6. Categories of Diagrams

6.1The Yoneda Lemma

Throughout this section all categories are considered to be locally small.

定义 6.1.1. The covariant Yoneda functor is the functor$$y:\mathcal{C}\to \mathrm{Fun}\left({\mathcal{C}}^{\mathrm{op}},\mathbf{Set}\right),$$where $\mathrm{Fun}\left({\mathcal{C}}^{\mathrm{op}},\mathbf{Set}\right)$ is the category of contravariant functors from $\mathcal{C}$ to $\mathbf{Set}$ (also called presheaves), defined on objects $C$ of $\mathcal{C}$ by (the contravariant functor)$$y(C)={\mathrm{Hom}}_{\mathcal{C}}(-,C):\mathcal{C}\to \mathbf{Set}$$and on arrows $f:C\to D$ in $\mathcal{C}$ by (the natural transformation)$$y(f)={\mathrm{Hom}}_{\mathcal{C}}(-,f):y(C)={\mathrm{Hom}}_{\mathcal{C}}(-,C)\to y(D)={\mathrm{Hom}}_{\mathcal{C}}(-,D).$$

注 6.1.2. Let us give the explicit definition of the covariant Yoneda functor, which will be useful for the proof and the consequences of the Yoneda lemma.

Let $C$ be an object of $\mathcal{C}$ and let $f:C\to D$ be an arrow in $\mathcal{C}$.

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The contravariant functor $$y(C)={\mathrm{Hom}}_{\mathcal{C}}(-,C):\mathcal{C}\to \mathbf{Set}$$is defined on objects $C^{\prime }$ of $\mathcal{C}$ by $$(y(C))\left(C^{\prime }\right)={\mathrm{Hom}}_{\mathcal{C}}\left(C^{\prime },C\right)$$and on arrows $g:C^{\prime }\to C^{\prime \prime }$ by $$(y(C))(g)={\mathrm{Hom}}_{\mathcal{C}}(g,C):{\mathrm{Hom}}_{\mathcal{C}}\left(C^{\prime \prime },C\right)\to {\mathrm{Hom}}_{\mathcal{C}}\left(C^{\prime },C\right),$$where for every $h\in {\mathrm{Hom}}_{\mathcal{C}}\left(C^{\prime \prime },C\right)$ we have: $$[(y(C))(g)](h)=\left[{\mathrm{Hom}}_{\mathcal{C}}(g,C)\right](h)=h\circ g.$$

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The component of the natural transformation $$y(f)={\mathrm{Hom}}_{\mathcal{C}}(-,f):{\mathrm{Hom}}_{\mathcal{C}}(-,C)\to {\mathrm{Hom}}_{\mathcal{C}}(-,D)$$at an object $C^{\prime }$ of $\mathcal{C}$ is defined by $$y(f{)}_{C^{\prime }}={\mathrm{Hom}}_{\mathcal{C}}\left(C^{\prime },f\right):{\mathrm{Hom}}_{\mathcal{C}}\left(C^{\prime },C\right)\to {\mathrm{Hom}}_{\mathcal{C}}\left(C^{\prime },D\right),$$where for every $h\in {\mathrm{Hom}}_{\mathcal{C}}\left(C^{\prime },C\right)$ we have: $$\left[y(f{)}_{C^{\prime }}\right](h)=\left[{\mathrm{Hom}}_{\mathcal{C}}\left(C^{\prime },f\right)\right](h)=f\circ h.$$

引理 6.1.3 (Yoneda). Let $\mathcal{C}$ be a locally small category. Then for every object $C$ of $\mathcal{C}$ and for every object $F$ of $\mathrm{Fun}\left({\mathcal{C}}^{\mathrm{op}},\mathbf{Set}\right)$, there is an isomorphism in $\mathbf{Set}$:$${\mathrm{Hom}}_{\mathrm{Fun}\left({\mathcal{C}}^{\mathrm{op}},\mathbf{Set}\right)}(y(C),F)\cong F(C),$$which is natural in both $C$ and $F$.

证明. For the sake of simplicity, we omit to write the index of ${\mathrm{Hom}}_{\mathrm{Fun}\left({\mathcal{C}}^{\mathrm{op}},\mathbf{Set}\right)}(y(C),F)$.

Step 1: Define two natural transformations.

Let $C$ be an object of $\mathcal{C}$ and let $F$ be an object of $\mathrm{Fun}({\mathcal{C}}^{\mathrm{op}},\mathbf{Set})$.

