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I put the theorem about presheaves on overcategories and overcategories of presheaves that had its own page at functors and comma categories into the Properties-section at category of presheaves: Presheaves on over-categories and over-categories of presheaves.
Then I added the analogous proposition for (oo,1)-presheaves at (∞,1)-category of (∞,1)-presheaves -- Interaction with overcategories
Incidentally, there is some bug on the nLab that might be related to the one Toby just pointed out in the thread on scrollboxes: Trying to put links to subsections of nLab entries into nLab entries is often troublesome. The Markup-code for links gets mixed up by the hash-sign, usually. Then usually the html-code will work. But at the moment at category of presheaves I cant get that to work either...
What is a good reference for learning this stuff from a physicist’s perspective? Say, something similar to Nakahara or Frankel for differential geometry?
Sorry, the nLab is not yet there yet :)
There is no such thing as a physicist's perspective on this. It is the most basic of the basic of category-theory stuff. I mean, you could ask a biologist for his views on relativity, and he might even be able to give you a decent answer, but you'd just be getting a second-hand source for the information.
You can read Mac Lane or Kashiwara and Schapira (I think the latter would probably be more on-point, since it is called "categories and sheaves"). Vistoli's notes also go over this stuff in chapters 1 and 2, which I suggested earlier.
Eric, I'm going to be frank with you here, but I think that you're wasting your time doing this the way you are. If you really want to learn mathematics (category theory being only a part of it), you have to start from the basics. You know a lot of smart guys from this community who have probably forgotten that category theory being a comfortable place to work is based on being comfortable with other algebraic constructions. If you're serious about this, learning about presheaves and overcategories and presheaves on overcategories is not the right thing to be doing.
I don't know how comfortable you are with algebra, but I would like to suggest the following books to you. If any of them is too basic for you, you should move on.
Algebra by Michael Artin
Algebra by Serge Lang
Algebra books ch 1-3 and ch 4-7 by Nicolas Bourbaki
The first book is a book for undergraduates, so you may not need to read it, but the other two books are very serious. I can guarantee to you that you stand to learn a lot more from them (at your current level of experience) than any book on category theory. Note that Lang actually introduces category theory at the end of chapter 1 and uses it throughout. If you can get through a pretty sizable portion of this reading, then I would say that you're ready to read what Urs is talking about. You seem to be completely averse to learning the skills that mathematicians consider basic and fundamental, instead opting for constructions that are useless on their own. That's your perogative, but the only way you'll be able to retain any of it is to understand what's really going on.
Edit: Thanks Harry. This comment was written when your previous comment consisted of only the first two paragraphs :)
For some reason, declaring something is “basic” doesn’t seem to make it much clearer to me :)
What I mean is, let’s assume I don’t like mathematics enough to learn it for its own sake (shock horror). I’m really interested in physics. I’ve got enough math kung fu to get through, say, the first halves of both Nakahara and Frankel. I recognize that sheaves come up enough in the literature on stuff I am interested in, e.g. smooth spaces, that it seems prudent to try to learn about them. I’ve stared at page 17 and 18 of my homework assignment for at least 4 train rides so far and still feel like I’m not much closer to getting it.
I’m sure the concept is really basic and I’m sure I already do understand it. It’s just I haven’t found the Rossetta stone to translate the terminology to concepts I know already.
For a specific example, let’s say I know what a manifold is. Is it possible to describe a manifold using lingua sheafia?
Is it possible to describe a manifold using lingua sheafia?
Yes, in several ways. You can look at them as sheaves on CartSp satisfying a covering condition, you can consider them as topological spaces with structure sheaves, you can construct them as locally ringed grothendieck toposes (small and large, depending on your taste).
However, all of these other descriptions require you to have a better understanding of descent. Descent is an important idea in geometry that is based on this language of category theory, but to understand category theory, you really need some experience in modern algebra.
What I mean is, let's assume I don't like mathematics enough to learn it for its own sake (shock horror). I'm really interested in physics. I've got enough math kung fu to get through, say, the first halves of both Nakahara and Frankel. I recognize that sheaves come up enough in the literature on stuff I am interested in, e.g. smooth spaces, that it seems prudent to try to learn about them. I've stared at page 17 and 18 of my homework assignment for at least 4 train rides so far and still feel like I'm not much closer to getting it.
