According to Wikipedia,

• A dual object in a monoidal category is analogous to the idea of a dual vector space.

• Infinite dimensional vector spaces are not dualizable.

• An object is often dualizable when it has a finiteness or compactness property.

• A category in which all objects have a dual is called autonomous or rigid.

• The category of finite-dimensional vector spaces with the tensor product is rigid.

• A dual vector space $V^*$ (dual to vector space $V$ over field $K$) has the following property: for field $K$, any two vector spaces over $K$ - $U$ and $W$ - there is an adjunction between $Hom(U \otimes V, W)$ and $Hom(U, V^* \otimes W)$. ‘This expression makes sense in any category with an appropriate replacement for the tensor product of vector spaces.’

1. What is a monoidal category? Does it simply mean there is a monoidal product defined on the objects of the category?

2. What is the definition of, and intuition behind, a dual vector space?

3. Why aren’t infinite vector spaces dualizable?

4. What compactness properties are often associated with a vector space being dualizable?

5. What interesting properties do rigid categories have?

6. What is the definition of the tensor product?

7. What is an adjunction?

8. What is the significance of the formula $Hom(U \otimes V, W) = Hom(U, V^* \otimes W)$? What is it really saying? Is the dual object something like an inverse element in a monoid - that there is a loose kind of equivalence or correspondence (and adjunction) between $UV$ and $W$, vs. $U$ and $V^*W$? Is there a name for variations on this relationship, such as an adjunction between $Hom(U \otimes V, W)$ and $(U, V \otimes W)$, or between $Hom(U \otimes V, W)$ and $(U, W \otimes V^*)$?

posted 3 months ago

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Julius H.‭

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