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

In introductory texts on category theory, it seems like the majority of examples come from algebraic topology, algebra, and logic.

What are some good examples of Kan extensions, adjunctions, and (co)monads in analysis, Lie theory, and differential geometry?

Since limits and colimits can be characterized as Kan extensions or adjunctions, we have the obvious standard constructions: (co)products, (co)equalizers, etc., but these are common to a lot of categories we work with. Are there any examples more specific to analysis and differential geometry?

• I'll move this to an answer once the question is CW, but an immediate example from analysis can be found in my answer mathoverflow.net/a/186205/763 to another MO question – Yemon Choi Jan 22 '15 at 2:49
• Also my favourite example: the Gelfand representation for commutative unital Banach algebras – Yemon Choi Jan 22 '15 at 2:49
• Universal enveloping algebras of Lie algebras? It's algebra, but at least it's Lie theory. – Allen Knutson Jan 22 '15 at 11:11

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.
• The second example also reminds me of an interesting result due to Schanuel, described here: ncatlab.org/nlab/show/bornological+set#properties – Todd Trimble Jan 23 '15 at 20:33
• Qiaochu, for the HolFC example you might want to add a link to mathoverflow.net/questions/132802 since this seems to be where you show that the holo functions form all the $n$-ary operations you can put on a commutative Banach algebra (unless I have misread) – Yemon Choi Jan 25 '15 at 13:06

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.

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.

The usual functor $L:\text{LieGroups}\to\text{LieAlgebras}$ has a left adjoint $\Gamma$. The unit $A\to L\Gamma A$ is an isomorphism, but the counit $\Gamma L G\to G$ is the universal covering of the base component of $G$.

The functor $\pi_0$ is left adjoint to the inclusion of discrete groups in Lie groups.

The category of tori is equivalent to the category of lattices (ie finitely generated free abelian groups), by any of the equivalent functors $\pi_1$,$H_1$ or $\text{Hom}(S^1,-)$. It is also useful to consider the functor $\text{Hom}(-,S^1)$ on tori, and the functor $\text{Hom}(-,\mathbb{Z})$ on lattices.

You can consider quotients of the category of Lie groups, where morphisms are identified if they are

• conjugate; or
• conjugate by an element in the identity component; or
• connected by a path in the space of homomorphisms.

There are various possible categorical approaches to the theory of root systems and so on, but I do not know a really nice way to make everything work. One should probably start with a category of triples $(G,T,A)$, where $G$ is a compact Lie group, $T$ is a maximal torus, and $A$ is a Weyl chamber.

• I assume that the right-adjoint functor $\Gamma: \mathsf{LieAlg} \to \mathsf{LieGrp}$ is the usual mapping of a Lie algebra to its simply connected Lie group? – ಠ_ಠ Jan 20 '16 at 8:07
• @ಠ_ಠ Well, what you describe gives a an adjoint pair so it must be the one by uniqueness. – Saal Hardali Apr 21 '16 at 19:58

The induced representation from a subgroup $H$ to a group $G$ is the right adjoint functor to the restriction functor of a representation of $G$ to $H$. This is called sometimes Frobenius reciprocity. For finite groups it is literally true. In the Lie group case one should be slightly more careful in defining the class of representations and the induced representation (some continuity should be assumed). This construction is very useful in the Lie theory and is a rich and standard source of representations of a Lie group $G$.

Moreover this construction also appears in differential geometry. Assume that a Lie group $G$ acts transitively on a manifold $X=G/H$ and $E$ is a $G$-equivariant vector bundle. Then the representation of $G$ in the space of continuous sections of $E$ is isomoprhic to the induced representation of the representation of $H$ in the fiber of $E$ over a fixed point $x_0\in X$.

Special cases of the above situation appear e.g. in convex and integral geometry when $G=GL(n,\mathbb{R})$, $X$ is the Grassmannian of linear $k$-subspaces in $\mathbb{R}^n$, and $E$ is some equivariant line bundle over the Grassmannian $X$. For example, the Radon transform between two pairs of Grassmannians can be rewritten as a $GL(n,\mathbb{R})$-equivariant operator between spaces of sections of appropriately chosen line bundles (this was observed by Gelfand, Graev, and Rosu in 1984).

Just saw this post. It turns out that continuity for functions between topological spaces corresponds exactly to adjointness for certain naturally arising functors.

