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I think the line between the two types of Kan extension (weak versus pointwise) is drawn at the wrong place. Am I missing something?
I tried to respond to Mike's now rather old observation about the 'local' definition of Kan extensions in terms of universal transformations. The entry claimed that this definition was of 'weak' extensions, contrasted with 'pointwise' ones, but that was misleading at best -- the 'global' extensions are also 'weak' in Kelly's sense, as the warning lower down says.
Maybe it would be better to forget the global/local distinction and concentrate on weak/pointwise.
I've reorganised Kan extension to better distinguish (I think) between weak and pointwise extensions.
I left the 'in terms of (co)ends' header in because there is an {#anchor} there and I don't know if anything else links to it.
Thanks.
Though I am wondering: the term "global Kan extension" seems to be much more descriptive than "weak Kan extension".
In fact, I find the term "weak Kan extension" quite misleading as it makes me think of higher-categorical Kan extension (similary for weak limit).
But if it's the established terminology, so be it.
I agree that 'weak' is not a good name, because of the potential for confusion with 'weak' as in higher categories or with 'weak' connoting non-unique existence.
I think the global/local distinction is not an essential one in that it's a special case of that for adjoints. It's probably least ambiguous to call non-pointwise extensions just that -- 'non-pointwise', unless anyone has a more suggestive term. I've edited the entry a bit along these lines.
Thanks everyone. I tried to clarify the terminology and organization a bit, and added some comments about geometric morphisms to the discussion of notation. I also removed an old query box.
added to Kan extension the statement and proof (one out of many) that left Kan extension along $F$ is given on representables by $F$. In a new section Properties.
What the following sentence mean ?
The upshot is that if a pointwise extension exists then it is the same as the corresponding non-pointwise extension, but the converse does not always hold.
I can not imagine a converse to the statement written. I mean logically the converse would be if it is the same that it exists, but how can be the same if it does not exist. Probably somebody wanted to say that some extensions are not pointwise, but this is not a converse.
I’m pretty sure I wrote that, and that I meant it to be parsed as
if a functor is the pointwise extension of blah along bluh, then it is the non-pointwise extension, but …
If you think it’s unclear, then you know what to do… (perhaps you could just delete the phrase ’the same as’).
I have added at Properties also the statement and the similar proof that for $F$ full and faithful we have $Lan_F \circ F^* \simeq id$.
Hey, by the way: What in the world does this symbol mean: ⋔
@Urs: I’ve got another fun property:
Let $F:C'\to C$ be a fully faithful functor whose essential image is a sieve (resp. cosieve) in $C$. Then if $M$ is a category with a terminal object $1$ (resp. initial object $0$) , then the the right (resp. left) kan extension along $F$, $F_*$ (resp. $F_!$) is precisely the extension by $1$ (resp. the extension by $0$).
Also, does anybody else here think that the notation “⋔” for cotensoring sucks?
I’ve been trying to find a good notation for cotensors for years. The only advantage to $\pitchfork$ that I can see is that once you know what it means, you’re unlikely to be in danger of confusing it with anything else.
@Zoran: If we slightly rephrase that sentence as “…if a pointwise extension exists, then so does the non-pointwise extension and the pointwise extension is the non-pointwise one” (which is equivalent, since saying that something is a pointwise extension necessarily entails the existence of a pointwise extension) then its converse would be “if a non-pointwise extension exists, then so does the pointwise extension and the non-pointwise extension is the pointwise one” which is meaningful and false.
@Urs: I’ve got another fun property:
You are kindly invited to add whatever insight you think is worth recording to the $n$Lab.
For instance the facts that I just added are elementary and standard, but I kept referrig to them elsewhere in some computations and so I found it indicated to explcitly record them and link to them, for the convenience of whoever might search around the $n$Lab.
@Mike: If you want to make the notation stick, you should really give an explanatory note on notation at least once per page. Since it is impossible to search the nlab for symbols (as far as I can tell), you should say something like “where ⋔ means the cotensor”.
Before chaning it, I would still discuss it:
Not all Kan extensions in the universal-transformation sense defined above are pointwise Kan extensions, i.e. computed as weighted (co)limits.
Pointwise is a property, in fact pointwise is defined in MacLane as such Kan extensions which is preserved under all corespresentable functors. If it is pointwise than it can be computed via the colimit formula; in fact the converse of that fact is true, but that point of view is less enlightening (the formula does not tell you that it is the same when it exists, while the property of preservation makes the sameness obvious). So complicated converse/upshot statementsin the whole paragraph are in my view at least superfluous (not only confusing). Please correct me if I am factually wrong here with the meaning of pointwise. I’d be happy to change the pragraph if my point of view is judged positively by others ?
