Let $G: \mathcal{A} \to \mathsf{Set}$ be a functor. Recall that the category of elements $\mathsf{el}G$ is defined by the comma square

$\require{AMScd}$ \begin{CD} \mathsf{el}G @>!>> \ast \\ @V U_G VV \Downarrow @VV \Delta_\ast V\\ \mathcal{A} @>>G> \mathsf{Set} \end{CD}

where $\ast$ is used to label the terminal category and $\Delta_\ast$ is the functor that is constant at the terminal set.

But there is another key universal property of this diagram: it exhibits the left Kan extension $ G = \operatorname{Lan}_{U_G} (\Delta_\ast !)$. This fact is key in the reduction of weighted colimits to conical ones (as explained here).

It's easy to verify that this is a Kan extension directly, but it's a bit mysterious, because arbitrary comma squares are not Kan extensions (for example, the same square in $\mathsf{Cat}^{\mathrm{co}}$ doesn't seem to be a left Kan extension). So what is going on here? Why, conceptually (by which I suppose I mean: 2-categorically) is this particular comma square also a Kan extension triangle?


The conceptual explanation is that a comma square exhibits the bottom morphhism as a left Kan extension if the right morphism is dense.

There is, of course, a problem what we mean by a "dense morphism" in an arbitrary 2-category. If we take the usual definition and say that a morphism is dense if its pointwise left Kan extension along itself exists and is the identity, then the above follows almost by the definition of density and the definition of "pointwiseness" in the sense of R. Street (i.e. stability under comma squares).

In your example $\Delta_* \colon 1 \rightarrow \mathbf{Set}$ is obviously dense (in the classical sense) --- therefore its left Kan extension along itself is the identity:

$$\begin{CD} \ast @>\Delta_\ast>> \mathsf{Set} \\ @V \Delta_\ast VV \Downarrow @| \\ \mathsf{Set} @>>\mathit{id} = \mathit{Lan}_{\Delta_\ast}\Delta_\ast> \mathsf{Set} \end{CD}$$

and when you draw the above left extension diagram on the right side of your comma square, then the fact that the bottom morphism is the left extension follows directly from the definition of pointwiseness.

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    $\begingroup$ Of course, the reason why Street's definition of pointwiseness involves comma squares is precisely that it is what works classically... $\endgroup$ – Zhen Lin Sep 27 '15 at 9:39
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    $\begingroup$ The fact that Street's definition of pointwise Kan extension doesn't typically work in the enriched case has me interested in getting a double-categorical perspective on this. There is a definition of pointwise Kan extension in an equipment, which is supposed to specialize to Street's definition in appropriate cases and to Dubuc's definition in the enriched case. $\endgroup$ – Tim Campion Sep 29 '15 at 18:29
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    $\begingroup$ Hi Tim, two quick comments. 1) There is a well-known definition of pointwise Kan extension in 2-categories equipped with proarrows, which dates back to early 80s and is due to R. J. Wood (in fact it mimics the idea from Yoneda structures introduced by Street and Walters in 70s). This definition can be restated in the language of double-categories. (cont...) $\endgroup$ – Michal R. Przybylek Sep 29 '15 at 23:04
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    $\begingroup$ 2) In my opinion, the biggest problem with Street's definition of pointwise Kan extension is actually not in the condition of stability under comma objects, but in the fact that the usual definition of comma object usually does not work properly in general 2-categories (it does work in so-called representable 2-categories). $\endgroup$ – Michal R. Przybylek Sep 29 '15 at 23:04

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