We define a natural transformation ${\eta }_{C,F}:\mathrm{Hom}(y(C),F)\to F(C)$ as follows. For every $\nu \in \mathrm{Hom}(y(C),F)$ (natural transformation), take$${\eta }_{C,F}(\nu )={\nu }_C\left(1_C\right),$$where ${\nu }_C:(y(C))(C)={\mathrm{Hom}}_{\mathcal{C}}(C,C)\to F(C)$. Note that ${\nu }_C\left(1_C\right)\in F(C)$.

Also, we define a natural tranformation ${\mu }_{C,F}:F(C)\to \mathrm{Hom}(y(C),F)$ as follows. For every $a\in F(C)$, let ${\nu }_a:y(C)\to F$ be given by$${\left({\nu }_a\right)}_{C^{\prime }}:(y(C))\left(C^{\prime }\right)={\mathrm{Hom}}_{\mathcal{C}}\left(C^{\prime },C\right)\to F\left(C^{\prime }\right)$$where ${\left({\nu }_a\right)}_{C^{\prime }}(h)=[F(h)](a)$ for every $h\in {\mathrm{Hom}}_{\mathcal{C}}\left(C^{\prime },C\right)$. Note that $F(h):F(C)\to F\left(C^{\prime }\right)$. Then we take ${\mu }_{C,F}(a)={\nu }_a$.

We show that ${\nu }_a:y(C)\to F$ is a natural transformation. Let $f:C^{\prime \prime }\to C^{\prime }$ be an arrow in $\mathcal{C}$. Consider the diagram

For every $h\in {\mathrm{Hom}}_{\mathcal{C}}\left(C^{\prime },C\right)$, we have:$$\begin{array}{rl}\left[{\left({\nu }_a\right)}_{C^{\prime \prime }}\circ {\mathrm{Hom}}_{\mathcal{C}}(f,C)\right](h)& ={\left({\nu }_a\right)}_{C^{\prime \prime }}\left({\mathrm{Hom}}_{\mathcal{C}}(f,C)(h)\right)={\left({\nu }_a\right)}_{C^{\prime \prime }}(h\circ f)\\ & =[F(h\circ f)](a)=(F(f)\circ F(h))(a)=F(f)[(F(h))(a)]\\ & =(F(f))\left[{\left({\nu }_a\right)}_{C^{\prime }}(h)\right]=\left[F(f)\circ {\left({\nu }_a\right)}_{C^{\prime }}\right](h),\end{array}$$hence the above diagram is commutative. Therefore, ${\nu }_a$ is a natural transformation, and thus ${\nu }_a\in \mathrm{Hom}(y(C),F)$.

Step 2: Show that their components are isomorphisms.

We show that ${\mu }_{C,F}\circ {\eta }_{C,F}=1_{\mathrm{Hom}(y(C),F)}$. Let $\nu \in \mathrm{Hom}(y(C),F)$ and $h\in {\mathrm{Hom}}_{\mathcal{C}}\left(C^{\prime },C\right)$. We have:$$\left({\mu }_{C,F}\circ {\eta }_{C,F}\right)(\nu )={\mu }_{C,F}\left({\nu }_C\left(1_C\right)\right)={\nu }_{{\nu }_C\left(1_C\right)}.$$We need to prove that ${\nu }_{{\nu }_C\left(1_C\right)}=\nu$. Since $\nu$ is a natural transformation, the following diagram is commutative:

hence $F(h)\circ {\nu }_C={\nu }_{C^{\prime }}\circ [y(C)](h)$. By using also Remark 6.1.2, it follows that:$$\begin{array}{rl}{\left({\nu }_{{\nu }_C\left(1_C\right)}\right)}_{C^{\prime }}(h)& =[F(h)]\left({\nu }_C\left(1_C\right)\right)=\left(F(h)\circ {\nu }_C\right)\left(1_C\right)=\left({\nu }_{C^{\prime }}\circ (y(C))(h)\right)\left(1_C\right)\\ & ={\nu }_{C^{\prime }}\left[((y(C))(h))\left(1_C\right)\right]={\nu }_{C^{\prime }}\left(1_C\circ h\right)={\nu }_{C^{\prime }}(h).\end{array}$$This shows that ${\nu }_{{\nu }_C\left(1_C\right)}=\nu$. It follows that ${\mu }_{C,F}\circ {\eta }_{C,F}=1_{\mathrm{Hom}(y(C),F)}$.

