Added redirects for absolute Kan extensions.

]]>Added the original reference (Kan’s paper, again).

]]>Added

For a treatment of left Kan extensions as ’partial colimits’, see

- Paolo Perrone, Walter Tholen,
*Kan extensions are partial colimits*, (arXiv:2101.04531)

Included the formulation of pointwise extensions via representing objects, obtained by unfolding the definition via weighted (co)limits.

]]>- Ross Street,
*Pointwise extensions and sketches in bicategories*, arXiv:1409.6427

have equipped the (co)end formulas with pointers to page and verse in Kelly’s text

]]>Just to say that the core (“uncheatable”) statement is recorded (and attributed and proven ) at *complete small category*. (Would make sense to point to that.)

Its relevance, as per that MO discussion, is also highlighted at the beginning of *adjoint functor theorem*. (Would make sense to cross-link with that.)

Finally, notice that Mike has an article all about this topic arXiv:0810.1279. (Would be worth citing here.)

[ edit: found time to add these pointers ]

]]>Yes, that was a very nice answer that Ivan gave. Direct link to the addition: Kan extension#existence.

]]>Transferred a MathOverflow answer.

]]>Let $Y : \text{Cat}^{op} \rightarrow [\text{Cat}, \text{Cat}]$ be the Yoneda embedding. This is a strict $2$-functor of strict $2$-categories. Write $\rightarrow$ for $1$-morphisms and $\implies$ for $2$-morphisms. Note that $1$-morphisms in $[\text{Cat}, \text{Cat}]$ are natural transformations, which again we will write by $\eta : F \rightarrow G$.

Consider $F : C \rightarrow D$ in $\text{Cat}$ and consider $YF : YD \rightarrow YC$, the contravariant Yoneda embedding. Suppose that $$\text{Lan} F$ and $\text{Ran} F$ exist. $\text{Lan} F, \text{Ran} F : YC \rightarrow YD$ are $1$-morphisms (natural transformations); $(\text{Lan} F)_C (K)$ is my notation for the left Kan extension of $K$ along $F$. $\text{Lan} F, \text{Ran} F : YC \rightarrow YD$ are such that

$\text{Lan} F \adjoint YF \adjoint \text{Ran} F$I call that “Kan extensions terms of adjoints”.

$(\text{Lan} F)_C (1_C) \adjoint F \adjoint (\text{Ran} F)_C(1_C)$I call that “Adjoints in terms of Kan extensions”.

I call this “primordial ooze” since neither Kan extension nor adjoint is fundamental.

I notice that ${}_C (1_C)$, and of course $Y$ also occur in the proof of the Yoneda lemma. Can someone clarify the connection here?

]]>Inserted reference for how to compute Kan extensions along Cartesian fibration in the context of quasicategories. (This might rather belong on the page for (infinity,1)-Kan extensions, but it also seemed to fit here as well since it has the proposition proved as a special case.)

Jan Steinebrunner

]]>Marc, you are of course right.

]]>@Urs: only if the site has finite limits, I think.

]]>Ahm, I feel a bit weird about this, but doesn’t this statement immediately imply that over an infinity-cohesive site the shape modality in fact preserves all finite $\infty$-limits?!

]]>added the statement that left Kan extension of a functor into a topos along the Yoneda embedding preserves left exactness; at *Kan extension – Properties – left Kan extension preserving certain limits*.

(This statement must be discussed in other entries, too, i suppose. I have added a pointer to *classifying topos* but we should add more cross-links, probably.)

Added remark about left Kan extension along small opfibrations *here*.

Oops; I’ve just read Urs comment above. I also prefer $f$ as generic 1-cell, but I’d rather have the diagrams with the same notation. I’d happily revert it to $f$’s all around if you disagree

]]>added absolute extensions. Also split preservation to a section, and switched to $\mathop{Lan}_p$ everywhere; the diagrams have $p$’s and I think is better to have a uniform notation throughout the page. What do you think?

]]>slightly expanded in the Properties section the statement that left Kan extension along full and faithful functors is itself full and faithful

]]>I decided to change the name of the functor that we are extending along from “p” to “f”.

I think the notation which is most suggestive to me is like in some references $i : I\to J$ for the morphism along which we extend, because it suggests that it is intuitively like some *small* category of *indices*, or something akin to site, as the most typical case is that we extend something in underlying site to a category of presheaves. Like $i$ induces $i_*$ an dits left adjoint is left Kan extension.

@Urs #36:

It was I who put those diagrams in, ages ago. Maybe a temporary fix would be to convert them to codecogs diagrams (not that I know how to do that, but Joyal has lots of them on his web). Here is the source:

First one:

```
\[
\xymatrix{
C \ar[rr]^F_{}="1" \ar[dr]_p & & D \\
& C' \ar[ur]_{\Lan F} \ar@{=>}"1";[]_(.4){\eta_F}
}
\]
```

Second one:

```
\[
\xymatrix{
C \ar[rr]^F_{}="1" \ar[dr]_p & & D \\
& C' \ar[ur]_G \ar@{=>}"1";[]
}
\quad = \quad
\xymatrix{
C \ar[rr]^F_{}="1" \ar[dr]_p & & D \\
& C' \ar[ur]_{}="2" \ar@{=>}"1";[]_(.4){\eta_F} \ar@/_1.5pc/[ur]_G^{}="3"
\ar@{=>}^{!}"2";"3"
}
\]
```

Third one:

```
\[
\xymatrix{
C \ar[rr]^F_{}="1" \ar[dr]_p & & D \\
& C' \ar[ur]_{\Ran F} \ar@{=>}[];"1"^(.6){\epsilon_F}
}
\]
```

]]>
@Urs. You said:

I am imagining that if more regulars here would add more of their daily notes (course notes or whatever) to the Lab, it would grow more quickly and be useful to a wider audience.

That has been my aim but with the profinite monograph and the menagerie ongoing I am finding it hard to do that.

]]>I realize that I have added one notational inconsistency that I cannot now easily fix:

I decided to change the name of the functor that we are extending along from “$p$” to “$f$”. (Because the former reminds one of projections, whereas the latter should make one think of a generic morphism).

I think I made this change consistently throughout the text, but then I realized that the notation is also in the diagrams there which are included as *pictures* .

Not sure what to do. Of course I could in principle undo my change of notation. But if somebody could instead change the notation in these pictures…?

This is a problem with picture diagrams. We should maybe redo those in another way, one that can be easily edited.

]]>I have added to the section Pointwise extensions by conical colimits an indication of the proof of the conical colimit formula (I say how to construct the universal morphism, but don’t (yet) go through the demonstration that the construction is well defined and that the result has indeed the required universal property).

]]>