Universal Properties, Representables and the Yoneda Lemma

Let $\mathcal{C}$ be a locally small category, so every hom-collection $\mathcal{C}(a,b)$ is a set (Category, object, morphism, domain, codomain, identity, composition, and hom-collection, Small, locally small, and large categories). Since sets and functions form $\mathbf{S}\mathbf{e}\mathbf{t}$ (Sets and functions form the large locally small category $\mathbf{S}\mathbf{e}\mathbf{t}$), the following assignments take values in $\mathbf{S}\mathbf{e}\mathbf{t}$.

For an object $a$, the covariant hom-assignment $\mathcal{C}(a,-)$ sends an object $b$ to $\mathcal{C}(a,b)$ and a morphism $u:b\to c$ to the function

$u_{\ast }:\mathcal{C}(a,b)⟶\mathcal{C}(a,c),f⟼u\circ f\mathrm{.}$

The contravariant hom-assignment $\mathcal{C}(-,a)$ sends $b$ to $\mathcal{C}(b,a)$ and $u:b\to c$ to

$u^{\ast }:\mathcal{C}(c,a)⟶\mathcal{C}(b,a),g⟼g\circ u\mathrm{.}$

Equivalently, the latter is an assignment on ${\mathcal{C}}^{\mathrm{o}\mathrm{p}}$ (Opposite category ${\mathcal{C}}^{\mathrm{o}\mathrm{p}}$). Their functor laws are proved in The assignments $\mathcal{C}(a,-)$ and $\mathcal{C}(-,a)$ are functors to $\mathbf{S}\mathbf{e}\mathbf{t}$ ↗.

The two variables combine into the hom-assignment

$\mathcal{C}(-,-):{\mathcal{C}}^{\mathrm{o}\mathrm{p}}\times \mathcal{C}⟶\mathbf{S}\mathbf{e}\mathbf{t}\mathrm{.}$

It sends $(a,b)$ to $\mathcal{C}(a,b)$. A morphism $(a,b)\to (a^{\mathrm{\prime }},b^{\mathrm{\prime }})$ in the product category consists of $h:a^{\mathrm{\prime }}\to a$ and $u:b\to b^{\mathrm{\prime }}$ in $\mathcal{C}$ (Product category and its projection functors), and its action is

$\mathcal{C}(h,u):\mathcal{C}(a,b)⟶\mathcal{C}(a^{\mathrm{\prime }},b^{\mathrm{\prime }}),f⟼u\circ f\circ h\mathrm{.}$

That this assignment is a functor, and hence a bifunctor, is proved in The hom-assignment $\mathcal{C}(-,-):{\mathcal{C}}^{\mathrm{o}\mathrm{p}}\times \mathcal{C}\to \mathbf{S}\mathbf{e}\mathbf{t}$ is a bifunctor ↗.

Let $\mathcal{C}$ be a locally small category and let $a$ be an object. The covariant hom-assignment $\mathcal{C}(a,-):\mathcal{C}\to \mathbf{S}\mathbf{e}\mathbf{t}$ and the contravariant hom-assignment $\mathcal{C}(-,a):{\mathcal{C}}^{\mathrm{o}\mathrm{p}}\to \mathbf{S}\mathbf{e}\mathbf{t}$ of The covariant and contravariant hom-assignments and the hom-bifunctor of a locally small category are functors.

For every locally small category $\mathcal{C}$, the hom-assignment

$\mathcal{C}(-,-):{\mathcal{C}}^{\mathrm{o}\mathrm{p}}\times \mathcal{C}⟶\mathbf{S}\mathbf{e}\mathbf{t}$

of The covariant and contravariant hom-assignments and the hom-bifunctor of a locally small category is a functor. Its restrictions in the two variables are the contravariant and covariant hom-functors of The assignments $\mathcal{C}(a,-)$ and $\mathcal{C}(-,a)$ are functors to $\mathbf{S}\mathbf{e}\mathbf{t}$.

Let $A$ and $S$ be objects of a category $\mathcal{C}$ (Category, object, morphism, domain, codomain, identity, composition, and hom-collection). A generalized element of $A$ of shape $S$ is a morphism $x:S\to A$. The domain $S$ is its shape.

For a set $A$, ordinary elements are exactly generalized elements of singleton shape: a function $1\to A$ is determined by the image of the unique point, and every element of $A$ determines such a function. Other shapes can record more structure than singleton-shaped elements do.

Let $\mathcal{C}$ be a locally small category. A presheaf on $\mathcal{C}$ is a functor $P:{\mathcal{C}}^{\mathrm{o}\mathrm{p}}\to \mathbf{S}\mathbf{e}\mathbf{t}$, with opposite category as in Opposite category ${\mathcal{C}}^{\mathrm{o}\mathrm{p}}$ and $\mathbf{S}\mathbf{e}\mathbf{t}$ as in Sets and functions form the large locally small category $\mathbf{S}\mathbf{e}\mathbf{t}$.