We show that ${\eta }_{C,F}\circ {\mu }_{C,F}=1_{F(C)}$. Let $a\in F(C)$. We have:$$\begin{array}{rl}\left({\eta }_{C,F}\circ {\mu }_{C,F}\right)(a)& ={\eta }_{C,F}\left({\mu }_{C,F}(a)\right)={\eta }_{C,F}\left({\nu }_a\right)\\ & ={\left({\nu }_a\right)}_C\left(1_C\right)=\left[F\left(1_C\right)\right](a)=1_{F(C)}(a)=a.\end{array}$$This shows that ${\eta }_{C,F}\circ {\mu }_{C,F}=1_{F(C)}$. Hence ${\eta }_{C,F}$ and ${\mu }_{C,F}$ are inverse to each other.

Step 3: Show naturality in $\mathbf{C}$.

Let $f:C^{\prime }\to C$ be an arrow in $\mathcal{C}$ and let $F$ be an object of $\mathrm{Fun}({\mathcal{C}}^{\mathrm{op}},\mathbf{Set})$. We show that the following diagram is commutative:

hence $F(f)\circ {\nu }_C={\nu }_{C^{\prime }}\circ (y(C))(f)$. By using also Remark 6.1.2, for every $\nu \in \mathrm{Hom}(y(C),F)$, we have:$$\begin{array}{rl}\left({\eta }_{C^{\prime },F}\circ \mathrm{Hom}(y(f),F)\right)(\nu )& ={\eta }_{C^{\prime },F}[\mathrm{Hom}(y(f),F)(\nu )]={\eta }_{C^{\prime },F}(\nu \circ y(f))\\ & =(\nu \circ y(f){)}_{C^{\prime }}\left(1_{C^{\prime }}\right)=\left({\nu }_{C^{\prime }}\circ y(f{)}_{C^{\prime }}\right)\left(1_{C^{\prime }}\right)\\ & ={\nu }_{C^{\prime }}\left(y(f{)}_{C^{\prime }}\left(1_{C^{\prime }}\right)\right)={\nu }_{C^{\prime }}\left(1_{C^{\prime }}\circ f\right)={\nu }_{C^{\prime }}(f)\\ & ={\nu }_{C^{\prime }}\left(f\circ 1_C\right)={\nu }_{C^{\prime }}\left[((y(C))(f))\left(1_C\right)\right]\\ & =\left[{\nu }_{C^{\prime }}\circ (y(C))(f)\right]\left(1_C\right)=\left(F(f)\circ {\nu }_C\right)\left(1_C\right)\\ & =F(f)\left({\nu }_C\left(1_C\right)\right)=F(f)\left({\eta }_{C,F}(\nu )\right)\\ & =\left(F(f)\circ {\eta }_{C,F}\right)(\nu ).\end{array}$$This shows that the above diagram is commutative, and thus we have naturality in $C$.

Step 4: Show naturality in $\mathbf{F}$.

Let $\Phi :F\to F^{\prime }$ be an arrow in $\mathrm{Fun}({\mathcal{C}}^{\mathrm{op}},\mathbf{Set})$ and let $C$ be an object of $\mathcal{C}$. We show that the following diagram is commutative:

By using also Remark 6.1.2, for every $\nu \in \mathrm{Hom}(y(C),F)$, we have$$\begin{array}{rl}\left({\eta }_{C,F^{\prime }}\circ \mathrm{Hom}(y(C),\Phi )\right)(\nu )& ={\eta }_{C,F^{\prime }}[\mathrm{Hom}(y(C),\Phi )(\nu )]={\eta }_{C,F^{\prime }}(\Phi \circ \nu )\\ & =(\Phi \circ \nu {)}_C\left(1_C\right)=\left({\Phi }_C\circ {\nu }_C\right)\left(1_C\right)\\ & ={\Phi }_C\left({\nu }_C\left(1_C\right)\right)={\Phi }_C\left({\eta }_{C,F}(\nu )\right)=\left({\Phi }_C\circ {\eta }_{C,F}\right)(\nu )\end{array}$$This shows that the above diagram is commutative, and thus we have naturality in $F$.

$\square$

定义 6.1.4. A functor $F:\mathcal{C}\to \mathcal{D}$ is called an embedding if $F$ is fully faithful and injective on objects in the sense that if $A,B$ are objects of $\mathcal{C}$ such that $F(A)=F(B)$, then $A=B$.

推论 6.1.5. The covariant Yoneda functor is an embedding.