By the way, Urs cites Barwick and Kan as a recent result that the categories with weak equivalences give after localization all (infinity,1)-categories. Joyal’s 2008, 2007 and some other notes quote this as alerady well known fact without mentioning any author, not as a new discovery in last 3-4 years. On the other hand, what about 2 out of 6 property, what if I get out of that ? I mean derived functors still make sense, while it is my impression that those with 2 out of 6 are enough for all (infinity,1)-categories. Are we possibly getting out of (infinity,1)-setup with looking at more than the localizers with 2 out of 6 ?
Rosenberg has some unpublished work in progress on homological algebra based on fibered categories instead of categories with weak equivalences, (co)fibrations and alike choices. Something like fiber over a point in a fibered category is a generalization of all homotopy-sense fibrations into that object. Then chasing cartesian diagrams replaces chasing elements in usual homological algebra. I am really puzzled with what is the minimum to make sense of derived functors as this setup is also enough, if certain limits exist.
Harry,
notice that the notation is in Kelly’s Basic concepts of enriched category theory hence not something exotic that Mike is promoting. Secondly, I believe in the same time it takes you to tell Mike to add a limk to some page you would have added that link yourself.
@Harry: To add to what Urs said — I don’t like the $\pitchfork$ notation and I have no desire to “make it stick.” I was just commenting on the sole advantage to it that I know of, mainly to draw attention to the fact that I can think of no other advantages to it! That said, it is of course always good to clarify the meaning of notation, whoever uses it.
that point of view is less enlightening
I disagree. I think that the defining property of pointwise Kan extensions is that they are computed by a colimit pointwise, and the fact that they turn out to also have the universal property of non-pointwise Kan extensions is a consequence. I think it’s not clear that non-pointwise Kan extensions have any real use, or that they should have been dignified with the unadorned name, thereby requiring us to add the adjective “pointwise” to characterize the interesting notion.
When I say that it is less enlightening is that the very definition does not make it obvious that they agree with the usual Kan extensions. If something is a property it is simpler to define it as a property, rather than changing the definition. No concept whatsoever has a use without knowing in addition main properties/theorems/formulas, so of course one needs to know its uses via the formula via the colimit. In any case, the way it is written in the nlab entry is very confusing to me, linguistically and mathematically.
Mike, if you want to make a point out of most important case, then one should work (in the theory of derived functors) indeed with absolute Kan extensions, which are preserved under ALL functors. This is stronger than pointwise, and almost all examples in homotopy theory are such, and it is simpler to assume them in theorems. Don’t you agree ?
Kelly’s point of view, which I basically agree with, is that “pointwise Kan extension” is the useful notion, and should be defined first, and perhaps even only, rather than defining “non-pointwise Kan extension” and then adding an additional property to it to obtain the pointwise ones. I’m sure the entry could be clarified, however.
What do you mean by “almost all examples in homotopy theory” are absolute Kan extensions? Most Kan extensions that I work with are not absolute. Are you referring to the fact that derived functors have the universal property of a Kan extension? I have yet to be convinced that that fact is particularly important, or a good way to define “derived functor.”
@Mike: Can’t we define an ordinary limit/colimit in terms of a non-pointwise Kan extension? If so, then I disagree with Kelly, since we could just define everything in sight to be a particular kind of not-necessarily-pointwise Kan extension, then say that a Kan extension is pointwise iff it decomposes as one of these other kinds of simpler Kan extensions iff it has the preservation-by-representables property.
At least, that seems pretty reasonable to me.
Are you referring to the fact that derived functors have the universal property of a Kan extension?
I am referring to (maybe my wrong impression) that most derived functors are in fact absolute Kan extensions.
@Harry: Yes in the unenriched case, no in the enriched case.
@Zoran: I did not know that most derived functors were absolute Kan extensions; where can I find a proof? But regardless, as I said, I have yet to be convinced that the fact that most derived functors are Kan extensions (of any sort) is a useful point of view.
Let me clarify/qualify what I said about derived functors: it can sometimes be useful to know that they are Kan extensions, but what I don’t believe is that that property is a good definition of “derived functor.” I realize now that this actually doesn’t have anything to do with what you were saying; I jumped too hastily onto an old argument. You were just saying that if derived functors are actually absolute Kan extensions, not just Kan extensions, then we should say so and work with them that way — right?
