Want to take part in these discussions? Sign in if you have an account, or apply for one below
Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.
started a section Introductions to category theory in physics at the woefully imperfect entry higher category theory and physics. So far this contains mostly th expository articles by Bob Coecke.
Great page. As I begin reading Coecke’s “Categories for the Practicing Physicist” (aside from feeling a “little” talked down to in Section 0… in fact I think I have seen this paper before, but put it down after reading about potatos. If it is for physicists, why not start with something a little more meaningful, e.g. evolution of a state vector, or something?), I still don’t think physicists will latch onto this and say “Ah ha!” this is why I should learn category theory.
I know it sounds silly to you guys, but I believe all that effort on functor was not in vain. I honestly do think physicists could “latch on” to the idea of commuting diagrams, so I really do think it is worthwhile trying to restructure some of the very basic notions in category to be a little less abstract. Is there somewhere a catalog of different, yet equivalent, ways to define a category?
What I would like to see is a discussion beginning with directed graphs interpreting the edges as morphisms from one object to another. Rather than just declare associativity and units as axioms (which is 100% fine from a mathematics POV), give some motivation. In my opinion, finding a statement in terms of commuting diagrams would go a long way. I’m sure this is not quite right, but as a start, I’d try something like
“A category is a set of morphisms in which each path admits a commuting diagram.”
This gets you composition and I’m pretty sure this would get you associativity, but I’m not sure if it gets you a unit. Really, from what I’ve learned, this (i.e. commuting diagrams) gets at the heart of what a category is and would be more easily digested by most physicists.
Once this definition in terms of commuting diagrams were ironed out, we could say that the basic “structure” of a category is its commuting diagrams. A functor preserves structure. Hence, a functor preserves commuting diagrams. A natural transformation preserves structure of things that preserve structures, etc.
I can appreciate (and even anticipate) some frustration at the suggestion, but I honestly believe that something like this would open the doors to many potential researchers from the physics community and possibly help bridge a communication divide (believe me, there is one and I’m the poster child).
I’d be all in favor of adding to the Lab whatever exposition of categories is useful for whatever group of people.
I do have to say, though, that the communication divide that you mention seems to go in both directions! :-) I don’t quite see what the alternative definitions of category that you suggest buy one, or why they are more physical. The standard concept of category seems to be so obvious that I don’t quite see what one could find troubling:
thinking of morphisms as processes is a great analogy. So a category is a bunch of states (objects) and a bunch of processes going between these states, and two processes can be carried out after each other and this gives a composition operation on them that does not depend on the order of the composition. And for each state we kee in mind the trivial process that does nothing.
That’s it. Isn’t that very simple and also very intuitively accessible? I (still) don’t quite understand what troubles you about this.
The standard concept of category seems to be so obvious that I don’t quite see what one could find troubling:
True. But you (and most others here) are special. The concept is not obvious to everyone otherwise everyone would be using categories every day in their work. Think about that.
As physicists, we (I use the term “we” loosely) try to get at the essence of something. Axioms are not essence. For some practicing physicists even groups are not 100% obvious. In the old days, e.g. back when I learned about groups, they were presented as 4 axioms. Now, I think of a group as a groupoid with 1 object. It’s not about the number of axioms, but about the essence contained in those axioms.
There is nothing troubling to me about the definition of a category other than the fact that I can understand completely that it is not something others can easily latch on to. Just as some physicists’ eyes begin to glaze over when reading the 4 axioms of a group, those same physicists’ eyes would begin to glaze over even reading the definition of category. Some people just do not deal well with abstraction.
What you say about processes is nice, but it is also a little wordy and doesn’t (in my opinion) get to the essence. What is the essence of a category? If the essence of a category is associativity and unit, fine, but that is a little unsatisfactory to me. I think the essence of a category has more to do with commuting diagrams, of which composition and units are special cases.
Perhaps this is more of a philosophical questions, but “What is the essence of a category?”
As physicists, we (I use the term "we" loosely) try to get at the essence of something. Axioms are not essence. For some practicing physicists even groups are not 100% obvious. In the old days, e.g. back when I learned about groups, they were presented as 4 axioms. Now, I think of a group as a groupoid with 1 object. It's not about the number of axioms, but about the essence contained in those axioms.
This is an example of missing the point, not an example of "finding the essence".
