Colimits of representable functors

Question

I'm trying to understand the proof "each functor $D \rightarrow \mathbf{Set}$ is a colimit of representable functors" in MacLane (CFWM), but like with all things Yoneda I'm having problems.

1.) After he takes $L$ to be vertex of some other cone over $M$ (say $\psi$), he says that we have $z$ and $z'$ s.t. $y^{-1} z = \psi_{(d,x)}$ and $y^{-1} z' = \psi_{(d',x')}$. But I lose him in the next part, where he claims that $z' = f z$ (I suppose this means $z' = L(f)(z)$). Why is this so?

2.) He then constructs $\theta_d$ that maps $x \in Kd$ to $z$ such that $y^{-1}z$ is in the cone of $L$. If you were to take $\eta \in D(d', u)$, then the following needs to hold: $$ \theta_u \circ (y^{-1}x')_u (\eta) = (y^{-1}z')_u(\eta) $$ Why does this hold?

Thanks.

Answer 1

Note that there is a typo in that version of CWM: The diagram should be $$M:\mathcal{J}^{\mathrm{op}} \to \textbf{Sets}^{\mathcal{D}}$$ instead of $$M:\mathcal{J}^{\mathcal{D}}\to \textbf{Sets}^{\mathcal{D}}$$ If you really want to learn the proof of this fact, I would recommend realizing it as a consequence of the fact that if $\mathcal{C}$ is a small category, $\mathcal{D}$ a cocomplete category and $F:\mathcal{C}\to\mathcal{D}$ a functor, then the functor \begin{eqnarray*} R:\mathcal{D}&\to&[\mathcal{C}^{\text{op}},\mathbf{Sets}]\\ D&\mapsto& R(D) \end{eqnarray*} where \begin{eqnarray*} R(D):\mathcal{C}^{\text{op}}&\to&\mathbf{Sets}\\ C&\mapsto& \text{Hom}_{\mathcal{D}}(F(C),D) \end{eqnarray*} has a left adjoint. Because it is a much more useful result with many other applications. And the proof is really quite similar in spirit to the one in CWM. So, two birds with one stone :)

This is Theorem 2 of Section I of MacLane and Moerdijk's "Sheaves in Geometry and Logic", where the result you are asking for appears as Corollary 3.

Answer 2

Let's decrypt the CWM proof:

But first some naming conventions:

1. Write $\int_DK$ for the category of elements, although I prefer $\sum_D K$ or $(d:D)\times K(d)$ as it is really a dependent sum ($K$ unpacked).
2. $y_d:\text{Nat}(D(d,-),K)\to K(d)$ is the natural isomorphism in the Yoneda lemma.
3. the $M:(\int_DK)^{\text{op}}\to\text{Set}^D$ functor (diagram) is the composite:

$$\left(\int_DK\right)^{\text{op}}\xrightarrow{\pi_1}D^{\text{op}}\xrightarrow{Y}\text{Set}^D$$

where $\pi_1$ is projection on first coordinate and $Y=\operatorname{hom}$ is the Yoneda embedding.

Now CWM claim that the colimit of $M$ is $K$ together with its cocone $\varphi_{d,x}=y^{-1}_d(x):M(d,x)=D(d,-)\to K$, as it is the only choice we have. Then $y_d(\varphi_{d,x})=x$.

Let $L,\psi_{d,x}:M(d,x)=D(d,-)\to L$ be another cocone, by the Yoneda lemma, each arrow $\psi_{d,x}:M(d,x)=D(d,-)\to K$ in the cocone corresponds 1-1 with an element $z=\psi'_{d,x}=y_d(\psi_{d,x})$ in the set $K(d)$ and vice versa.

Furthermore, let $f:d\to d'$ in $D$ such that $K(f)(x)=x'$, i.e., an arrow in $\int_D K$, then $M(f)=D(f,-):D(d',-)\to D(d,-)$ is an arrow in the base of the cocone. By the naturality of the isomorphism $y$ in the Yoneda lemma, we can correspond 1-1 the cocone equation $\psi_{d,x}D(f,-)=\psi_{d',x'}$ with the equation $L(f)(\psi'_{d,x})=\psi'_{d',x'}$ and vice versa (same is true for $K$ as well, but that is just the equation $K(f)(x)=x'$). This is the equation $L(f)(z)=z'$ in CWM.

$\require{AMScd}$ \begin{CD} \text{Nat}(D(d,-),K) @>\varphi\mapsto\varphi D(f,-)>> \text{Nat}(D(d',-),K) \\ @VV y_d V @VV y_{d'} V\\ K(d) @>> K(f) > K(d') \end{CD}

Using these information, we define the natural transformation $\theta:K\to L$ on each component $\theta_d:K(d)\to L(d)$ by realizing it is essentially the same transformation on the corresponding functors:

$$\theta_d':\text{Nat}(D(d,-),K)\to \text{Nat}(D(d,-),L)$$

$$\theta_d'(\varphi)=\psi_{d,y(\varphi)}$$

Then we see $\theta_d'(\varphi_{d,x})=\psi_{d,y(\varphi_{d,x})}=\psi_{d,x}$.

so $\theta_d=y_d\theta_d'y_d^{-1}:K(d)\to L(d)$, and $\theta_d(x)=y(\psi_{d,x})=\psi'_{d,x}=z$.

To show it is a cocone morphism, i.e., $\theta\varphi_{d,x}=\psi_{d,x}$. Let $f:d\to d'$ with $K(f)(x)=x'$ as before, we compute component-wise: $\theta_{d'}(\varphi_{d,x})_{d'}(f)=\theta_{d'}(y^{-1}_d(x))_{d'}(f)=\theta_{d'}(K(f)(x))=\theta_{d'}(x')=\psi'_{d',x'}=z'$. On the other hand, $(\psi_{d,x})_{d'}(f)=(y_d^{-1}(z))_{d'}(f)=L(f)(z)=z'$. So $\theta$ is a well-defined cocone morphism.

The equation $L(f)(\theta_d(x))=L(f)(z)=z'=\theta_{d'}(x')=\theta_{d'}(K(f)(x))$ shows $\theta$ is natural.

Finally, to show $\theta$ is unique, let $\sigma:K\to L$ be another cocone morphism. Notice the Yoneda isomorphism $y_d$ gives us a 1-1 correspondence between $\text{Nat}(D(d,-),K)$ and $K(d)$, so any $\varphi:D(d,-)\to K$ can be written as $y_d^{-1}(x)$ for some $x\in K(d)$, so the cocone morphism equation $\sigma\varphi_{d,x}=\psi_{d,x}$ determines it uniquely.