Exact squares and exactness of comma squares

Question

An interesting property of square 2-cells of functors is exactness. The nlab offers a characterization of exact squares which is opaque to me. Fixing a 2-cell $vf\overset{\psi}\Rightarrow gu$ $$\require{AMScd} \begin{CD} \mathsf A @>f>> \mathsf B\\ @VuVV @VVvV\\ \mathsf C @>>g> \mathsf D \end{CD}$$ the characterization revolves around the following functor. Given objects $b\in \mathsf B,c\in \mathsf C$ consider the functor $$(b/\mathsf A/c)\longrightarrow \mathsf D(vb,gc)$$ taking a triple $(a,b\overset{\beta}\rightarrow fa,ua\overset{\alpha}\rightarrow c)$ to the composite $$vb\overset{v\beta}\rightarrow vfa\overset{\psi_a}\rightarrow gua\overset{g\alpha}\rightarrow gc.$$ If I understand correctly, the characterization is as follows.

Proposition. The square (2-cell) $\psi$ is exact iff for any pair $b\in \mathsf B,c\in \mathsf C$ the functor $(b/\mathsf A/c)\longrightarrow \mathsf D(vb,gc)$ induces a bijection on connected components.

The latter encapsulates conditions 1,2 in the profunctor proof (which I don't understand). After the comma object proof (which I do more-or-less understand) it is remarked that in derivators (which I plan on meeting) exactness is characterized by a refinement of inducing a bijection on connected components.

Question 1. What is the intuition behind the above characterization of exact squares (2-cells) of functors?

Question 2. Where can I find a detailed proof of this characterization in the spirit of the comma object proof linked above?

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Given functors $\mathsf A\overset{F}\rightarrow \mathsf C \overset{G}\leftarrow \mathsf B$ one may consider the 2-cell given by the (non-commutative) comma square below. $$\require{AMScd} \begin{CD} F/G @>>> \mathsf B\\ @VVV @VVV\\ \mathsf A @>>> \mathsf C \end{CD}$$

It is claimed that all such comma squares are exact.

Question 3. How does this follow from the charcterization?

Question 4. Is there some reference to a direct proof of this fact? I tried to prove it using the colimit formula for left Kan extensions but didn't get very far.

Answer 1

1. I think of exactness as saying that $A$ is, in a very weak sense, equivalent to $v/g$ as a span over $B$ and $C$. Specifically, we can see $v/g$ as a bimodule, profunctor, or discrete two-sided fibration in that it's the Grothendieck construction of the functor $B^{\mathrm{op}}\times C\to \mathbf{Set}$ sending $b,c\mapsto D(vb,gc)$. Now there's a universal construction $A''$ of a bimodule over $B$ and $C$ associated to $A$, and asking for exactness of the square is exactly asking that $A''$ be equivalent to $v/g$ in the slice of categories over $B$ and $C$. To construct $A''$, we first turn $A$ from a span into a fibration, more specifically an opfibration over $C$ and a fibration over $B$, by setting $A'=u/ C\times_A B/f$. The point of this is that the strict fibers of $A'$ are the weak fibers of $A$: $A'_{b,c}=u/c\times_A b/f=b/A/c$. Now any map from $A'\to E$ over $B$ and $C$, where $E$ has discrete (strict) fibers, factors for each $b$ and $c$ through $\pi_0(b/A/c)$, so the universal discrete two-sided fibration associated to $A'$, and thus to $A$, is a category $A''$ over $B$ and $C$ with fibers $\pi_0(b/A/c)$. Constructing this category is perhaps most easily done by passing through the Grothendieck construction. So exactness is most meaningful when you're thinking of $A$ as modeling a profunctor between $B$ and $C$.

2. It's the same proof, but you can look in Section 3 here. If you want a proof that the colimit of a constant diagram in a derivator always factors through copowering with the nerve, you still have to read Cisinski's original paper.

3. The question is to compare $C(Fa,Gb)$ and $a/(F/G)/b$. For any $f:Fa\to Gb$, the fiber $[a/(F/G)/b]_f$ is given by all $(f':Fa'\to Gb', x:a\to a',y:b'\to b)$ such that the induced map $Fa\to Gb$ is $f$. Such an object maps to $(yf',x,1_b)$ via $y$, which admits a map from $(f,1_a,1_b)$ via $x$, which proves the fiber over $f$ is connected.

4. To calculate something about a right Kan extension out of $F/G$ (for the left Kan extension we'd be talking about $G/F$), you need to be able to calculate limits over categories of the form $a/(F/G)$. These are closely related to the categories $Fa/G$ which appear in Kan's formula for the eastern/southern route around the square. Specifically, we have a map $Fa/G\to a/(F/G)$ sending $(b,Fa\to Gb)$ to $(a,b,a=a,Fa\to Gb)$ which admits a right adjoint sending $(a',b', a\to a',Fa'\to Gb')$ to $(b',Fa\to Fa'\to Gb')$. That means that to evaluate a limit over $a/(F/G)$ we can first restrict to $Fa/G$ and calculate the limit there: left adjoints are initial functors. This proves the claim: pulling back from $B$ to $F/G$ and then pushing forward to $A$ is given by the limit over $Fa/G$, just as is pushing forward to $C$ and pulling back to $A$. The general reason why there's this adjunction between the lax and the strict fibers of $F/G$ above $a$ is that the projection $F/G\to A$ is a Grothendieck fibration, and in fact similar arguments allow you to prove that pullback squares with an appropriate leg a fibration or opfibration are always exact.