If you're not willing to put time into understanding the fundamentals from mathematics, you will never be able to learn the much more difficult and abstract stuff arising from physics. There are books about mathematics for physicists, but most people here in the n-Community understand mathematics as mathematicians (even the people with doctorates in physics), not as physicists. The question here is: Do you want to understand things like Urs/Zoran or like Ian Durham? If you want to understand things like Urs or Zoran, you need to engage the material with a mathematical perspective, which is sometimes more difficult, but vastly more rewarding. If you want to understand things like Ian Durham, you can avoid all of this and not actually learn anything.
Thanks again Harry.
Yes, in several ways.
That is nice. I hope to get to the point where those examples make sense.
However, all of these other descriptions require you to have a better understanding of descent.
I’ll have a look at Vistoli and see where I’m missing more fundamentals and work backward from there.
If you’re not willing to put time into understanding the fundamentals from mathematics, you will never be able to learn the much more difficult and abstract stuff arising from physics.
I’m willing! :) But if I am going to commit to something, I’d like it to be a little more focused than covering a complete undergraduate maths curriculum. I admit to struggling with abstract algebra. It was one of the most challenging courses I’ve ever taken and I, unfortunately, didn’t come out with a very good understanding of the basics despite an entire semester of bashing my head against the wall.
As I stated on the very first revision of my personal wiki web:
My ultimate goal is to follow what others at the nLab are doing and reformulate it in a way I can understand (which usually means that I need to “discretize” everything). In particular, I would like to one day understand space and quantity and be able to understand how
$discrete space \Longleftrightarrow discrete differential forms$works (if it does).
Note: Something has changed in the math formatting and it appears I cannot (or I don’t know how to) quote equations properly anymore.
I hope to understand this duality between spaces and algebras. So I’m willing to put in whatever effort is necessary to get me there, but I’m hoping the course can be somewhat focused.
I guess you mean well, Harry…
You know a lot of smart guys from this community who have probably forgotten that category theory being a comfortable place to work is based on being comfortable with other algebraic constructions.
:-D :-D :-D
(Maybe you’ve forgotten that some smart guys from this community either are or have been teachers, at many different levels?)
If you want to understand things like (name omitted), you can avoid all of this and not actually learn anything.
That is an incredibly rude thing to say.
Do you believe that by directly insulting other people, you make your point effectively? I mean, imagine if you were a teacher, and you pointed your finger at someone in the class as an ignorant person, and someone to avoid emulating. How sympathetic would the class be to that opinion? How effective a teaching strategy would that be?
I see the point you’re trying to make regarding prerequisites, and on the whole it’s a good point. I would give similar advice to anyone who wants to study categories seriously: in order to place category theory in context, one should have ideally absorbed a ton of math. The specific references you give are very fine for this purpose. Of course, in order for someone like Eric, who works for a living and has a family, to learn by following the path you are suggesting, he needs to be convinced that this path really is the most efficient way to get where he wants to go. That sort of conviction (and fascination with the subject matter) is prerequisite to all, and I doubt browbeating will help him much.
I'm willing! :) But if I am going to commit to something, I'd like it to be a little more focused than covering a complete undergraduate maths curriculum. I admit to struggling with abstract algebra. It was one of the most challenging courses I've ever taken and I, unfortunately, didn't come out with a very good understanding of the basics despite an entire semester of bashing my head against the wall.
Try Artin's book. Some of the material is more important than other parts, but it's an excellent introduction that has a lot of interesting special topics that are related to physics and geometry (the geometry of SU(2), which is important in physics, for instance). Artin also focuses a lot on matrix groups/linear algebraic groups and much less on the permutation groups and other finite groups.
You're too fixated on doing things "your way" because you never learned how to do things the "right way". Things that should be easy appear to be extremely difficult for you because you're learning them with the improper background. Mathematics is a rich subject that branches out in many directions and converges again to build a magnificent interconnected edifice, but this entire structure is built on top of a basic knowledge of algebra and topology, which are where you can develop your algebraic and geometric intuition. It seems like you're trying to jump into fields that require a solid understanding of algebra without that knowledge, so you're forced to try to get people to translate algebra into geometry. One is not a substitute for the other. You need both.
Edit: To address Todd's last paragraph (since it is relevant to this post), it seems like Eric's heart is in the right place, and that he really is interested in the subject matter. I just question some of the guidance that people here give him some of the time. If someone has obvious deficiencies in his knowledge of algebra or topology, I would be very careful to explain that learning category theory (let alone higher category theory) will probably not be a fruitful endeavor. I would instead suggest that he go back and resolve those issues before continuing. I don't think that telling him to read papers on presheaves and sheaves is going to actually improve his knowledge, and I'm sure that it's really frustrating for him to read the same page over and over again, progressing only when he's willing to ask a question here.