证明. We show that $y$ is fully faithful. Let $C$ and $D$ be objects of $\mathcal{C}$. By using the Yoneda lemma with $F=y(D)$ and Remark 6.1.2 we have:$${\mathrm{Hom}}_{\mathcal{C}}(C,D)\cong (y(D))(C)\cong {\mathrm{Hom}}_{\mathrm{Fun}({\mathcal{C}}^{\mathrm{op}},\mathbf{Set})}(y(C),y(D)).$$We need to show that this isomorphism is induced by $y$. For every $h\in (y(D))(C)={\mathrm{Hom}}_{\mathcal{C}}(C,D)$, the isomorphism yields a natural transformation ${\nu }_h:y(C)\to y(D)$ defined as follows. For every arrow $f:C^{\prime }\to C$ in $\mathcal{C}$, we have:$${\left({\nu }_h\right)}_{C^{\prime }}(f)=[y(D)(f)](h)=h\circ f=(y(h){)}_{C^{\prime }}(f),$$where $y(h):y(C)\to y(D)$ and$$(y(h){)}_{C^{\prime }}={\mathrm{Hom}}_{\mathcal{C}}\left(C^{\prime },h\right):(y(C))\left(C^{\prime }\right)={\mathrm{Hom}}_{\mathcal{C}}\left(C^{\prime },C\right)\to (y(D))\left(C^{\prime }\right)={\mathrm{Hom}}_{\mathcal{C}}\left(C^{\prime },D\right).$$Hence ${\nu }_h=y(h)$, which shows that the above isomorphism is induced by $y$. Therefore, $y$ is fully faithful.

We show that $y$ is injective on objects. Let $C$ and $D$ be objects of $\mathcal{C}$ such that $y(C)=y(D)$. We have:$$1_C\in {\mathrm{Hom}}_{\mathcal{C}}(C,C)=(y(C))(C)=(y(D))(C)={\mathrm{Hom}}_{\mathcal{C}}(C,D),$$which implies that $C=D$. Hence $y$ is injective on objects.

Therefore, $y$ is an embedding.

$\square$

注 6.1.6.  

(1)	If $\mathcal{C}$ is small, then $\mathrm{Fun}\left({\mathcal{C}}^{\mathrm{op}},\mathbf{Set}\right)$ is locally small, and thus $\mathrm{Hom}(y(C),F)\in \mathrm{Fun}\left({\mathcal{C}}^{\mathrm{op}},\mathbf{Set}\right)$ is a set.
(2)	If $\mathcal{C}$ is locally small, then $\mathrm{Fun}\left({\mathcal{C}}^{\mathrm{op}},\mathbf{Set}\right)$ need not be locally small. In this case, the Yoneda lemma tells us that $\mathrm{Hom}(y(C),F)\in \mathrm{Fun}\left({\mathcal{C}}^{\mathrm{op}},\mathbf{Set}\right)$ is always a set.
(3)	If $\mathcal{C}$ is not locally small, then $y:\mathcal{C}\to \mathrm{Fun}({\mathcal{C}}^{\mathrm{op}},\mathbf{Set})$ will not even be defined, and thus the Yoneda lemma does not apply.
(4)	Taking $F=y(D)$ in the Yoneda lemma, we have:

$$\mathrm{Hom}({\mathrm{Hom}}_{\mathcal{C}}(-,C),{\mathrm{Hom}}_{\mathcal{C}}(-,D))\cong \mathrm{Hom}(y(C),y(D))\cong (y(D))(C)={\mathrm{Hom}}_{\mathcal{C}}(C,D).$$Dually, there is a contravariant Yoneda functor and the corresponding Yoneda lemma.

定义 6.1.7. The contravariant Yoneda functor is the functor$$y:{\mathcal{C}}^{\mathrm{op}}\to \mathrm{Fun}(\mathcal{C},\mathbf{Set}),$$where $\mathrm{Fun}(\mathcal{C},\mathbf{Set})$ is the category of covariant functors from $\mathcal{C}$ to $\mathbf{Set}$, defined on objects $C$ of $\mathcal{C}$ by (the covariant functor)$$y(C)={\mathrm{Hom}}_{\mathcal{C}}(C,-):\mathcal{C}\to \mathbf{Set}$$and on arrows $f:C\to D$ in $\mathcal{C}$ by (the natural transformation)$$y(f)={\mathrm{Hom}}_{\mathcal{C}}(f,-):y(D)={\mathrm{Hom}}_{\mathcal{C}}(D,-)\to y(C)={\mathrm{Hom}}_{\mathcal{C}}(C,-).$$

引理 6.1.8 (Yoneda). Let $\mathcal{C}$ be a locally small category. Then for every object $C$ of $\mathcal{C}$ and for every object $F$ of $\mathrm{Fun}(\mathcal{C},\mathbf{Set})$, there is an isomorphism in $\mathbf{Set}$:$${\mathrm{Hom}}_{\mathrm{Fun}(\mathcal{C},\mathbf{Set})}(y(C),F)\cong F(C),$$which is natural in both $C$ and $F$. In particular, we have:$$\mathrm{Hom}\left({\mathrm{Hom}}_{\mathcal{C}}(C,-),{\mathrm{Hom}}_{\mathcal{C}}(D,-)\right)\cong \mathrm{Hom}(y(C),y(D))\cong (y(D))(C)={\mathrm{Hom}}_{\mathcal{C}}(D,C).$$

推论 6.1.9. The contravariant Yoneda functor is an embedding.

