I'm an economist, not a mathematician. I've been trying to make sense of some concepts in functional analysis: dual, bidual, adjoint, natural mapping. The definitions of these notions come out of nowhere and I see no intuition. I thought that I should read category theory to see the big picture.
Well, I read quite a bit of category theory, but I'm unable to see the connection between the categorical notion of adjunction and dual spaces/adjoints in vector spaces. In particular:
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In the context of the category of vector spaces over some field F, what are the two functors and the two transposition assignments that are part of the definition of adjunction in category theory? What are the unit and co-unit?
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Supposedly, adjunctions in category theory allow us to compare an object from one category to an object from another category? But, in vector spaces, both the source and target categories are the category of vector spaces over F (right?). So, why do we need an adjunction?
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According to Wikipedia, adjunctions provide a way to find a universal solution to a diagram (if I'm reading this correctly). In the context of vector spaces, dual spaces/adjoints allow us to find what universal solution to what diagram?
Also, can anyone recommend further reading about this. All the stuff I read on category theory does not say much about adjunctions in vector spaces.
Thanks a lot!
Best Answer
A functor $F : C \to D$ is called left adjoint to a functor $G : D \to C$ if there are natural bijections $\hom(F(X),Y) \cong \hom(X,G(Y))$, where $X \in C, Y \in D$.
There is an equivalent description (which follows directly from the Yoneda Lemma): There are natural transformations $\eta : \mathrm{id}_C \to GF$ (unit) and $\varepsilon : FG \to \mathrm{id}_D$ (counit) which are "inverse" to each other in the sense that the triangle identities are satisfied (see Wikipedia for instance).
Now consider the category $C=D=\mathsf{Vect}$ of vector spaces over a fixed field $K$. Let $V$ be a finite-dimensional vector space. Let $V^*$ be its dual space. Then the functor $V \otimes -$ is left adjoint to the functor $V^* \otimes -$. The counit is induced by the usual evaluation map $V \otimes V^* \to K$, $v \otimes \omega \mapsto \omega(v)$. The unit is induced by the map $K \to V^* \otimes V$ which sends $1 \in K$ to $\sum_i v_i^* \otimes v_i$, where $(v_i)$ is a basis of $V$ and $(v_i^*)$ is its dual basis. By the way, one can show that if $V$ is a vector space, then $V \otimes -$ has a left adjoint if and only if $V$ is finite-dimensional. This provides a category-theoretic characterization of finite-dimensionality and dual vector spaces. (And the story doesn't end here.)
The notions of adjoint operators and adjoint functors coincide for $\mathsf{Hilb}$-enriched categories. See J. Baez, Higher Dimensional Algebra II, arXiv.