Representing functors by objects

We say a functor $F : \mathcal C^{op} \to \operatorname{Set}$ is representable if there exists $X \in \mathcal C$ and a natural isomorphism $h^X \cong F$.

Constructing a functor and then seeking representability is a basic technique in category theory. In general, you can dream up functors more freely than objects. Being the representable object then gives a strong understanding of how $X$ interacts with all objects in $\mathcal C$.

Examples

We will look at some common choices for $F$ and the construction of representing objects in $\operatorname{Set}, \operatorname{Mod-}R,$ and $R\operatorname{-Alg}$ where the last one is the category of commutative $R$-algebras for a commutative ring $R$.

• Let $\ast$ be a one-element set. We can make a functor \(\begin{aligned} FX & := \ast \\ F(f) & := 1_{\ast} \end{aligned}\) On its face this functor seems silly. If $h^X \cong F$, then we know that \(\operatorname{Hom}_{\mathcal C}(Y,X) = \ast\) for any $Y \in \mathcal C$. In other words, for any $Y$ there is a unique map $X \to Y$. We call such $X$ a terminal object of $\mathcal C$ if it exists.

In $\operatorname{Set}$, the terminal object is $\ast$.

In $\operatorname{Mod-}R$, the terminal object is $0$.

In $R\operatorname{-Alg}$, the terminal object is $0$ as a $R$-algebra.

• Assume we have a collection of objects $X_i \in \mathcal C$ indexed by a set $I$. Then, we can set \(\begin{aligned} FX & := \prod_{i \in I} \operatorname{Hom}_{\mathcal C}(X,X_i) \\ F(f) & := \prod_{i \in I} \operatorname{Hom}_{\mathcal C}(f,X_i) \end{aligned}\) Here representing objects are called products and denoted by $\prod_{i \in I} X_i$.

In $\operatorname{Set}$, $\prod_{i \in I} X_i$ is the Cartesian product.

In $\operatorname{Mod-}R$, $\prod_{i \in I} M_i$ is the Cartesian product with component-wise addition and action of $R$.

In $R\operatorname{-Alg}$, $\prod_{i \in I} R_i$ is the Cartesian product with component-wise addition and multiplication and with \(\begin{aligned} R & \to \prod_i R_i \\ r & \mapsto \prod_i r \end{aligned}\)

• Let $I$ be a partially-ordered set. We can view $I$ as a category with \(\operatorname{Ob} I := I\) and \(\operatorname{Hom}_I(i,j) := \begin{cases} \ast & i > j \\ \empty & \text{ otherwise} \end{cases}\) Assume we have a functor $\phi : I \to \mathcal C$. This is the data of objects $X_i$ for $i \in I$ and maps $\phi_{ij} : X_i \to X_j$ if $i > j$ satisfying the condition that \(\phi_{ij} \phi_{jk} = \phi_{ik}\) Often this called a diagram of shape $I$ in $\mathcal C$.

Recall the (inverse) limit of a diagram of sets $X_i$ is \(\operatorname{lim}_I X_i := \left\lbrace x \in \prod_{i \in I} X_i \mid \phi_{ij}(x_i) = x_j \right\rbrace\) Using this, for a general $I$-diagram in $\mathcal C$, we can consider \(\begin{aligned} FX := & \lim \operatorname{Hom}_{\mathcal C}(X,X_i) \\ Ff := & \lim \operatorname{Hom}_{\mathcal C}(f,X_i) \end{aligned}\)

We already saw the construction of limits in $\operatorname{Set}$. In $\operatorname{Mod} R$ and $R\operatorname{-Alg}$, the set-theoretic construction carries a natural structure of an $R$-module or an $R$-algebra.

One particularly common diagram is

The limit over this diagram is called the fiber product or pullback of the diagram.

One might think that we can always reduce to an “underlying set-theoretic” construction for representing objects. This is not always the case. Beyond the fact that functors do not need to be representable, the representing object is not necessarily induced from the construction when we are able to forget down to sets.

The opposite category of commutative rings is equivalent a category of topological spaces with extra structure and the product in this category does not forget to the set-theoretic product.

We can also consider functors $F : \mathcal C \to \operatorname{Set}$ that are represented in the other slot: $h_X \cong F$ for some $X \in \mathcal C$. Each of the functors described above as co version.

• If we have an object (co-)representing the functor $FX = \ast$, then it is called the initial object.

For $\operatorname{Set}$, the initial object is $\empty$.

For $\operatorname{Mod} R$, it is $0$ again.

Finally for $R\operatorname{-Alg}$, the initial object is $R$ itself.

• Given a collection of objects $X_i \in \mathcal C$ for $i \in I$ we have \(FX := \prod_{i \in I} \operatorname{Hom}_{\mathcal C}(X_i,X)\) The object representing this functor is a called the coproduct of the $X_i$. It is often denoted by $\bigsqcup_{i \in I} X_i$.

For $\operatorname{Set}$, it is the disjoint union. For $\operatorname{Mod} R$, it is the direct sum $\bigoplus_i M_i$.

