Connection between categorical notion of adjunction and dual space/adjoint in vector spaces

Question

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:

• 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?

• 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?

• 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!

Answer 1

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.

Answer 2

Too long for comment but probably addresses the background of the problem for OP.

Approaching duality in functional analysis from the category theoretic point of view is a terrible idea in the beginning. When a functional analyst speaks of, say, the adjoint operator, he's not really talking about the adjoint functor between some categories but rather the simple pull-back map.

A casual sample of pretty and concrete duality results which can, and should, be viewed without the categorical perspective:

1. The Riesz representation theorem for Hilbert spaces.

2. The Riesz representation theorem for positive linear functionals on $C_0(X)$ where $X$ is LCH. In view of this, the von Neumann-Morgenstern expected utility is a pairing between the dual and and double dual.

3. The Hanh-Banach theorem for Banach spaces. (A corollary is the hyperplane separation theorem. Economists quote this all the time, usually unnecessarily for Hilbert spaces. )

4. The Banach-Alaoglu theorem. This says the unit ball in the dual is weak-$^*$ compact. A consequence of this is that a Markov chain, where the state space is not necessarily finite, has a stationary distribution---an infinite dimensional Perron-Frobenius theorem, if you like. In some macroeconomic models, this means the economy has a steady state.

5. The Krein-Milman theorem. (Not about the dual a priori but applications usually make use of weak-* compactness given by Banach-Alaoglu.)

6. The Choquet representation theorem: value of an affine map $\phi$ at a point $x$ in a convex set $C$ is given by an integral $\int_{\delta C} \phi dp_x$ for some probability measure $p_x$ on the boundary $\delta C$ of $C$.

7. Not quite a theorem but a particular identification: $(l^1)^* = l^{\infty}$ and $(l^{\infty})^* = c_0$. This is an example of non-reflexivity for infinite-dimensional spaces. General welfare theorems usually take $l^{\infty}$ as space of prices and $l^1$ the space of bundles.

8. The Gelfand-Naimark theorem for abelian $C^*$-algebras. After linearization, the Pontryagin dual becomes part of the unit sphere in the TVS dual. The resulting dual map generalizes the Fourier transform.

Added Reference

The Handbook of Mathematical Economics series, edited by Arrow. In particular volume 4 is functional analysis-centric. For instance, Chap 34 Equilibrium theory in infinite dimensional spaces uses duality as a functional analyst would: weak-topology, weak-$^*$ topology, Mackey topology, Riesz spaces, etc.

Answer 3

The duality adjunction you are referring to, is explained in some detail in MacLane's CWM 2nd ed. page 88. It is an adjunction between $Vect$ and $Vect^{op}$.

The statement:"adjunctions in category theory allow us to compare an object from one category to an object from another category" should be taken as a very general indication.

I'd rather say:"adjunctions in category theory allow us to relate an object from one category to an object from another category".

Also I like to think that an adjunction is a relation between 2 functors. The 2 functors are sort of "pseudo-inverses" of each other. What you do with one functor, you can more or less undo with the other.