What is a dual object? [closed]
Closed as too generic by Peter Taylor on Feb 19, 2024 at 08:25
This post contains multiple questions or has many possible indistinguishable correct answers or requires extraordinary long answers.
This question was closed; new answers can no longer be added. Users with the reopen privilege may vote to reopen this question if it has been improved or closed incorrectly.
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 finitedimensional 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.’

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

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

Why aren’t infinite vector spaces dualizable?

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

What interesting properties do rigid categories have?

What is the definition of the tensor product?

What is an adjunction?

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 $(U, W \otimes V^*)$?
1 comment thread