For finite coproducts in $R\operatorname{-Alg}$, the construction is the tensor product $\bigotimes_R R_i$. Let’s take a moment to see why.

For each $R_i$, we have a homomorphism \(\begin{aligned} \ell_i : R_i & \to R_1 \otimes_R \cdots \otimes_R R_s \\ r_i & \to 1 \otimes \cdots \otimes r_i \otimes \cdots \otimes 1 \end{aligned}\) Given $f : \bigotimes_R R_i \to S$, we get $f \circ \ell_i : R_i \to S$.

In the inverse direction, if we have $f_i : R_i \to S$, we can set \(f\left( \sum r_1 \otimes \cdots \otimes r_s \right) := \sum f_1(r_1) \cdots f(r_s)\)

Infinite coproducts in $R\operatorname{-Alg}$ also exist. It is the subring of the infinite tensor product generated by elementary tensors with all but finitely entries equal to $1$.

• Given a poset $I$ and a diagram $I^{op} \to \mathcal C$ we can consider \(FX := \lim \operatorname{Hom}_{\mathcal C}(X_i,X)\) which is represented by the colimit, when it exists. It is denoted by $\operatorname{colim}_I X_i$.

In $\operatorname{Set}$, the colimit is given by the quotient \(\left( \bigsqcup_{i \in I} X_i \right)/\sim\) where $\phi_{ij}(x_j) \sim x_i$.

In $\operatorname{Mod} R$, the colimit takes a similar form \(\bigoplus_i M_i / (\phi_{ij}(m_j)-m_i).\)

Finally, for $R\operatorname{-Alg}$, the analogous formula works.

Taking our diagram from from before and reversing the arrows (plus rotating a bit) ask us to fill in the missing node in the square

The resulting object is fiber coproduct or pushout of the diagram.

In $R\operatorname{-Alg}$, the pushout of

is given by the tensor product $S_1 \otimes_T S_2$.

Yoneda

Referring to “the” limit or “the” initial object is justified by the following theorem.

Lemma. The functor \(h^\bullet: \mathcal C \to \operatorname{Func}(\mathcal C^{op},\operatorname{Set})\) is fully-faithful.

Corollary. If there is a natural isomorphism $h^X \cong h^Y$, then there is an isomorphism $X \cong Y$.

Proof (of Yoneda Lemma). We first look at the space of natural transformation $\nu : h^X \to F$ for a general $F$. Evaluating at $X$ gives a function \(\nu_X : \operatorname{Hom}_{\mathcal C}(X,X) \to F(X)\) We set $\phi_X := \nu_X(1_X)$.

This defines a map \(\phi : \operatorname{Hom}(h^X,F) \to F(X).\)

Given $f : Y \to X$, we want to understand $\nu_Y(f)$. We have a commutative diagram

Tracing $1_X$ through says that \(\nu_Y(f) = F(f)(\phi_X)\) Thus $\nu$ is uniquely determined by $\phi_X$. Moreover, given $\phi_X \in F(X)$ and defining \(\nu_Y(f) := F(f)(\phi_X)\) gives a natural transformation $\nu : h^X \to F$. Thus, $\phi$ is a bijection.

Now, let’s look at the case $F = h^Y$. Then $h^Y(X) = \operatorname{Hom}(X,Y)$ and we see that the only natural transformations $h^X \to h^Y$ are $h^f$ for $f : X \to Y$.

Revisting adjunctions

Say we start with a functor $F: \mathcal C \to \mathcal D$. Then we can make \(h_{F,Y} := \operatorname{Hom}_{\mathcal D}(F(-),Y) : \mathcal C^{op} \to \operatorname{Set}\) Is this representable?

If so, then there is some $G(Y) \in \mathcal D$ with \(\operatorname{Hom}_{\mathcal D}(F(-),Y) \cong \operatorname{Hom}_{\mathcal D}(-,G(Y))\)

If we can represent this functor for any $Y$, we get pretty close to an adjunction.

Lemma. $F$ has a right adjoint if and only if $h_{F,Y}$ is representable for any $Y$. The analogous statement holds for left adjoints.

No proof is given here but you should think about the details.

Finally, let’s note that adjoints necessarily commute with some universal construction. For example, let’s assume we have a diagram $I^{op} \to \mathcal C$ and its limit $\lim X_i$. Then, we can use adjunction and the definition of representability to produce a natural isomorphism \(\begin{aligned} \lim \operatorname{Hom}(X,G(X_i)) & \cong \lim \operatorname{Hom}(F(X),X_i) \\ & \cong \operatorname{Hom}(F(X),\lim X_i) \\ & \cong \operatorname{Hom}(X,G(\lim X_i)) \end{aligned}\) Thus, $\lim G(X_i) \to G(\lim X_i)$ is an isomorphism.

Similarly, left adjoints automatically commute with formation of colimits.

In fact, this observation can be taken further. Under appropriate and reasonable conditions, all your need for a right/left adjoint is commutation colimits/limits. For examples, see these notes for a proof of one such version.