Comma categories and Kan Extensions

Question

I believe a copresheaf $W:\mathbf{J} \to \mathsf{Set}$ is the left Kan extension of $\Delta \{ *\}:\operatorname{Elts}W \to \mathsf{Set}$ along the projection $\operatorname{Elts}W \xrightarrow{\pi} \mathbf{J}$.

Also $\operatorname{Elts}W$ happens to be the comma category of the diagram $\mathbf{1} \xrightarrow{\{*\}}\mathsf{Set} \xleftarrow{W} \mathbf{J}$.

The weak pullback property of the comma category sounds a lot like the Kan extension property. (See here, where I have asked the question with better diagrams than I can draw here.)

1. Does the weak pullback property of the comma category implies the left Kan extension property in this case?

2. Given functors $\mathscr{A} \xrightarrow{F} \mathscr{B} \xleftarrow{G} \mathscr{C}$ (between small categories), is it always the case that $G$ is the left Kan extension of $F\downarrow G \xrightarrow{\pi}\mathscr{A} \xrightarrow{F}\mathscr{B}$ along $F \downarrow G \xrightarrow{\pi'}\mathscr{C}$?

A reference would be helpful, if the full story is hard to tell.

Answer 1

Regarding the second question, the answer is no for a somewhat trivial reason: any left Kan extension of a functor picking out initial objects is a functor picking out initial objects. This is because a functor picking out initial object is a left Kan extension of the empty diagram, and so the left Kan extension of that is still a left Kan extension of the empty diagram.

---

A less trivial reason is the formula for computing pointwise left Kan extensions. I will refer to the following diagram of comma categories.

$$\require{AMScd} \begin{CD} (\Pi_G\downarrow B_0) @>>> \mathbf 1\\ @V\Pi_{\Pi_G}VV \overset{\tau^{B_0}}\Rightarrow@VB_0VV\\ (F\downarrow G) @>\Pi_G>> \mathbf B\\ @V\Pi_FVV \overset\tau\Rightarrow @VVGV\\ \mathbf A @>F>>\mathbf C \end{CD}$$

The colimit formula for pointwise left Kan extensions says that, if for each object $B_0$ in $\mathbf B$ a colimit of $(\Pi_G\downarrow B_0)\xrightarrow{F\Pi_F\Pi_{\Pi_G}}\mathbf C$ exists, then a choice of these colimits assembles into a left Kan extension of $(F\downarrow G)\xrightarrow{F\Pi_F}\mathbf C$ along $(F\downarrow G)\xrightarrow{\Pi_G}\mathbf B$, and that if the left Kan extension is pointwise (i.e. preserved by representable functors when $\mathbf C$ is locally small), its values have to be given by those colimits.

Let us compute and compare these colimits with $GB_0$.

1. An object of the category $(\Pi_G\downarrow B)$ consists of an object of $(F\downarrow G)$, which is to say a morphism $FA\xrightarrow{h}GB$ in $\mathbf C$, together with a morphism $B\xrightarrow{g} B_0$ in $\mathbf B$.

2. A morphism in $(\Pi_G\downarrow B_0)$ consists of a pair of morphisms $A_1\xrightarrow{f}A_2$ in $\mathbf A$ and $B_1\xrightarrow{g}B_2$ in $\mathbf B$ making the diagrams $$\begin{CD} FA_1 @>h_1>> GB_1 @. B_1 @>g_1>> B_0\\ @VFfVV @VGgVV @VgVV @|\\ FA_2 @>h_2>> GB_2 @. B_2 @>g_2>> B_0 \end{CD}$$ commute. Note that every object $FA\xrightarrow{h}GB,B\xrightarrow{g}B_0$ has a morphism $$\begin{CD} FA @>h>> GB @. B @>g>> B_0\\ @VF\mathrm{id}_AVV @VGgVV @VgVV @|\\ FA @>Gg\circ h>> GB_0 @. B_0 @>\mathrm{id}_{B_0}>> B_0 \end{CD}$$ to the object $FA\xrightarrow{Gg\circ h}GB_0,B_0\xrightarrow{\mathrm{id}}B_0$.

3. A cocone under the diagram $(\Pi_G\downarrow B_)\xrightarrow{F\Pi_f\Pi_{\Pi_G}}\mathbf C$ with vertex $C$ consists of morphisms $FA\xrightarrow{(h,g)}C$ indexed by pairs $FA\xrightarrow{h}GB,B\xrightarrow{g}B_0$ and compatible with the above morphisms.

4. If limiting cocones exist for each $B_0$ of $\mathbf B$, then the components of the unit $F\Pi_F\Rightarrow \mathrm{Lan}_{F\Pi_F}\Pi_G$ are the morphisms $F\Pi_F(FA\xrightarrow{h}GB_0)=FA\xrightarrow{\eta_h} \mathrm{colim}\Pi_G\Pi_{\Pi_G}$ given by the components $FA\xrightarrow{(h,\mathrm{id}_{B_0})}\mathrm{colim}\Pi_G\Pi_{\Pi_G}$ of the limiting cocones.

5. If limiting cocones exist for each $B_0$, then the action of the left Kan extension on morphisms is determined by the fact that post-composition with a morphism $B_0\to B'_0$ gives a functor $(\Pi_G\downarrow B_1)\to(\Pi_G\downarrow B_0)$ which makes the limiting cocone over the $B_0$ diagram a cocone over the $B_1$ diagram, hence gives a unique morphism from one vertex to the other vertex.

In the case where $\mathbf A$ is discrete (only identity morphisms), we see that every object $FA\xrightarrow{h}GB,B\xrightarrow{g}B_0$ has a unique morphism to an object with second morphism $B_0\xrightarrow{\mathrm{id}_{B_0}}B_0$. This implies (I think?) that the colimit is actually the coproduct $\bigsqcup_{A\in\mathrm{Ob}(\mathbf A),\mathrm{Hom}_{\mathbf C}(FA,GB_0)}FA$. In particular, if $\mathbf A$ is small and $\mathbf C$ is locally small and cocomplete, these coproducts exist, hence must be the values of a left Kan extension at $B_0$. Such a value could also be written as $\bigsqcup_{A\in\mathrm{Ob}(\mathbf A)}\mathrm{Hom}_{\mathbf C}(FA,GB_0)\times FA$ where $\mathsf{Set}\times\mathbf C\xrightarrow{-\times-}\mathbf C$ is the copower.

In particular, the left Kan extension is indeed $\mathbf B\xrightarrow{G}\mathsf{Set}$ in the case of $\mathbf A\xrightarrow{F}\mathsf{Set}$ being $\mathbf 1\xrightarrow{\{*\}}\mathsf{Set}$, but it certainly does not have to be $G$ as $F$ varies. I think you can also check that the unit of the Kan extension is indeed the tautological natural transformation of the comma category.

---