A covariant functor $F:\mathcal{C}\to \mathbf{S}\mathbf{e}\mathbf{t}$ is covariantly representable when there are an object $R$ and a natural isomorphism

$\theta :\mathcal{C}(R,-)\stackrel{\cong }{\Rightarrow }F\mathrm{.}$

The pair $(R,\theta )$ is a representation of $F$, and $R$ is a representing object.

A presheaf $P:{\mathcal{C}}^{\mathrm{o}\mathrm{p}}\to \mathbf{S}\mathbf{e}\mathbf{t}$ is contravariantly representable when there are an object $R$ and a natural isomorphism

$\theta :\mathcal{C}(-,R)\stackrel{\cong }{\Rightarrow }P\mathrm{.}$

The same terms are used for $(R,\theta )$ and $R$. The hom-functors exist by The assignments $\mathcal{C}(a,-)$ and $\mathcal{C}(-,a)$ are functors to $\mathbf{S}\mathbf{e}\mathbf{t}$, and natural transformations and natural isomorphisms have the meanings of Natural transformation and its components and Natural isomorphism. When the variance is clear, representable is used without an adjective.

Let $\mathcal{C}$ be a locally small category and let $1:\mathcal{C}\to \mathbf{S}\mathbf{e}\mathbf{t}$ and $1:{\mathcal{C}}^{\mathrm{o}\mathrm{p}}\to \mathbf{S}\mathbf{e}\mathbf{t}$ denote the constant functors at a singleton set.

An object $I$ is initial if and only if $I$ represents the covariant constant singleton functor, and an object $T$ is terminal if and only if $T$ represents the contravariant constant singleton functor. Consequently the covariant functor is representable exactly when $\mathcal{C}$ has an initial object, and the presheaf is representable exactly when $\mathcal{C}$ has a terminal object.

Any two initial objects in a category are joined by a unique isomorphism. Any two terminal objects are likewise joined by a unique isomorphism.

Let $\mathcal{C}$ be a locally small category, let $a$ be an object, and let $F:\mathcal{C}\to \mathbf{S}\mathbf{e}\mathbf{t}$ be a functor. Evaluation at the identity defines a bijection

$E_{a,F}:\mathrm{Nat}(\mathcal{C}(a,-),F)⟶F(a),E_{a,F}(\alpha )={\alpha }_a(1_a)\mathrm{.}$

Its inverse sends $x\in F(a)$ to the natural transformation ${\alpha }^x$ whose component at $c$ is

${\alpha }_c^x:\mathcal{C}(a,c)⟶F(c),{\alpha }_c^x(f)=F(f)(x)\mathrm{.}$

The explicit parametrization by the set $F(a)$ proves that the natural-transformation collection in the display is a set. The construction makes no choice from a family of nonempty sets.

Under the hypotheses of Evaluation at the identity gives $\mathrm{Nat}(\mathcal{C}(a,-),F)\cong F(a)$ and proves that the natural-transformation collection is a set, the bijections

$E_{a,F}:\mathrm{Nat}(\mathcal{C}(a,-),F)\stackrel{\cong }{\to }F(a)$

are natural in both variables. Explicitly:

1. if $h:a\to a^{\mathrm{\prime }}$ and $\alpha :\mathcal{C}(a,-)\Rightarrow F$, then $E_{a^{\mathrm{\prime }},F}(\alpha \circ \mathcal{C}(h,-))=F(h)(E_{a,F}(\alpha ));$
2. if $\eta :F\Rightarrow G$, then $E_{a,G}(\eta \circ \alpha )={\eta }_a(E_{a,F}(\alpha ))\mathrm{.}$

Here $\mathcal{C}(h,-):\mathcal{C}(a^{\mathrm{\prime }},-)\Rightarrow \mathcal{C}(a,-)$ is precomposition by $h$. These equations make sense for every locally small $\mathcal{C}$ and do not require a functor category with source $\mathcal{C}$.

Let $\mathcal{C}$ be locally small, let $a$ be an object, and let $P:{\mathcal{C}}^{\mathrm{o}\mathrm{p}}\to \mathbf{S}\mathbf{e}\mathbf{t}$ be a presheaf. Evaluation at the identity gives a bijection

$E_P^a:\mathrm{Nat}(\mathcal{C}(-,a),P)\stackrel{\cong }{\to }P(a),E_P^a(\alpha )={\alpha }_a(1_a),$

whose inverse sends $x\in P(a)$ to the natural transformation with $c$-component $f:c\to a\mapsto P(f)(x)$. It is natural in both variables. In particular, for $h:a\to a^{\mathrm{\prime }}$ and $\beta :\mathcal{C}(-,a^{\mathrm{\prime }})\Rightarrow P$,

$E_P^a(\beta \circ \mathcal{C}(-,h))=P(h)(E_P^{a^{\mathrm{\prime }}}(\beta )),$

and postcomposition by a natural transformation $P\Rightarrow Q$ corresponds to its component at $a$.