I did not realize that derived functors were absolute Kan extensions. But now that you point it out, I can see that the standard proof that they are Kan extensions works just as well to show that they are absolute. I guess that means that they are in particular pointwise, since they are then preserved by representables (as well as all other functors)? For some reason I thought derived functors were not pointwise Kan extensions (perhaps because the homotopy categories in which they land do not in general have limits and colimits), but I guess actually they are something stronger than pointwise!
But now that you point it out, I can see that the standard proof that they are Kan extensions works just as well to show that they are absolute. I guess that means that they are in particular pointwise, since they are then preserved by representables (as well as all other functors)?
Right, if you start with some other definition of derived functors in a more concrete setup (than in the full generality of a localizer where you have nothing than abstract usual Kan), than you typically can prove that you in fact have an absolute extension (and in particular pointwise). Now once you know from the start that they are in fact absolute that property may possibly simplify some later proofs which may be more difficult if this were unnoticed. You will probably come easier than me with the examples of the latter phenomenon.
I’d think the punchline is that there is not one definition of Kan extension more fundamental than the other, but that the complex of concepts going by global/pointwise Kan extension, limits/colimits, representable functors and adjoint functors all reflect on each other, are all useful in some context with none of them being really more fundamental than the other but rather all of them being facets of one single phenomenon (which maybe is conceivable only through these facets and not by reduction to one single fundamental notion).
I’d agree with that Urs! but I also think that it iis very useful that in practice some versions are more typical than others. Such emphasis is useful to know.
BTW, Urs I am coming to Bonn on March 2, I count on meeting you few times within March…let us email on the details these days. This is the first time that I got the permission to leave Zagreb to Germany after Spetmeber 2009, so I can dedicate for the first time to things we planned before to finish.
I’ve tried to clarify/correct the statements at Kan extension and derived functor.
I am playing with the thought of bringing the entry Kan extension into a more systematic state that might allow using it alongside a course. Not sure if I find enough energy, though, but I’ll have a try.
The first thing that struck me was that the current Examples-section must give a weirdly inappropriate impression to any layman looking at it in hope of getting an idea on what’s going on. As a first step towards improving on this situation I decided to add here and in the other entries on universal construction the following public announcement
The central point about examples of Kan extensions is:
Kan extensions are ubiquitous .
To a fair extent, category theory is all about Kan extensions and the other universal constructions: limits, adjoint functors, representable functors, which are all special cases of Kan extensions – and Kan extensions are special cases of these.
Listing examples of Kan extensions in category theory is much like, say, listing examples of integrals in analysis: one can and does fill books with these. (In fact, that analogy has more to it than meets the casual eye: see coend for more).
Keeping that in mind, we do list here some special cases and special classes of examples that are useful to know. But any list is necessarily wildly incomplete.
second step in my attempt to clean up the entry: I reorganized the lead-in to the Definitions-section. See here. Trying to give a non-confuing list of the various different notions. Check if you agree.
third step: touched the section on global Kan extensions. Spelled out the argument that Kan extensions to the point are (co)limits.
@Urs Whilst I would not want a proliferation of Lab pages on related subjects, I have thought for some time that some more explanatory pages would be useful. This would be useful for courses as you are giving, but the usual lab-page entry style is a bit stark for introducing a topic to students so perhaps a separate page with more examples, a bit more philosophy and motivation before the definitions may be useful, not just at Kan extensions but in many places.
Sure. Lots of entries deserve lots of additions and variants. I add stuff as I need it.
Since I have no official teaching duties, I am not teaching much (currently only because I ran into a bunch of students who asked me to lecture them) and therefore most of my additions to the $n$Lab concern my research. But right now I need some more pedagogical bits, so I add them.
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.
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).
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.
@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.
@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:
\[
\xymatrix{
C \ar[rr]^F_{}="1" \ar[dr]_p & & D \\
& C' \ar[ur]_{\Ran F} \ar@{=>}[];"1"^(.6){\epsilon_F}
}
\]
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.
slightly expanded in the Properties section the statement that left Kan extension along full and faithful functors is itself full and faithful
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?
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 remark about left Kan extension along small opfibrations here.
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.)
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?!
@Urs: only if the site has finite limits, I think.
Marc, you are of course right.
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
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?
Yes, that was a very nice answer that Ivan gave. Direct link to the addition: Kan extension#existence.
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 ]
Added
For a treatment of left Kan extensions as ’partial colimits’, see
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