6.2Applications of the Yoneda Lemma

We have seen that every fully faithful functor is “essentially injective” on objects. Hence we have the following consequence.

推论 6.2.1 (Yoneda principle). Let $\mathcal{C}$ be a locally small category, and let $A,B$ be objects of $\mathcal{C}$ such that $y(A)\cong y(B)$. Then $A\cong B$.

例 6.2.2. Let $\mathcal{C}$ be a locally small category with binary products, binary coproducts and exponentials, in the sense that any two objects $B$ and $C$ of $\mathcal{C}$ have an exponential $C^B$ (e.g., $\mathbf{Set}$). Then it is known that there is an isomorphism (bijection)$${\mathrm{Hom}}_{\mathbf{Set}}(A\times B,C)\cong {\mathrm{Hom}}_{\mathbf{Set}}\left(A,C^B\right),$$which is natural in $A$, $B$ and $C$. Let us show that:$$(A\oplus B)\times C\cong (A\times C)\oplus (B\times C).$$One option is to get the isomorphism by using universal mapping property of the product and the coproduct. Alternatively, one may use the Yoneda principle as follows. We first show that for every object $X$ of $\mathcal{C}$, we have the natural isomorphism:$${\mathrm{Hom}}_{\mathbf{Set}}((A\oplus B)\times C,X)\cong {\mathrm{Hom}}_{\mathbf{Set}}((A\times C)\oplus (B\times C),X).$$To this end, note that we have the natural isomorphisms:$$\begin{array}{rl}{\mathrm{Hom}}_{\mathbf{Set}}((A\oplus B)\times C,X)& \cong {\mathrm{Hom}}_{\mathbf{Set}}\left(A\oplus B,X^C\right)\\ & \cong {\mathrm{Hom}}_{\mathbf{Set}}\left(A,X^C\right)\oplus {\mathrm{Hom}}_{\mathbf{Set}}\left(B,X^C\right)\\ & \cong {\mathrm{Hom}}_{\mathbf{Set}}(A\times C,X)\oplus {\mathrm{Hom}}_{\mathbf{Set}}(B\times C,X)\\ & \cong {\mathrm{Hom}}_{\mathbf{Set}}((A\times C)\oplus (B\times C),X).\end{array}$$Now the Yoneda principle implies that $(A\oplus B)\times C\cong (A\times C)\oplus (B\times C)$.

例 6.2.3 (Yoneda meets Cayley). Recall that Cayley Theorem states that every group is isomorphic to a subgroup of a symmetric group. Let $(G,\cdot )$ be a group and consider the symmetric group$$S_G=\{g:G\to G\mid g\text{is bijective}\}.$$For every $a\in G$, define$$t_a:G\to G\text{by}{t}_a(x)=ax,\forall x\in G.$$One proves that $t_a\in S_G$, that is, $t_a$ is bijective. We may now define$$f:G\to S_G\text{by}f(a)=t_a,\forall a\in G.$$One shows that $f$ is an injective group homomorphism. By the first isomorphism theorem, it follows that $G$ is isomorphic to the subgroup $\mathrm{Im}f$ of $S_G$.

Recall that every monoid $(M,\cdot )$ has an associated category, which has only one object $M$ and has as arrows the elements of $M$. In particular, we may consider the associated category $\mathcal{C}$ of a group $(G,\cdot )$. Now consider the Yoneda embedding $y:\mathcal{C}\to \mathrm{Fun}({\mathcal{C}}^{\mathrm{op}},\mathbf{Set})$ defined on objects $C$ of $\mathcal{C}$ by (the contravariant functor)$$y(C)={\mathrm{Hom}}_{\mathcal{C}}(-,C):\mathcal{C}\to \mathbf{Set}$$and on arrows $f:C\to D$ in $\mathcal{C}$ by (the natural transformation)$$y(f)={\mathrm{Hom}}_{\mathcal{C}}(-,f):y(C)={\mathrm{Hom}}_{\mathcal{C}}(-,C)\to y(D)={\mathrm{Hom}}_{\mathcal{C}}(-,D).$$

But the category $\mathcal{C}$ has only one object, namely $(G,\cdot )$. Let $a$ be an arrow in $\mathcal{C}$, that is, $a\in {\mathrm{Hom}}_{\mathcal{C}}(G,G)=G$.

