Representable Functors

This post was machine-translated from the Korean original by Marvin (via Kimi). It may contain errors or awkward phrasing — the Korean original is the source of truth.

Yoneda lemma

For any object \(A\) in a (locally small) category \(\mathcal{A}\), we define two functors

\[\Hom_\mathcal{A}(A,-):\mathcal{A}\rightarrow\Set,\qquad \Hom_\mathcal{A}(-,A):\mathcal{A}\rightarrow\Set\]

the first covariant and the second contravariant. (§Functors, ⁋Example 4)

Definition 1 Let a category \(\mathcal{A}\) be given.

1. A covariant functor \(F:\mathcal{A}\rightarrow\Set\) is called a representable functor if there exists an object \(A\in\obj(\mathcal{A})\) such that \(F\) and \(\Hom_\mathcal{A}(A,-)\) are naturally isomorphic.
2. A contravariant functor \(F:\mathcal{A}\rightarrow\Set\) is called a representable functor if there exists an object \(A\in\obj(\mathcal{A})\) such that \(F\) and \(\Hom_\mathcal{A}(-,A)\) are naturally isomorphic.

For any functor \(F\), we call the choice of an object \(A\in\obj(\mathcal{A})\) satisfying the above condition together with a natural isomorphism a representation of \(F\).

Example 2 For instance, \(\id_\Set:\Set \rightarrow \Set\) is representable. This is because for any singleton \(\ast\), the natural isomorphism

\[\id_\Set\cong\Hom_\Set(\ast,-)\]

holds. For any set \(A\), the bijection

\[\id_\Set(A)=A\rightarrow\Hom_\Set(\ast,A)\]

sends each element \(a\in A\) to the function \(a:\ast\rightarrow A\) whose image is \(\{a\}\); conversely, the image of any function \(\ast\rightarrow A\) yields an element of \(A\). The naturality of this correspondence follows because for any function \(f:A \rightarrow B\) and any \(a\in A\), writing \(b=f(a)\), the map \(\id_\Set(B)\rightarrow\Hom_\Set(\ast,B)\) sends this to the function \(b:\ast \rightarrow B\) with image \(\{b\}\), which is precisely the composite \(\ast\overset{a}{\longrightarrow}A\overset{f}{\longrightarrow}B\).

The most important theorem in this context is the Yoneda lemma.

Theorem 3 (Yoneda) For any functor \(F:\mathcal{A}\rightarrow\Set\) and any \(A\in\obj(\mathcal{A})\), there exists a bijection of sets

\[\Phi:\{\text{natural transformations from $\Hom_\mathcal{A}(A,-)$ to $F$}\}\rightarrow F(A);\qquad \alpha\mapsto \alpha_A(\id_A)\]

Proof

First, let us briefly examine how the above function operates. A natural transformation from \(\Hom_\mathcal{A}(A,-)\) to \(F\) assigns, to each object \(X\), a function \(\alpha_X:\Hom_\mathcal{A}(A,X)\rightarrow F(X)\). In particular, for \(X=A\), the function \(\alpha_A\) maps \(\Hom_\mathcal{A}(A,A)\) to \(F(A)\), and since \(\id_A\in\Hom_\mathcal{A}(A,A)\), we have \(\alpha_A(\id_A)\in F(A)\).

To show that this function is a bijection, it suffices to construct an inverse. That is, from an arbitrary element \(x\in F(A)\) we must produce a natural transformation \(\Psi(x)\), which is given, for each object \(X\) in \(\mathcal{A}\), by a function \(\Psi(x)_X:\Hom_\mathcal{A}(A,X)\rightarrow F(X)\). Now, if \(\Psi(x)\) is a natural transformation, the following diagram must commute.

Consider again \(\id_A\in\Hom_\mathcal{A}(A,A)\). Traversing the upper-right path yields \(F(f)(\Psi(x)_A(\id_A))\), while the lower-left path gives \(\Psi(x)_X(f)\). Hence

\[\Psi(x)_X(f)=F(f)(\Psi(x)_A(\id_A))\]

must hold. On the other hand, for \(\Psi\) to be the inverse of \(\Phi\) we require \((\Psi\circ\Phi)(x)=x\), so from the definition of \(\Psi\) we see that \(\Psi(x)_A(\id_A)\) must equal \(x\) exactly. Thus we must define \(\Psi(x)\) by the formula

\[\Psi(x)_X(f)=F(f)(x)\]

We must additionally verify that \(\Psi\) so defined is indeed a natural transformation, but this is straightforward.

Moreover, regarding both sides as functors from \(\mathcal{A}\times\Set^\mathcal{A}\) to \(\Set\), this bijection is natural in each component of \(\mathcal{A}\) and \(\Set^\mathcal{A}\). We will not use this fact immediately, so we mention it only in passing; its proof is likewise not difficult, much as above. Also, by duality there is a Yoneda lemma for contravariant functors.

Theorem 4 (Yoneda) For any contravariant functor \(F:\mathcal{A}\rightarrow\Set\) and any \(A\in\obj(\mathcal{A})\), there exists a bijection of sets

\[\Phi:\{\text{natural transformations from $\Hom_\mathcal{A}(-,A)$ to $F$}\}\rightarrow F(A);\qquad \alpha\mapsto \alpha_A(\id_A)\]

For ease of exposition, in the remainder of this post we treat only the covariant case; however, the same statements hold for contravariant functors in an obvious manner.

Universal property

Looking at Definition 1, we agreed to call the choice of an object \(A\) together with a natural isomorphism \(F\cong\Hom_\mathcal{A}(A,-)\) a representation. Yet by Theorem 3 (Yoneda), choosing a natural isomorphism is equivalent to selecting a suitable element of \(F(A)\). We formalize this as follows.

