Nice and small presheaves

Proof. The collection of nice presheaves is clearly closed under small colimits of presheaves, since colimits commute with colimits and colimits of presheaves are computed objectwise. Furthermore, we have seen above that representable presheaves are nice. It follows that every small presheaf is nice.

Conversely, assume that $F$ is a nice presheaf and choose an exceptional set $S_F$. To show that $F$ is small, we will show that its category of elements $\int F$ has a small final subcategory. Let $\D$ be the full subcategory of $\int F$ consisting of the objects

• $(A,a)$ for $a \in F(A)$,
• $(B,b)$ for $b \in F(B)$,
• $(X,x)$ for $x \in F(X)$ and $X \in S_F$.

This is a small category since $S_F$ is a set.

We need to show that, for every object $T \in \int F$, the comma category $T \downarrow \D$ is connected. This is trivial for objects $T$ of $\D$. It remains to check this for $T = (X,x)$, where $x \in F(X)$ and $X \notin S_F$. By the definition of $S_F$, the diagram $$F(B) ~\overset{f^*}{\underset{g^*}{\rightrightarrows}}~ F(A) \xrightarrow{u_X^*} F(X) \tag{1}$$ is a coequalizer diagram. In particular, $u_X^*$ is surjective, so there is some $a \in F(A)$ with $u_X^*(a) = x$. Then $u_X : (X,x) \to (A,a)$ is a morphism in $\int F$, showing that $(X,x) \downarrow \D$ is non-empty.

There are two types of objects in $(X,x) \downarrow \D$. The first type consists of morphisms $$u_X : (X,x) \to (A,a),$$ where $a \in F(A)$ satisfies $u_X^*(a)=x$. The second type consists of morphisms $$v_X : (X,x) \to (B,b),$$ where $b \in F(B)$ satisfies $v_X^*(b)=x$. Every object of the second type is connected to an object of the first type, since $f : (A,f^*(b)) \to (B,b)$ is a morphism with $f \circ u_X = v_X$.

It remains to show that every two objects $$(X,x) \to (A,a), \quad (X,x) \to (A,a')$$ are connected, where $a,a' \in F(A)$ satisfy $u_X^*(a)=u_X^*(a')=x$. Since the diagram $(1)$ is a coequalizer diagram, there is a finite sequence of elements $a_0,\dotsc,a_n$ in $F(A)$, where $a_0=a$ and $a_n=a'$, and a finite sequence of elements $b_0,\dotsc,b_{n-1} \in F(B)$ such that, for every $0 \leq i < n$, either $$a_i=f^*(b_i), \quad a_{i+1}=g^*(b_i),$$ or $$a_i=g^*(b_i), \quad a_{i+1}=f^*(b_i).$$ It suffices to show that $(X,x) \to (A,a_i)$ and $(X,x) \to (A,a_{i+1})$ are connected. We may assume without loss of generality that $$a_i=f^*(b_i), \quad a_{i+1}=g^*(b_i).$$ But then both are connected to $(X,x) \to (B,b_i)$ via the morphisms

$$f : (A,a_i) \to (B,b_i), \quad g : (A,a_{i+1}) \to (B,b_i),$$

respectively. $\square$

Proof. We define the presheaf $$F \coloneqq \coprod_{a \in G(A)} \Hom(-,A) \sqcup \coprod_{b \in G(B)} \Hom(-,B) \sqcup \coprod_{X \in S, \, x \in G(X)} \Hom(-,X).$$ As a coproduct of representable functors, $F$ is a small presheaf. By the Yoneda Lemma, there is a morphism $\alpha : F \to G$ characterized by

• $\alpha_A(i_a(\id_A)) = a$ for $a \in G(A)$
• $\alpha_B(i_b(\id_B)) = b$ for $b \in G(B)$
• $\alpha_X(i_x(\id_X)) = x$ for $X \in S$, $x \in G(X)$

In particular, by construction, $\alpha$ hits all elements of $G$ except possibly those $x \in G(X)$ where $X \notin S$. But in this case, $u_X^* : G(A) \to G(X)$ is surjective, and since $\alpha$ hits all elements of $G(A)$, it also hits $x$. Thus, $\alpha$ is an epimorphism. $\square$