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The contravariant functor $y(G)={\mathrm{Hom}}_{\mathcal{C}}(-,G):\mathcal{C}\to \mathbf{Set}$ is defined on the object $G$ of $\mathcal{C}$ by $$(y(G))(G)={\mathrm{Hom}}_{\mathcal{C}}(G,G)=G$$and on arrows $a\in {\mathrm{Hom}}_{\mathcal{C}}(G,G)=G$ by $$(y(G))(a)={\mathrm{Hom}}_{\mathcal{C}}(a,G):{\mathrm{Hom}}_{\mathcal{C}}(G,G)=G\to {\mathrm{Hom}}_{\mathcal{C}}(G,G)=G,$$where for every $x\in {\mathrm{Hom}}_{\mathcal{C}}(G,G)=G$ we have: $$[(y(G))(a)](x)=\left[{\mathrm{Hom}}_{\mathcal{C}}(a,G)\right](x)=x\cdot a.$$

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The component of the natural transformation $$y(a)={\mathrm{Hom}}_{\mathcal{C}}(-,a):{\mathrm{Hom}}_{\mathcal{C}}(-,G)\to {\mathrm{Hom}}_{\mathcal{C}}(-,G)$$at the object $G$ of $\mathcal{C}$ is defined by $$y(a{)}_G={\mathrm{Hom}}_{\mathcal{C}}(G,a):{\mathrm{Hom}}_{\mathcal{C}}(G,G)=G\to {\mathrm{Hom}}_{\mathcal{C}}(G,G)=G,$$where for every $x\in {\mathrm{Hom}}_{\mathcal{C}}(G,G)=G$ we have: $$\left[y(a{)}_G\right](x)=\left[{\mathrm{Hom}}_{\mathcal{C}}(G,a)\right](x)=a\cdot x,$$Note that $y(a{)}_G=t_a$.

By the Yoneda lemma and the Yoneda embedding, we have the following (natural) isomorphism $f$:$$G={\mathrm{Hom}}_{\mathcal{C}}(G,G)\cong \mathrm{Hom}\left({\mathrm{Hom}}_{\mathcal{C}}(-,G),{\mathrm{Hom}}_{\mathcal{C}}(-,G)\right),$$which is given by $f(a)=y(a)$. The natural transformation $y(a)$ has only one component, namely $y(a{)}_G=t_a$.

例 6.2.4 (Yoneda meets Linear Algebra). Let $K$ be a field. Consider the category $\mathbf{Mat}(K)$, denoted simply by $\mathbf{Mat}$, defined as follows:

•	objects: the non-zero natural numbers
•	arrows: the matrices in $M_{m,n}(K)$ for $m,n\in {\mathbb{N}}^{\ast }$
•	composition: the multiplication of matrices
•	identity arrow: the identity matrix

For every $k\in {\mathbb{N}}^{\ast }$, the set of all matrices with $k$ columns is organized by the data of the $k$-column functor $h_k:\mathbf{Mat}\to \mathbf{Set}$ defined on objects $n\in {\mathbb{N}}^{\ast }$ by the set$$h_k(n)=M_{n,k}(K)$$and on arrows $A:m\to n$ in $\mathbf{Mat}$, that is, $A\in M_{m,n}(K)$ by the function given by the left multiplication by $A$:$$h_k(A):h_k(n)=M_{n,k}(K)\to h_k(m)=M_{m,k}(K),\left[h_k(A)\right](C)=A\cdot C.$$In this way we get a contravariant functor $h_k:\mathbf{Mat}\to \mathbf{Set}$.

Now let $k,j\in {\mathbb{N}}^{\ast }$. A natural transformation $\alpha :h_k\to h_j$ consists of a family$${\left({\alpha }_n:h_k(n)=M_{n,k}(K)\to h_j(n)=M_{n,j}(K)\right)}_{n\in {\mathbb{N}}^{\ast }}$$of functions such that for every arrow $A\in M_{m,n}(K)$ the following diagram is commutative:

Let us classify all natural transformations $\alpha :h_k\to h_j$ or, in other words, all naturallydefined column operations that transform $k$-column matrices to $j$-column matrices.

Now let us see how this setting matches the one of the Yoneda lemma.