Definition 5 Let a representable functor \(F:\mathcal{A}\rightarrow\Set\) be given. For a natural isomorphism \(\Hom_\mathcal{A}(-,A)\cong F\), we call an element \(x\in F(A)\) a universal element, and we call the pair \((A,x)\) a universal property.

The following example makes this more intuitive.

Example 6 Fixing two \(k\)-vector spaces \(V,W\), define the functor \(\operatorname{Bilin}(V,W;-)\) from the category \(\Vect_k\) to \(\Set\) by

\[\operatorname{Bilin}(V,W;U)=\{\text{bilinear maps from $V\times W$ to $U$}\}\]

It is well known that this functor is representable. That is, there exists a suitable \(k\)-vector space \(V\otimes W\) for which the natural isomorphism

\[\Hom_{\Vect_k}(V\otimes W,-)\cong\operatorname{Bilin}(V,W;-)\]

holds. In this case, by the Yoneda lemma the natural isomorphism is determined by an element of \(\operatorname{Bilin}(V,W;V\otimes W)\), i.e., by a bilinear map \(V\times W\rightarrow V\otimes W\).

In other words, the universal property of the tensor product consists of the object \(V\otimes W\) together with the universal element \(V\times W\rightarrow V\otimes W\), and what the above natural isomorphism asserts is precisely that whenever a bilinear map \(V\times W\rightarrow U\) is given (right-hand side), there is a unique \(k\)-linear map \(V\otimes W\rightarrow U\) (left-hand side).

Through this example we can see that objects defined via universal properties in various fields are indeed of this form. However, from a purely category-theoretic standpoint, so far the only justification for calling these universal properties is that we chose the name in Definition 5.
To justify the terminology, let us call an object \(I\) of a category \(\mathcal{A}\) an initial object if for every object \(A\) there exists a unique morphism \(I\rightarrow A\). Dually, we define a terminal object. Then Proposition 8 provides an appropriate answer: all such objects can be regarded as initial (or terminal) objects in suitable categories. To explain this, we need the following definition.

Definition 7 The category of elements of a functor \(F: \mathcal{A}\rightarrow \Set\) is the category \(\int F\) defined by the following data.

• The objects of \(\int F\) are pairs \((A,x)\) with \(A\in \mathcal{A}\) and \(x\in F(A)\).
• A morphism \((A_1,x_1) \rightarrow (A_2, x_2)\) in \(\int F\) is a morphism \(f\) in \(\mathcal{A}\) satisfying \(F(f)(x_1)=x_2\).

For example, the category of elements of \(\Hom_{\mathcal{A}}(A,-):\mathcal{A}\rightarrow\Set\) is given by the following data.

• The objects of \(\int \Hom_\mathcal{A}(A,-)\) are pairs \((X,\pi)\) with \(X\in \mathcal{A}\) and \(\pi\in \Hom_\mathcal{A}(A,X)\).
• A morphism \(f:(X_1,\pi_1)\rightarrow(X_2,\pi_2)\) in \(\int \Hom_\mathcal{A}(A,-)\) is a morphism in \(\mathcal{A}\) satisfying \(\pi_2=\Hom_\mathcal{A}(A,f)(\pi_1)=f\circ\pi_1\).

That is, \(\int\Hom_\mathcal{A}(A,-)\) is the under category \({}_{A/}\mathcal{A}\).

We are now ready to prove the following proposition.

Proposition 8 A functor \(F:\mathcal{A}\rightarrow\Set\) is representable if and only if \(\int F\) has an initial object.

Proof

If \(F\) is representable, there exist a suitable object \(A\) and a natural isomorphism \(\alpha\) such that \(F\cong\Hom_\mathcal{A}(A,-)\). From this we obtain an isomorphism \((X,x)\mapsto (X,\alpha_X(x))\) from \(\int F\) to \(\int\Hom_\mathcal{A}(A,-)\). But \(\int\Hom_\mathcal{A}(A,-)={}_{A/}\mathcal{A}\) has the initial object \(\id_A\).

Now suppose \(\int F\) has an initial object \((A,x)\); we must construct from this a natural isomorphism \(\Hom_\mathcal{A}(A,-)\Rightarrow F\). First, from Theorem 3 (Yoneda) we know that the bijection

\[\Phi:\{\text{natural transformations from $\Hom_\mathcal{A}(A,-)$ to $F$}\}\rightarrow F(A)\]

exists, and in proving that it is a bijection we defined, for each \(x\in F(A)\), a natural transformation \(\Psi(x):\Hom_\mathcal{A}(A,-)\Rightarrow F\) by

\[\Psi(x)_X(f)=F(f)(x)\]

Meanwhile, in \(\int F\), the condition that \((A,x)\) is initial means that for any \((X,y)\in\int F\) there exists a unique morphism \(f:A \rightarrow X\) in \(\mathcal{A}\) with \(F(f)(x)=y\in F(X)\). By the formula above, \(F(f)(x)=\Psi(x)_X(f)\); and since \(y\) can be chosen arbitrarily in \(F(X)\) once \(X\) is fixed, this says that for any given \(y\in F(X)\) there is always a unique \(f\in\Hom_\mathcal{A}(A,X)\) satisfying \(y=\Psi(x)_X(f)\). Hence \(\Psi(x)_X\) is an isomorphism for every \(X\), and since \(X\) is arbitrary, \(\Psi(x)\) defines a natural isomorphism from \(\Hom_\mathcal{A}(A,-)\) to \(F\).

Since the initial object of any category is uniquely determined up to unique isomorphism, the universal property is also uniquely determined up to unique isomorphism.

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References

[Rie] Emily Riehl. Category Theory in Context. Dover Publications, 2016.

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