Local smallness says that each individual hom-collection is a set (Small, locally small, and large categories). It does not, by itself, turn an object-indexed family of components over a proper class of objects into set-coded data. For this reason Functor category $[\mathcal{C},\mathcal{D}]$ forms $[\mathcal{C},\mathcal{D}]$ as a category only when $\mathcal{C}$ is small, and If $\mathcal{C}$ is small and $\mathcal{D}$ is locally small then $[\mathcal{C},\mathcal{D}]$ is locally small; if both are small it is small obtains local smallness from a small source and a locally small target.

The representable-source case has additional structure. For an object $a$ and $F:\mathcal{C}\to \mathbf{S}\mathbf{e}\mathbf{t}$, the explicit formulas of Evaluation at the identity gives $\mathrm{Nat}(\mathcal{C}(a,-),F)\cong F(a)$ and proves that the natural-transformation collection is a set parametrize every natural transformation $\mathcal{C}(a,-)\Rightarrow F$ by the set $F(a)$. Thus this particular natural-transformation collection is a set even when $\mathcal{C}$ is large and locally small. No global proper-class counterexample is asserted here.

Let $\mathcal{C}$ be locally small. The Yoneda assignment $y$ sends an object $a$ to the representable presheaf

$y(a):=\mathcal{C}(-,a):{\mathcal{C}}^{\mathrm{o}\mathrm{p}}\to \mathbf{S}\mathbf{e}\mathbf{t}$

and sends a morphism $f:a\to b$ to the natural transformation $y(f):y(a)\Rightarrow y(b)$ whose component at $c$ is postcomposition by $f$:

$y(f{)}_c:\mathcal{C}(c,a)⟶\mathcal{C}(c,b),g⟼f\circ g\mathrm{.}$

The functoriality and naturality of these formulas are instances of the hom-bifunctor The hom-assignment $\mathcal{C}(-,-):{\mathcal{C}}^{\mathrm{o}\mathrm{p}}\times \mathcal{C}\to \mathbf{S}\mathbf{e}\mathbf{t}$ is a bifunctor.

When $\mathcal{C}$ is small, Functor category $[\mathcal{C},\mathcal{D}]$ and If $\mathcal{C}$ is small and $\mathcal{D}$ is locally small then $[\mathcal{C},\mathcal{D}]$ is locally small; if both are small it is small form the presheaf category and the assignment is the functor

$y:\mathcal{C}⟶[{\mathcal{C}}^{\mathrm{o}\mathrm{p}},\mathbf{S}\mathbf{e}\mathbf{t}],$

traditionally called the Yoneda embedding. For an arbitrary large locally small $\mathcal{C}$, the same formulas are called the Yoneda assignment; no large-source functor category is silently formed. Full faithfulness is proved in The Yoneda functor is fully faithful, and it is a full embedding when its object map is injective. Under the terminology of Embedding and full embedding of categories, a fully faithful functor is a full embedding only if its object map is also injective.

Let $\mathcal{C}$ be locally small. For all objects $a,b$, the Yoneda assignment induces a bijection

$\mathcal{C}(a,b)\stackrel{\cong }{\to }\mathrm{Nat}(\mathcal{C}(-,a),\mathcal{C}(-,b)),f⟼y(f),$

where $y(f{)}_c(g)=f\circ g$. Thus, when $\mathcal{C}$ is small and $y$ is the functor of The Yoneda assignment and the small-source Yoneda functor, traditionally called the Yoneda embedding, it is fully faithful. If its object map is also injective, it is a full embedding in the library's stronger sense.

Let $a$ and $b$ be objects of a locally small category $\mathcal{C}$. Then

$a\cong b⟺\mathcal{C}(-,a)\cong \mathcal{C}(-,b)$

by a natural isomorphism of presheaves.

Let $\mathcal{C}$ be locally small. For a functor $F:\mathcal{C}\to \mathbf{S}\mathbf{e}\mathbf{t}$, a universal element is a pair $(R,u)$ with $R\in \mathcal{C}$ and $u\in F(R)$ such that the maps

${\theta }_c^u:\mathcal{C}(R,c)⟶F(c),{\theta }_c^u(f)=F(f)(u)$

are the components of a natural isomorphism ${\theta }^u:\mathcal{C}(R,-)\stackrel{\cong }{\Rightarrow }F$.