To provide some more detail: Let X and Y be topological spaces, and let f:X->Y be a (not necessarily continuous) function. Let Closed(X) denote the category of closed sets of X, with morphisms given by inclusions, and let Closed(Y) denote the category of closed sets of Y. (Categories of open sets work just as well.) We obtain two functors:

1. A functor F from Closed(X) to Closed(Y), sending each closed subset U of X to the closure of f(U).

2. A functor G from Closed(Y) to Closed(X), sending each closed subset V of Y to the closure of f^{-1}(V).

It turns out that f is a continuous function if and only if (F,G) is an adjoint pair.

The proof is pretty straightforward (I think), and can be found in http://arxiv.org/abs/1408.2596, which recently appeared in the American Mathematical Monthly. (Apologies for plugging my own paper.)

This and similar phenomena appear in various more specialized contexts, but I believe the fully general version has not appeared before.

Best regards, Ed

• That's really interesting! Thanks for sharing. – ಠ_ಠ Apr 10 '15 at 2:13

You'll see more category theory, including adjunctions, relating to differential geometry by departing from standard topics and treatments, e.g., by looking to synthetic differential geometry or higher differential geometry.

Many of the interesting spaces of analyis---spaces of measures, Schwartzian distributions, analytic functionals---whose existence was not immediately obvious historically can be obtained as objects with suitable universal properties which, in the language of category theory, correspond to them arising as adjoints of forgetful functors. The basic example is the free vector space over a set which corresponds to the adjoint of the forgetful functor from the category of vector spaces to that of sets. It can be constructed directly or its existence deduced from the Freyd adjoint functor theorem. There is no analysis present here, of course, but this can easily be added. Thus in the context of locally convex spaces we can supply the free vector space over a completely regular space with the topology induced by the family of those seminorms on it whose restrictions to the original space are continuous. This describes the free locally space over such a topological space as the adjoint to the forgetful functor from the category of locally convex spaces to that of the completely regular spaces. One usually carries the construction one step further by composing with the functor of completion. In the simple case where we start with the unit interval, the resulting space is that of the Radon measures thereon. This basic situation can be varied in a plethora of ways---instead of categories of topological spaces, one can consider that of uniform spaces or less familiar ones (e.g., the compactological spaces of Waelbroeck and Buchwalter) and instead of (complete) locally convex spaces Walbroeck spaces, Saks spaces (Orlicz) or CoSaks spaces. This provides (with hindsight) a unified approach to such topics as uniform measures (Pachl), bounded Radon measures and suggests some useful extensions to further analogous spaces (distributions, analytic functionals,...).

The Schwartz kernel theorem, e.g, that all continuous linear maps from the space $\cal S(\mathbb R^m)$ of Schwartz functions on $\mathbb R^m$ to tempered distributions $\cal S'(\mathbb R^n)$ on $\mathbb R^n$ are given by "kernels" $K(,)$ in $\cal S'(\mathbb R^{m+n})$, is ${\rm Hom}(\cal S\otimes \cal S,\mathbb C)\approx {\rm Hom}(\cal S, \cal S')$, assuming/when the tensor product exists in the category of locally convex topological vector spaces. (Emphatically, the usual "projective" and/or "injective" tensor products are not tensor products in the full categorical sense.) This adjunction is an instance of the Cartan-Eilenberg ${\rm Hom}(A\otimes B,C)\approx {\rm Hom}(A,{\rm Hom}(B,C))$. Existence of the genuine tensor product uses the nuclearity of $\cal S$, that is, the demonstrable fact that $\cal S$ is a countable projective limit of Hilbert spaces with transition maps that are Hilbert-Schmidt. (And "trace-class" might best be characterized as being the composition of two Hilbert-Schmidt operators.)

Examples from Lie theory:

• The universal enveloping algebra functor $\mathcal{U}$ from Lie algebras to unital associative algebras is the left adjoint of the functor which assigns to each unital associative algebra a corresponding Lie algebra with bracket given by the commutator. This adjunction shows that the category of representations of a Lie algebra $\mathfrak{g}$ is in fact isomorphic to the category of left $\mathcal{U}(\mathfrak{g})$-modules.

• What is usually called Frobenius reciprocity in the representation theory of Lie groups and Lie algebras is actually a special case of the extension $\dashv$ restriction $\dashv$ coextension of scalars adjoint triple. This can easily be seen using the group algebra and universal enveloping algebra.