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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 herehere).

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?

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?

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?

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Tim Campion
Tim Campion
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Relationship between two universal properties of the category of elements?

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?