The covariant Yoneda functor is the functor$$y:\mathbf{Mat}\to \mathrm{Fun}\left({\mathbf{Mat}}^{\mathrm{op}},\mathbf{Set}\right)$$defined on objects $n$ of $\mathbf{Mat}$ by (the contravariant functor)$$y(n)={\mathrm{Hom}}_{\mathbf{Mat}}(-,n):\mathbf{Mat}\to \mathbf{Set}$$and on arrows $A:m\to n$ in $\mathbf{Mat}$, that is, $A\in M_{m,n}(K)$ by (the natural transformation)$$y(A)={\mathrm{Hom}}_{\mathbf{Mat}}(-,A):y(m)={\mathrm{Hom}}_{\mathbf{Mat}}(-,m)\to y(n)={\mathrm{Hom}}_{\mathbf{Mat}}(-,n).$$

Explicitly, we have the following.

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Let $k$ be an object of $\mathbf{Mat}$. The contravariant functor $y(k)={\mathrm{Hom}}_{\mathbf{Mat}}(-,k):\mathbf{Mat}\to \mathbf{Set}$ is defined on objects $n$ of $\mathbf{Mat}$ by $$(y(k))(n)={\mathrm{Hom}}_{\mathbf{Mat}}(n,k)=M_{n,k}(K)$$and on arrows $A\in {\mathrm{Hom}}_{\mathbf{Mat}}(m,n)=M_{m,n}(K)$ in $\mathbf{Mat}$ by $$(y(k))(A)={\mathrm{Hom}}_{\mathbf{Mat}}(A,k):{\mathrm{Hom}}_{\mathbf{Mat}}(n,k)=M_{n,k}(K)\to {\mathrm{Hom}}_{\mathbf{Mat}}(m,k)=M_{m,k}(K),$$where for every $C\in {\mathrm{Hom}}_{\mathbf{Mat}}(n,k)=M_{n,k}(K)$ we have: $$[(y(k))(A)](C)=\left[{\mathrm{Hom}}_{\mathbf{Mat}}(A,k)\right](C)=A\cdot C.$$Hence we have the functor $y(k)=h_k:\mathbf{Mat}\to \mathbf{Set}$.

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Let $B\in M_{k,j}(K)$ be an arrow in $\mathbf{Mat}$. The component of the natural transformation $$y(B)={\mathrm{Hom}}_{\mathbf{Mat}}(-,B):{\mathrm{Hom}}_{\mathbf{Mat}}(-,k)=h_k\to {\mathrm{Hom}}_{\mathbf{Mat}}(-,j)=h_j$$at an object $n$ of $\mathbf{Mat}$ is defined by $$y(B{)}_n={\mathrm{Hom}}_{\mathbf{Mat}}(n,B):{\mathrm{Hom}}_{\mathbf{Mat}}(n,k)=M_{n,k}(K)\to {\mathrm{Hom}}_{\mathbf{Mat}}(n,j)=M_{n,j}(K),$$where for every $C\in {\mathrm{Hom}}_{\mathcal{C}}(n,k)=M_{n,k}(K)$ we have: $$\left[y(B{)}_n\right](C)=\left[{\mathrm{Hom}}_{\mathbf{Mat}}(n,B)\right](C)=C\cdot B.$$Hence we have the natural transformation $y(B)=\alpha :h_k\to h_j$.

By the Yoneda lemma we have:$$\mathrm{Hom}\left(h_k,h_j\right)=\mathrm{Hom}\left({\mathrm{Hom}}_{\mathbf{Mat}}(-,k),{\mathrm{Hom}}_{\mathbf{Mat}}(-,j)\right)\cong {\mathrm{Hom}}_{\mathbf{Mat}}(k,j)=M_{k,j}(K).$$Also using the proof of the Yoneda lemma, we obtain the following:

•	Every naturally-defined column operation $\alpha :h_k\to h_j$ is determined by a single matrix $B\in M_{k,j}(K)$.
•	We have $B={\alpha }_k\left(I_k\right)\in h_j(k)=M_{k,j}(K)$, that is, $B$ is obtained by applying the column operation $${\alpha }_k:h_k(k)=M_{k,k}(K)\to h_j(k)=M_{k,j}(K)$$to the identity matrix $I_k$.
•	The column operation $${\alpha }_n:h_k(n)=M_{n,k}(K)\to h_j(n)=M_{n,j}(K)$$is given by right multiplication by the matrix $B={\alpha }_k\left(I_k\right)$, that is, ${\alpha }_n(C)=C\cdot B$ for every matrix $C\in M_{n,k}(K)$.
•	Every matrix $B\in M_{k,j}(K)$ determines a naturally-defined column operation defined by right multiplication.