You know a lot of smart guys from this community who have probably forgotten that category theory being a comfortable place to work is based on being comfortable with other algebraic constructions.
:-D :-D :-D
(Maybe you've forgotten that some smart guys from this community either are or have been teachers, at many different levels?)
They may be teachers at many different levels, but I don't understand why people are encouraging someone to learn something from which he stands to gain very little. I disagree and instead think that Eric would be much better served by developing a firm understanding of the foundational material (I don't mean the foundations of mathematics, which have very little to do with actual mathematics. I mean material that is foundational for mathematicians working in the majority of fields).
If you want to understand things like (name omitted), you can avoid all of this and not actually learn anything.
That is an incredibly rude thing to say.
Do you believe that by directly insulting other people, you make your point effectively? I mean, imagine if you were a teacher, and you pointed your finger at someone in the class as an ignorant person, and someone to avoid emulating. How sympathetic would the class be to that opinion? How effective a teaching strategy would that be?
I feel like hiding my contempt for Ian at this point would be counterproductive. He's just such a good example of what you can become if you don't learn foundational material.
One day, a PhD advisor will be considering potential graduate students and will find their way to the n-Forum. There are clearly documented cases of students/teachers being rejected for consideration due to comments made on the web. Just saying…
PS: As a side note, I was once interviewing a student for a summer internship (at a reputable firm). Before the interview I “Googled” the person and found their Facebook page. On it they had a video posted of them in Cambodia with a group of militiamen he and some friends hired to let him fire a rocket launcher “for fun”. He was clearly stoned in the video and shot a rocket into the side of a mountain. I didn’t hire him :)
I have no respect for Ian Durham because he is incompetent and uses his position to intimidate people and prevent them from articulating their criticisms of his (rather limited and often incorrect) knowledge. There, now when they read this page, they'll hear the whole story.
I disagree and instead think that Eric would be much better served by developing a firm understanding of the foundational material (I don’t mean the foundations of mathematics, which have very little to do with actual mathematics. I mean material that is foundational for mathematicians working in the majority of fields).
Sure, I knew what you meant.
And that’s a perfectly valid opinion. I think in fact it’s quite a good point, and by making it more focused it gets better. I wouldn’t say, “Here, read Lang’s Algebra and the first seven books of Bourbaki’s Algebra”; I would instead suggest specific points of entry (as you have done elsewhere), adapted to current interests and fascinations. No one learns such masses of material by reading linearly.
I feel like hiding my contempt for Ian at this point would be counterproductive. He’s just such a good example of what you can become if you don’t learn foundational material.
Oh for chrissake. Take your contempt for Ian and stuff it, Harry. IT DOESN’T BELONG HERE!
Oh for chrissake. Take your contempt for Ian and stuff it, Harry. IT DOESN'T BELONG HERE!
**Contempt stuffed
And that's a perfectly valid opinion. I think in fact it's quite a good point, and by making it more focused it gets better. I wouldn't say, "Here, read Lang's Algebra and the first seven books of Bourbaki's Algebra"; I would instead suggest specific points of entry (as you have done elsewhere), adapted to current interests and fascinations. No one learns such masses of material by reading linearly.
Well, I didn't know how comfortable Eric was with algebra. I suggested three books, the first of which is an undergraduate textbook. I suspected that it would be the best fit, but I didn't want to be condescending in case I misjudged how comfortable Eric was with the subject, so I included two books that are substantially harder.
Artin's book seems like it will be the right level of depth, and it also discusses topics that are important for physicists like Minkowski spaces, representation theory, Lie groups, classical groups, etc. I would suggest reading through pretty much the whole book (when I was typing this out, I was looking through the book, and every chapter is useful and interesting except chapters 1 and 3, which are reviews of matrix computations and linear algebra respectively. At the very least, they're worth skimming to make sure that everything looks familiar).
If you're looking for sheer amount of stuff covered, Lang is definitely your best bet. It covers the majority of stuff in Atiyah-MacDonald, a lot of stuff about algebraic geometry, homology/cohomology, a very nice abstract approach to linear algebra over arbitrary CRings, Representation theory, etc.