For a presheaf $P:{\mathcal{C}}^{\mathrm{o}\mathrm{p}}\to \mathbf{S}\mathbf{e}\mathbf{t}$, a universal element is a pair $(R,u)$ with $u\in P(R)$ such that

${\theta }_c^u:\mathcal{C}(c,R)⟶P(c),{\theta }_c^u(f)=P(f)(u)$

are the components of a natural isomorphism ${\theta }^u:\mathcal{C}(-,R)\stackrel{\cong }{\Rightarrow }P$. Thus a universal element is a representation in the sense of Presheaves, covariantly and contravariantly representable functors, and representations, with its natural isomorphism specified by a distinguished element of the representing object's value.

Let $\mathcal{C}$ be locally small.

1. A pair $(R,u)$ with $u\in F(R)$ is universal for a functor $F:\mathcal{C}\to \mathbf{S}\mathbf{e}\mathbf{t}$ if and only if, for every object $c$ and every $x\in F(c)$, there is a unique morphism $f:R\to c$ such that $F(f)(u)=x$.
2. A pair $(R,u)$ with $u\in P(R)$ is universal for a presheaf $P:{\mathcal{C}}^{\mathrm{o}\mathrm{p}}\to \mathbf{S}\mathbf{e}\mathbf{t}$ if and only if, for every object $c$ and every $x\in P(c)$, there is a unique morphism $f:c\to R$ such that $P(f)(u)=x$.

Every representation $\theta$ determines its universal element by $u={\theta }_R(1_R)$, and the universal element recovers $\theta$ by the formulas in the statement. These correspondences are the Yoneda correspondences and are natural in the representing object and the set-valued functor.

Let $F:\mathcal{C}\to \mathbf{S}\mathbf{e}\mathbf{t}$ have universal elements $(R,u)$ and $(R^{\mathrm{\prime }},u^{\mathrm{\prime }})$. There is a unique isomorphism $i:R\to R^{\mathrm{\prime }}$ satisfying $F(i)(u)=u^{\mathrm{\prime }}$.

If instead $P:{\mathcal{C}}^{\mathrm{o}\mathrm{p}}\to \mathbf{S}\mathbf{e}\mathbf{t}$ has universal elements $(R,u)$ and $(R^{\mathrm{\prime }},u^{\mathrm{\prime }})$, there is a unique isomorphism $i:R\to R^{\mathrm{\prime }}$ satisfying $P(i)(u^{\mathrm{\prime }})=u$.

Hence a representing object is unique up to the unique isomorphism compatible with the chosen universal elements, in either variance.

Let $F:\mathcal{C}\to \mathbf{S}\mathbf{e}\mathbf{t}$ be a functor. Its category of elements $\int F$ has:

• objects $(c,x)$ with $c\in \mathcal{C}$ and $x\in F(c)$;
• a morphism $(c,x)\to (d,y)$ given by a morphism $f:c\to d$ in $\mathcal{C}$ satisfying $F(f)(x)=y$.

The identity of $(c,x)$ is $1_c$, and composition is composition in $\mathcal{C}$. The functor laws of Covariant functor, identity functor, composite functor, and contravariant functor make the identity and composite equations hold, so these data satisfy Category, object, morphism, domain, codomain, identity, composition, and hom-collection.

For a presheaf $P:{\mathcal{C}}^{\mathrm{o}\mathrm{p}}\to \mathbf{S}\mathbf{e}\mathbf{t}$, its category of elements $\int P$ again has objects $(c,x)$ with $x\in P(c)$, while a morphism $(c,x)\to (d,y)$ is a morphism $f:c\to d$ in $\mathcal{C}$ satisfying

$x=P(f)(y)\mathrm{.}$

This reversed equation comes from the opposite category Opposite category ${\mathcal{C}}^{\mathrm{o}\mathrm{p}}$; contravariant functoriality again supplies identities and composition. A universal element in the sense of Universal elements of covariant functors and presheaves is itself an object of the appropriate category of elements.

Let $\mathcal{C}$ be locally small.

1. For $F:\mathcal{C}\to \mathbf{S}\mathbf{e}\mathbf{t}$, a pair $(R,u)$ is a universal element if and only if $(R,u)$ is initial in $\int F$.
2. For $P:{\mathcal{C}}^{\mathrm{o}\mathrm{p}}\to \mathbf{S}\mathbf{e}\mathbf{t}$, a pair $(R,u)$ is a universal element if and only if $(R,u)$ is terminal in $\int P$.

Let $U:\mathcal{D}\to \mathcal{C}$ be a functor as in Covariant functor, identity functor, composite functor, and contravariant functor and let $X$ be an object of $\mathcal{C}$.

A universal arrow from $X$ to $U$ is a pair $(R,\eta )$ with $R\in \mathcal{D}$ and $\eta :X\to U(R)$ such that, for every $D\in \mathcal{D}$ and $f:X\to U(D)$, there is a unique $h:R\to D$ satisfying