a weak monoidal functor.

Q: What are some examples of monoidal functors?

A: Some examples of monoidal functors include:

• The forgetful functor from a monoidal category to the underlying category.
• The tensor product functor from a monoidal category to itself.
• The identity functor on a monoidal category.

Q: Can a monoidal functor be used to relate different monoidal categories?

A: Yes, a monoidal functor can be used to relate different monoidal categories. This is one of the key applications of monoidal functors in category theory.

Q: What are some open problems related to monoidal functors?

A: Some open problems related to monoidal functors include:

• Developing a theory of monoidal functors that do not preserve the associativity of the tensor product.
• Investigating the implications of the definition of a monoidal functor provided by Etingof, Gelaki, Nikshych, and Ostrik.
• Developing new techniques for constructing monoidal functors that do not preserve the associativity of the tensor product.

Q: What are some potential applications of monoidal functors?

A: Some potential applications of monoidal functors include:

• Relating different monoidal categories and their properties.
• Developing new techniques for constructing monoidal categories and functors.
• Investigating the properties of monoidal functors and their implications for the study of monoidal categories.

Q: What is the relationship between monoidal functors and other areas of mathematics?

A: Monoidal functors have connections to other of mathematics, including:

• Representation theory: Monoidal functors can be used to relate different representation categories.
• Algebraic geometry: Monoidal functors can be used to relate different algebraic geometry categories.
• Topology: Monoidal functors can be used to relate different topological categories.

Q: What are some resources for learning more about monoidal functors?

A: Some resources for learning more about monoidal functors include:

• The book "Tensor Categories" by Etingof, Gelaki, Nikshych, and Ostrik.
• The book "Categories for the Working Mathematician" by Mac Lane.
• Online lectures and courses on category theory and monoidal functors.