If you're looking for clarity of presentation and comprehensiveness of coverage, Bourbaki trumps every other book on the subject (this is also the case with Bourbaki's Topologie Generale, Algebre Commutative, and Groupes et Algebres de Lie). If you can't understand a proof in another book, there is probably a better one in Bourbaki.
I feel a little guilty because I told Urs I wouldn’t, but after 4 train rides, it became obvious I needed a little more background material to even begin answering his trainwork problem. So… I had a look at Vistoli gasp
I didn’t know what a scheme was, so I skipped that part, but the brief touch on category theory was very nice. I was VERY happy to see his use of $F^*$ and $F_*$. I am a very sensitive to notation and I like this notation. I only just started Chapter 2, but like what I saw.
Yeah, any of the stuff about schemes has to do with algebraic geometry.
Another thing, Vistoli has a habit of simply calling presheaves "contravariant functors" or sometimes even just "functors", but it is always clear from context when he means it to be a presheaf.
What is a good reference for learning this stuff from a physicist’s perspective?
Have a look at the expository articles by Bob Coecke that are now listed in the References-section at higher category theory and physics
Sorry, the nLab is not yet there yet :)
i know very well. The nLab isn’t yet where I would like it to be by far and in many areas and aspects.
I can’t invest much (more) energy into expositional stuff at the moment. I am hoping at some point we are lucky and somebody joins us who does. But maybe we can use constructive question-and-answer games on the forum to slowly but surely jointly produce useful expository content for suitable nLab entries.
But maybe we can use constructive question-and-answer games on the forum to slowly but surely jointly produce useful expository content for suitable nLab entries.
Sure sure. You can count on that. And I’ll try to “bring up the rear”. As I learn, I’ll pay it forward :)
Have a look at the expository articles by Bob Coecke that are now listed in the References-section at higher category theory and physics
Wow! Thanks. I haven’t even skimmed them, but this seems to be precisely what I was hoping for.
Believe it or not, this is really the first time since 2004 I’ve been able to spend ANY time actually studying ANYTHING non-work related.
What would be REALLY great is if we could get Professor Coecke to come here and help with the nLab. I think one of his students started, but not sure what happened.
I think one of his students started, but not sure what happened.
Yes, and I met him a few weeks back in Oxford, and he seemed determined to continue. But I don’t know what happened after that, either.
@Urs: I feel like you're misjudging Eric's level of experience and giving him bad advice for that reason. It doesn't affect me, but I think you should reconsider your reading suggestions.
In Vistoli, there is a statement:
Let $\mathcal{C}$ be a category. Consider functors from $\mathcal{C}^{op}$ to $(Set)$. These are the objects of a category, denoted by
$Hom(\mathcal{C}^{op}, (Set)),$in which the arrows are the natural transformations. From now on we will refer to natural transformations of contravariant functors on $\mathcal{C}$ as morphisms.
Let $X$ be an object of $\mathcal{C}$. There is a functor
$h_X : \mathcal{C}^{op}\to (Set)$to the category of sets, which sends an object $\mathcal{U}$ of $\mathcal{C}$ to the set
$h_X U = Hom_\mathcal{C} (\mathcal{U}, X).$I guess he is talking about representable functor. What would be an example of a non-representable functor $\mathcal{C}^{op}\to (Set)$?
PS: Sorry for the bad math mode. Not sure what is going on.
Yup, exactly. The first thing he's talking about is the presheaf category [C^op,Set], and then he introduces representable functors. The notation h_X is pretty much standard for representables.
@Eric: An example of a functor that isn't representable depends on your base category C.
Here’s an example of a non-representable functor that works every time: take every object $c$ of $C$ to the empty set. If you think about it, it’s clear this can’t be representable.
Thanks Todd. That should have been obvious to me, but makes perfect sense now that you said it.
Here’s another non-representable functor with more interesting content:
Take a compact Lie group $G$ and let $Diff_{fd,c}$ be the category of finite dimensional compact manifolds and smooth maps. Let $H^1(-,G): Diff_{fd,c}^{op} \to Set$ be the functor which sends a manifold in the category to the collection of isomorphism classes of $G$-bundles. A map $M \to N$ induces the map $H^1(N,G) \to H^1(M,G)$ by pullback. This is not (always) representable. To be concrete, take $G = SU(N)$ or something like that. The only cases I could think of when it might be representable involve the construction of classifying spaces of finite groups: say take a finite subgroup of $PSL(2,\mathbb{C})$ that acts freely on the upper half plane of $\mathbb{C}$.
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