Examples of adjoint 2-functors

Question

I hope this is not too vague, but I'm learning about 2-dimensional categories and I would like more examples of adjoint 2-functors to study. Could somebody tell me some interesting 2-adjunctions ? They don't have to involve strict 2-categories, but I prefer the simpler strict stuff.

Mostly I'm looking for interesting examples to study. References or detailed definitions would also be most welcome so I can follow along.

Answer 1

In 1-dimensional category theory, left adjoints can be created by freely adding stuff to a type of object. For example, the free group freely adds a group structure on a set. The same is true in 2-dimensional category theory, but the only difference is that here the hom-sets are actually categories and the bijections $$\hom(F(X),Y) \longrightarrow \hom(X,G(Y)) \quad (\ast)$$ will be replaced by equivalences of categories. (Some authors prefer to call this a bicategorical adjunction then.) Many examples exist for $2$-categories of structured categories (categories, monoidal categories, cocomplete categories, etc.).

For example, let's freely add finite coproducts to an arbitrary category $\mathcal{C}$. Objects of $F(\mathcal{C})$ are tuples $(X_1,\dotsc,X_n)$ of objects in $\mathcal{C}$. You should see this as a "formal" coproduct $X_1 \sqcup \dotsc \sqcup X_n$. A morphism $(X_1,\dotsc,X_n) \to (Y_1,\dotsc,Y_m)$ consists of a map $f : \{1,\dotsc,n\} \to \{1,\dotsc,m\}$ together with morphisms $g_i : X_i \to Y_{f(i)}$. It is easy to write down the composition, and the identity as well. So we get a category $F(\mathcal{C})$, and there is an evident functor $\iota : \mathcal{C} \to F(\mathcal{C})$, $X \mapsto (X)$. Notice that $F(\mathcal{C})$ has finite coproducts, the initial object is $(~)$ (empty tuple) and the binary coproducts are $(X_1,\dotsc,X_n) \sqcup (X'_1,\dotsc,X'_k) = (X_1,\dotsc,X_n,X'_1,\dotsc,X'_k)$, as the universal property can easily be verified. Now, if $\mathcal{D}$ is any category with finite coproducts, the functor $$\hom_{\sqcup}(F(\mathcal{C}),\mathcal{D}) \to \hom(\mathcal{C},\mathcal{D}), \quad G \mapsto G \circ \iota$$ is an equivalence of categories; here $\sqcup$ means the functors preserving finite coproducts. The pseudo-inverse maps a functor $H : \mathcal{C} \to \mathcal{D}$ to the functor $G : F(\mathcal{C}) \to \mathcal{D}$ defined by $G((X_1,\dotsc,X_n)) = H(X_1) \sqcup \cdots \sqcup H(X_n)$ on objects and $G(f,g) \circ \iota_i = \iota_{f(i)} \circ H(g_i)$ on morphisms. There won't be an isomorphism between $\hom_{\sqcup}(F(\mathcal{C}),\mathcal{D})$ and $\hom(\mathcal{C},\mathcal{D})$, since there is no canonical way of choosing the coproducts.

The same process also works for other types of categories. For example, we can form the free monoidal category on a category, also the free symmetric monoidal category on a category. We have the free symmetric monoidal category on a monoidal category (MO/117533). We can form the free cocompletion (under small colimits) of a category. Much more generally, if $T$ is a $2$-dimensional monad, we can create free $T$-algebras.

Here is another, quite simple example. Let $k$ be a commutative ring. If $\mathcal{C}$ is any category, we can form a $k$-linear category $k[\mathcal{C}]$ with the same objects of $\mathcal{C}$, but with morphisms $\hom_{k[\mathcal{C}]}(X,Y) = $ free $k$-vector space on $\hom_{\mathcal{C}}(X,Y)$. This construction generalizes the monoid algebra, which is the special case that $\mathcal{C}$ has exactly one object. If $\mathcal{D}$ is any $k$-linear category, any functor $\mathcal{C} \to \mathcal{D}$ uniquely extends to a $k$-linear functor $k[\mathcal{C}] \to \mathcal{D}$. This means that $k[\mathcal{C}]$ is indeed the free $k$-linear category on $\mathcal{C}$, and the equivalences $(\ast)$ are actually isomorphisms here.

In my thesis I have studied several adjunctions in the context of the $2$-category $\mathbf{Cat}_{c\otimes/k}$ of cocomplete symmetric monoidal $k$-linear categories together with cocontinuous symmetric monoidal $k$-linear functors. The simplest example is the following: If $R$ is a commutative $k$-algebra, then $\mathbf{Mod}_R$ is an object of $\mathbf{Cat}_{c\otimes/k}$. Conversely, if $\mathcal{C} \in \mathbf{Cat}_{c\otimes/k}$, then $\mathrm{End}(\mathbf{1}_{\mathcal{C}})$ is a commutative $k$-algebra. And in fact, we have a $2$-categorical adjunction $$\hom_{c\otimes/k}(\mathbf{Mod}_R,\mathcal{C}) \xrightarrow{\sim} \hom_{k}(R,\mathrm{End}(\mathbf{1}_{\mathcal{C}})).$$

If you want to learn more about $2$-categories, I can recommend the freely (sic!) available book 2-Dimensional Categories by Johnson-Yau.

Answer 2

Since I know that you are interested in categorical logic, let me mention one important (class) of examples.

Let $K$ be the 2-category whose 0-cells are cartesian closed categories, whose 1-cells are functors which preserve the structure up to invertible coherence cells and whose 2-cells are invertible transformation. Let $U: K\to Cat_{\cong}$ be the forgetful 2-functor. Then $U$ has a left pseudo-adjoint $F$ such that \begin{align*} K(FC,D) \simeq Cat_{\cong}(C,UD) \end{align*} pseudo-naturally at least. The category $FC$ can be explicitely defined via the simply-typed $\lambda$-calculus. It is the syntactic category created from the simply typed $\lambda$-calculus over the "signature" $C$. Its objects are lists of variable declarations and its morphisms are lists of well-typed terms. The fact that the equation \begin{align*} K(FC,D) \simeq Cat_{\cong}(C,UD) \end{align*} determines $FC$ only up to ("2-unique") equivalence and not up to unique isomorphisms means that you have some freedom in the definition of "simply typed $\lambda$-calculus". It will for example not matter which variable-numbering convention you choose or which type constructors such as $A,B\mapsto A\times B$ you include (of course you have to include the constructor $A,B\mapsto (A\to B)$ in one way or other).

A similar example is: Let $K$ be the 2-category whose 0-cells are locally cartesian closed categories, 1-cells functors which preserve the structure weakly and 2-cells invertible transformations. Let $U:K\to Cat_{\cong}$ again be the forgetful 2-functor. Then the left pseudo-adjoint $F:Cat_{\cong}\to K$ sends a category $C$ to the syntactic category $FC$ of Martin-Löf's dependent type theory over the "signature $C$".

Reference: John Power, 2-categories (lecture notes) and other texts by John Power.

Remark: It is possible to turn $F$ into a strict 2-functor, but the 2-adjunction $F\dashv U$ can never a strict one. If it were strict then $F(\varnothing)$ would be a strict 2-initial object in $K$, but $K$ does not have a strict 2-initial object (see John Power's text for details).