Examples of Kan extensions, adjunctions, and (co)monads in analysis, Lie theory, and differential geometry?

One way to understand $l^1(X)$ for a set $X$ with counting measure is that $l^1(-): Set \to Ban$ provides a left adjoint to the functor $\hom(k, -): Ban \to Set$. Here $k$ is the ground field and $Ban$ denotes the category of Banach spaces and linear maps $T: X \to Y$ with $\|T\| \leq 1$. Another way of saying this is that $l^1(X)$ is a coproduct of an $X$-indexed collection of copies of the ground field $k$. Similarly, $l^\infty(X)$ is a product of an $X$-indexed collection of copies of $k$.

Tom Leinster has given a neat description of $L^1[0, 1]$ in terms of universal properties: it is initial among Banach spaces $X$ equipped with maps $u: k \to X$ and $\xi: X \oplus X \to X$ such that $\xi(u, u) = u$. Details can be found here.

The following is really an adjunction between $2$-categories but I am going to ignore that subtlety. This blog post discusses everything in more detail and with a few more examples.

Consider on the one hand all concrete categories, by which I mean pairs $(C, U)$ of a category $C$ and a functor $U : C \to \text{Set}$ (the "underlying set") functor, and on the other hand the inclusion of those concrete categories which arise as the categories of models of a Lawvere theory $T$. Here by a Lawvere theory I mean for simplicity a category with finite products and objects $1, x, x^2, \dots$ for a distinguished object $x$, and by the category of models of a Lawvere theory I mean the category of product-preserving functors $T \to \text{Set}$, with the underlying set functor given by evaluation at $x$. Many familiar concrete categories of algebraic objects arise in this way, e.g. groups, rings, modules.

This inclusion has a left adjoint sending a concrete category $(C, U)$ to the "closest approximation" of that concrete category by the category of models of a Lawvere theory, which is the following. The full subcategory of the category of functors $C \to \text{Set}$ on the products $1, U, U^2, \dots$ of $U$ can be thought of as the category of operations on the objects of $C$ (e.g. natural transformations $U^n \to U$ correspond to the $n$-ary operations), and these naturally form a Lawvere theory which one can take the category of models of. Moreover, there is a natural functor of concrete categories from $(C, U)$ to the category of models of the Lawvere theory $T_U$ determined by $U$; this is the unit of the adjunction.

Alright, now for some examples in analysis, Lie theory, and differential geometry.

• Let $(C, U)$ be the concrete category of Banach spaces and weak contractions (maps of norm at most $1$), where $U$ sends a Banach space to its unit ball, and broaden the definition of a Lawvere theory to allow infinite products. Then the category of models of the infinitary Lawvere theory $T_U$ is the category of totally convex spaces.
• Let $(C, U)$ be the concrete category of commutative Banach algebras. Then the Lawvere theory $T_U$ is the Lawvere theory of holomorphic functions: it is equivalent to the category with objects $\mathbb{C}^n$ and morphisms holomorphic functions. This is a converse to the holomorphic functional calculus, and produces the holomorphic functional calculus itself from just the concrete category of commutative Banach algebras; in particular we did not have to know what a holomorphic function was in advance.
• Let $(C, U)$ be the concrete category of representations of a Lie algebra $\mathfrak{g}$. Then the category of models of the Lawvere theory $T_U$ is the category of representations of the universal enveloping algebra $U(\mathfrak{g})$. This is a fancy way of saying that on representations of $\mathfrak{g}$ one has more operations than just acting by elements of $\mathfrak{g}$: one can in fact act by elements of $U(\mathfrak{g})$. But it also says that all operations on representations of $\mathfrak{g}$ arise in this way.
• Let $(C, U)$ be the concrete category $\text{Man}^{op}$, the opposite of the category of finite-dimensional smooth manifolds, with $U$ given by sending a smooth manifold to its ring of smooth functions. Then the Lawvere theory $T_U$ is the Lawvere theory of smooth functions: it is equivalent to the category with objects $\mathbb{R}^n$ and morphisms smooth functions. The category of models of $T_U$ is the category of smooth algebras. The opposite of this category is the starting point of some approaches to synthetic differential geometry.

Let CBA denote the category of commutative, unital Banach algebras, with morphisms being the continuous unit-preserving homomorphisms. Let CHff denote the category of compact Hausdorff spaces, with morphisms being the continuous maps.

There's a contravariant functor $C$ from CHff to CBA (or if you prefer, a functor $C$ from CHff${}^{\rm op}$ to CBA) which assigns to each object $X$ in CHff the Banach algebra $C(X)$. I claim that $C$ has a left adjoint $\Phi$. If you use the "initial object in comma category" formulation, and remember the contravariance, this means that there is a morphism $\eta_A: A\to C(\Phi(A))$ in CBA, such that whenever I have some $X$ in CHff and some morphism $g:A \to C(X)$ in CBA, there exists a unique morphism $\gamma: X\to\Phi_A$ in CHff such that $C(\gamma)\circ \eta_A = g$.

In looser language, $\Phi(A)$ is universal among compact Hausdorff spaces on which you can hope to represent $A$ as an algebra of continuous functions.

In functional analysis one usually gives a concrete definition of $\Phi(A)$ as a particular such space, namely the set of characters on $A$ equipped with the relative weak-star topology; then (this realization of) $\Phi(A)$ is known as the character space or Gelfand spectrum of $A$, and the homomorphism $\eta_A : A \to C(\Phi(A))$ is the Gelfand transform or Gelfand representation of $A$. In the case where $A=\ell^1(G)$ for some (discrete) abelian group $G$, then $\Phi(A)$ can be identified with the Pontryagin dual $\widehat{G}$, and under this identification $\eta_A$ corresponds to the Fourier transform.