Other consequences are the following:

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A naturally-defined column operation $\alpha :h_k\to h_k$ is invertible if and only if the matrix $B={\alpha }_k\left(I_k\right)\in M_k(K)$ is invertible. Recall the elementary operations on matrices from linear algebra: interchange two columns, multiply a column by a non-zero scalar, multiply a column by a scalar and add it to another column. They are invertible, because the corresponding elementary matrices are invertible.

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The composite $\beta \circ \alpha :h_k\to h_m$ of two naturally-defined column operations $\beta :h_j\to h_m$ and $\alpha :h_k\to h_j$ is defined by the right multiplication of the product of the corresponding matrices: $$(\beta \circ \alpha {)}_k\left(I_k\right)={\alpha }_k\left(I_k\right)\cdot {\beta }_j\left(I_j\right).$$In this way elementary column operations generate all invertible column operations.

6.3Limits and Colimits in Categories of Diagrams

定义 6.3.1. A category $\mathcal{E}$ is called complete if it has all small limits, that is, for any small category $\mathcal{J}$ and functor $F:\mathcal{J}\to \mathcal{E}$, there is a limit for $F$.

Dually, a category $\mathcal{E}$ is called cocomplete if it has all small colimits, that is, for any small category $\mathcal{J}$ and functor $F:\mathcal{J}\to \mathcal{E}$, there is a colimit for $F$.

命题 6.3.2. For any locally small category $\mathcal{C}$, the functor category $\mathrm{Fun}({\mathcal{C}}^{\mathrm{op}},\mathbf{Set})$ is complete. Moreover, for a small category $\mathcal{J}$ and a functor $F:\mathcal{J}\to \mathrm{Fun}({\mathcal{C}}^{\mathrm{op}},\mathbf{Set})$, the limit of $F$ is computed pointwise in the sense that, for every object $C$ of $\mathcal{C}$, we have:$$(\underset{\overleftarrow{j\in J}}{\lim }{F}_j)(C)=\underset{\overleftarrow{j\in J}}{\lim }\left(F_j(C)\right).$$

证明. Let $\mathcal{J}$ be a small category and let $F:\mathcal{J}\to \mathrm{Fun}({\mathcal{C}}^{\mathrm{op}},\mathbf{Set})$ be a functor. Assume that a limit for $F$ does exist. Then it is an object of $\mathrm{Fun}({\mathcal{C}}^{\mathrm{op}},\mathbf{Set})$, hence it is a functor $L=\underset{\overleftarrow{j\in J}}{\lim }{F}_j:{\mathcal{C}}^{\text{op }}\to \mathbf{Set}$. By the Yoneda lemma and the fact that representable functors preserve limits, for every object $C$ of $\mathcal{C}$ we have the following (natural) isomorphisms in $\mathbf{Set}$:$$L(C)\cong \mathrm{Hom}(y(C),L)\cong \mathrm{Hom}(y(C),\underset{\overleftarrow{j\in J}}{\lim }{F}_j)\cong \underset{\overleftarrow{j\in J}}{\lim }\mathrm{Hom}\left(y(C),F_j\right)\cong \underset{\overleftarrow{j\in J}}{\lim }{F}_j(C).$$Hence the limit should be defined by $(\underset{\overleftarrow{j\in J}}{\lim }{F}_j)(C)=\underset{\overleftarrow{j\in J}}{\lim }{F}_j(C)$. One may show that this definition yields a limit for $F$.

$\square$

Dually, we have the following result.

命题 6.3.3. For any locally small category $\mathcal{C}$, the functor category $\mathrm{Fun}({\mathcal{C}}^{\mathrm{op}},\mathbf{Set})$ is cocomplete. Moreover, for a small category $\mathcal{J}$ and a functor $F:\mathcal{J}\to \mathrm{Fun}({\mathcal{C}}^{\mathrm{op}},\mathbf{Set})$, the colimit of $F$ is computed pointwise in the sense that, for every object $C$ of $\mathcal{C}$, we have:$$(\underset{\overrightarrow{j\in J}}{\lim }{F}_j)(C)=\underset{\overrightarrow{j\in J}}{\lim }(F_j(C)).$$

注 6.3.4. In particular, for any locally small category $\mathcal{C}$, the functor category $\mathrm{Fun}({\mathcal{C}}^{\mathrm{op}},\mathbf{Set})$ has products and coproducts, equalizers and coequalizers, terminal and initial objects, pullbacks and pushouts. Hence the functor category may have much richer properties than the original category $\mathcal{C}$. In order to prove a property in $\mathcal{C}$ one may use the Yoneda embedding to go to the associated functor category, prove the property in the functor category, and finally use the Yoneda lemma to get the corresponding property in $\mathcal{C}$.