What are some beautiful examples of adjunctions?

Question

Lately I've been very interested in finding examples of adjunctions. In particular, examples that are elementary enough for an undergrad like me to grasp. So I was wondering if perhaps you could share some of your favourite examples of adjunctions and maybe little hints and suggestions on how to explore them and their implications.

Although I'd be very interested in examples in real analysis, number theory, and topology (something that motivates the study of locales would be very nice), I don't wish to be too restrictive, so feel free. And don't pay much attention to "for an undergrad to grasp", since being unable to understand something is a temporary thing, I hope...

Answer 1

Here are some very basic examples, but which you don't find so often mentioned.

1. For real numbers $r$ and integers $z$ we have $$\iota(z) \leq r \Leftrightarrow z \leq \lfloor r \rfloor,$$ where $\iota$ is the inclusion map from integers to real numbers and $\lfloor - \rfloor$ is the floor function. That means that $\iota : (\mathbb{Z},\leq) \to (\mathbb{R},\leq)$ is left adjoint to $\lfloor - \rfloor : (\mathbb{R},\leq) \to (\mathbb{Z},\leq)$, where we regard preorders (and monotonic maps) as categories (and functors). Similarly, $\iota$ is right adjoint to the ceiling function $\lceil - \rceil: (\mathbb{R},\leq) \to (\mathbb{Z},\leq)$. (In the context of preorders, adjunctions are usually called Galois connections.)

2. Let $\mathsf{Ban}_1$ denote the category of Banach spaces with short linear maps. The forgetful functor $\mathsf{Ban}_1 \to \mathsf{Set}$ which maps a Banach space to its unit ball has a left adjoint $\ell^1 : \mathsf{Set} \to \mathsf{Ban}_1$ which maps a set $X$ to the Banach space $\ell^1(X)$ of summable functions on $X$.

3. Let's say a group $G$ is of exponent $n$ if $g^n=1$ for all $g \in G$. These groups constitute a full subcategory $\mathsf{Grp}_n \hookrightarrow \mathsf{Grp}$. The inclusion functor has a left adjoint: It maps a group $G$ to the quotient group $G/N$, where $N$ is the normal subgroup which is generated (as a normal subgroup) by all $g^n$, $g \in G$. This construction is connected to the famous Burnside problem.

4. For a map of sets $f : X \to Y$ we have $$f_*(A) \subseteq B \Leftrightarrow A \subseteq f^{-1}(B)$$ for subsets $A \subseteq X$ and $B \subseteq Y$. This means that $f_* : (\mathcal{P}(X),\subseteq) \to (\mathcal{P}(Y),\subseteq)$ is left adjoint to $f^{-1}: (\mathcal{P}(Y),\subseteq) \to (\mathcal{P}(X),\subseteq)$. Notice that this implies $f_*(\bigcup_i A_i) = \bigcup_i f_*(A_i)$, since left adjoints are cocontinuous (but not $f_*(\bigcap_i A_i) = \bigcap_i f_*(A_i)$, which only holds when $f$ is injective).

5. Let $\mathsf{Mor}(\mathsf{Grp})$ be the usual morphism category of the category of groups and let $\mathsf{Mono}(\mathsf{Grp})$ be the full subcategory consisting of monomorphisms of groups, i.e. injective homomorphisms. Then, the inclusion functor has a left adjoint, mapping a homomorphism $f : G \to H$ to the induced homomorphism $\overline{f}: G/\ker(f) \to H$.

6. The inclusion functor $\mathsf{Haus} \to \mathsf{Top}$ from Hausdorff spaces to topological spaces has a left adjoint, the "maximal Hausdorff quotient". Explicitly, this is $X \mapsto X/{\sim}$, where $\sim$ is the intersection of all equivalence relations $R$ such that $X/R$ is Hausdorff.

7. The inclusion functor from normed spaces to seminormed spaces has a left adjoint. It maps a seminormed space $(V,\lVert - \rVert)$ to $(V/K,\lVert - \rVert)$, where $K := \{v \in V : \lVert v \rVert = 0\}$ and $\lVert \overline{v} \rVert := \lVert v \rVert$. Exactly this construction is used in the definition of $L^p$-spaces in measure theory.

Answer 2

I'm no expert in category theory, but here are some easy examples I can think of:

1. Categories with closed monoidal structure are really great, because they give you a nicely behaved notion of a "mapping object". A closed monoidal category $\mathsf{C}$ is one where the functor $(-)\otimes X$ has a right adjoint $[X, (-)]$ called the internal hom. This essentially allows you to curry arrows out of $Y \otimes X$. For example, in the category of abelian groups, or more generally R-modules for a commutative ring R, the hom-sets have an obvious R-module structure, and this gives us an internal hom associated to the tensor product. If the monoidal product is the usual categorical product (i.e. the limit) then you get what we call a Cartesian closed structure. The category of sets is Cartesian closed, for example, and so is the category of compactly generated spaces, and the category of diffeological spaces.

2. Using adjoints, we can make precise the notion of a free functor in algebra, which you have probably already encountered in the form of free groups, free R-modules, and so on. A free functor is just one that is left-adjoint to a forgetful functor.

3. The category of topological spaces has a forgetful functor to the category of sets, which has both a left adjoint (discrete functor) and a right adjoint (indiscrete functor). This means that the forgetful functor preserves both limits and colimits, which is why limits and colimits in topology are pretty easy to compute.

Answer 3

Here's a fun example. Suppose $ H $ a subgroup of $ G $ and let $ \iota $ denote the inclusion $ H \hookrightarrow G $. Let $ k $ be a field and let $ \operatorname{Vect}_{k} $ denote the category of $ k $-vector spaces. Regarding $ G $ as a category, objects of the functor category $ \operatorname{Vect}_{k}^{G} $ are $ G $-respresentations over $ k $ and morphisms are $ G $-equivariant maps.

Pre-composition with $ \iota $ defines a restriction functor $ \operatorname{res} \colon \operatorname{Vect}_{k}^{G} \to \operatorname{Vect}_{k}^{H} $. This functor has both left and right adjoints — the left adjoint is the left Kan extension along $ \iota $ (regarded as a functor from $ H $ to $ G $ regarded as categories) and the right adjoint is the right Kan extension along $ \iota $. These are the induction functor $ \operatorname{ind}_{H}^{G} $ which sends a $ H $-representation to the induced $ G $-representation, and the (perhaps less well-known) coinduction functor $ \operatorname{coind}_{H}^{G} $.

The same thing happens for $ G $-sets, $ G $-spaces, based versions of these, and $ G $-objects in any category when both adjoints exist.

Answer 4

Here is a logic example, as a special case of Galois connections, from Lambek's "The Influence of Heraclitus on Modern Mathematics" (Scientific Philosophy Today, pp. 111-121). Lambek reports that it was first studied by Lawvere:

Let $\mathcal{P}$ be the category with formulas of propositional calculus as objects and "entailments" $\vdash$ as arrows (so that $\mathcal{P}$ is a preorder). Define $\forall p\in\operatorname{Obj}(\mathcal{P})$ endofunctors $F_p,G_p:\mathcal{P}\to\mathcal{P}$ by $F_p:q\mapsto (p \wedge q)$ and $G_p:q\mapsto (p\Rightarrow q)$ (the arrow functions are the expected ones). Then $F_p$ is a left adjoint to $G_p$, and the counit of the adjunction is the good old "modus ponens" (naturality in $p$ can be established by the so-called "parameter theorem").