Category theory - right adjoint problem

Thread starter razizi
Start date Jan 15, 2010
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Category theory Theory

In summary, the right adjoint problem in category theory seeks to find the most appropriate functor to pair with a given functor in order to form an adjunction. It is essentially the inverse of the left adjoint problem and is important in establishing adjunctions, which reveal deep connections between mathematical structures. Some techniques for solving the right adjoint problem include the method of representable functors, the Yoneda lemma, and Kan extensions. There are still open problems related to the right adjoint problem, such as the existence of a universal solution and the classification and uniqueness of solutions.

Jan 15, 2010

razizi

I'm stuck with the attached practice problems. does anyone with knowledge of category theory please have solutions?

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Jan 15, 2010

Hurkyl

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We are not an answer service -- we will help you work through the problem, but we will not give you answers.

That said, a lot of people (myself included) are turned off by having to open a *.doc file...

Jan 15, 2010

razizi

Thanks for offering to assist. I'll take note regarding attaching word documents.

1. What is the right adjoint problem in category theory?

The right adjoint problem in category theory is a fundamental problem that seeks to find the right adjoint functor for a given functor. In other words, it aims to find a functor that is the most appropriate to pair with a given functor in order to form an adjunction.

2. How does the right adjoint problem relate to the left adjoint problem?

The right adjoint problem is essentially the inverse of the left adjoint problem. While the left adjoint problem seeks to find the left adjoint for a given functor, the right adjoint problem seeks to find the right adjoint for a given functor. Both of these problems are important in establishing adjunctions between functors.

3. What is the significance of solving the right adjoint problem?

Solving the right adjoint problem is important because it allows us to establish adjunctions between functors, which are essential in many areas of mathematics. Adjunctions provide a way to relate different mathematical structures and often reveal deep connections between seemingly unrelated concepts.

4. What are some techniques for solving the right adjoint problem?

There are several techniques that can be used to solve the right adjoint problem, including the method of representable functors, the Yoneda lemma, and Kan extensions. Each of these techniques provides a different approach to finding the right adjoint and can be used in different situations depending on the specific problem at hand.

5. Are there any open problems related to the right adjoint problem?

Yes, there are still some open problems related to the right adjoint problem in category theory. One of the main open problems is the existence of a universal solution to the right adjoint problem, which would provide a general method for finding the right adjoint for any functor. Other open problems include the classification of right adjoints and the uniqueness of solutions to the right adjoint problem.