You should notice that there are a bunch of axioms for a category:
Associativity of composition
Left and right unit laws for composition
Hmm.. now if we add another axiom here for groupoids:
All arrows are invertible
Then we start to see what's going on here. Groups have the same axioms! By saying that a group is a groupoid with one object, we're not saving any axiomatic legwork. It just turns out that when we define groupoids axiomatically, they include groups as a special case! This isn't magical or even interesting, it's obvious and a direct consequence of the axioms. There's nothing profound in the statement that a group is a groupoid with one object or that a monoid is a category with one object. This tells us more things about categories and groupoids than it does about monoids and groups.
There is nothing troubling to me about the definition of a category other than the fact that I can understand completely that it is not something others can easily latch on to. Just as some physicists' eyes begin to glaze over when reading the 4 axioms of a group, those same physicists' eyes would begin to glaze over even reading the definition of category. Some people just do not deal well with abstraction.
I don't see why we should be catering to people whose eyes glaze over when reading the axioms for a group. We can't force people to use mathematics who don't want to learn about it.
What you say about processes is nice, but it is also a little wordy and doesn't (in my opinion) get to the essence. What is the essence of a category? If the essence of a category is associativity and unit, fine, but that is a little unsatisfactory to me. I think the essence of a category has more to do with commuting diagrams, of which composition and units are special cases.
No... Urs is right. This is what I've been trying to tell you for the past months. Category theory is an abstraction of the theory of algebraic structures and the homomorphisms between them. Its "essence" has nothing to do with commuting diagrams.
Here's what I think you're doing wrong. Instead of learning things at face value, you're trying to translate them to this completely nonsensical language of commuting diagrams. However, what you fail to realize again and again is that commuting diagrams can be represented algebraically.
You can represent associativity as a commutative diagram, sure, but all that the diagram says is:
a(bc)=(ab)c
whenever a,b,c are composable in that order.
It's rather a large waste of time to draw out the diagram when it's a hundred times easier to understand what is meant by writing the algebraic equation. If you're disappointed that a category is "just" some thing satisfying some axioms about associativity and units, then you should take it up with whoever told you differently. I'm pretty sure that nobody here on the nForum has ever tried to tell you any differently.
Edit: Once again, this comment was written when Harry’s previous comment end wth the paragraph “Then we start to see what’s going on here.” :) Try using preview before posting! Or if you double the length of your comment after posting, make it a new comment so my responses don’t seem so out of place :P
Right right. Once you are familiar with categories, groupoids are a cinch and groups almost look obvious (and to a certain extent not even fun anymore). But that is precisely my point. We haven’t really saved anything in terms of the number of axioms needed (that’s why I said the number of axioms is not important), but I do believe by understanding categories you get closer to the “essence” of a group. So now I’m trying to take that further by getting to the “essence” of a category.
Just as the 4 axioms defining a group are sufficient from a mathematical POV to study group theory, the axioms of composition and units are sufficient for studying category theory. Maybe I’m mistaken, but I think we can do better than that.
This conversation reminds me of the frustration suffered by me and many others before me trying to convince practitioners in computational physics that differentials forms were much more appropriate than vector calculus. The people I was talking to were fluent with vector calculus and nothing I could say would convince them because they had internalized the essence of vector calculus so deeply that any insight gained from differentials forms they already possessed through years of hard-fought experience.
You guys have internalized the essence of category theory so deeply that any insight I might be able to provide via formulating things in (perhaps a more “essential” way) you already possess (obviously). However, even though you have internalized the essence through hard work, I think there should be recognition that it is not easy to internalize this and I think one reason why it is not as easy as it should be (category theory SHOULD be easy) is due to the unnecessarily abstract presentation. I’m trying (without too much optimism but some hope) to find a better way.
Just as the 4 axioms defining a group are sufficient from a mathematical POV to study group theory, the axioms of composition and units are sufficient for studying category theory. Maybe I'm mistaken, but I think we can do better than that.
I don't. There may be a way to make the list of axioms shorter (say by giving an object-free formulation of the definition), but I would say that the standard definition of a category is the "best-possible" definition in terms of clarity.
This conversation reminds me of the frustration suffered by me and many others before me trying to convince practitioners in computational physics that differentials forms were much more appropriate than vector calculus. The people I was talking to were fluent with vector calculus and nothing I could say would convince them because they had internalized the essence of vector calculus so deeply that any insight gained from differentials forms they already possessed through years of hard-fought experience.
You guys have internalized the essence of category theory so deeply that any insight I might be able to provide via formulating things in (perhaps a more "essential" way) you already possess (obviously). However, even though you have internalized the essence through hard work, I think there should be recognition that it is not easy to internalize this and I think one reason why it is not as easy as it should be (category theory SHOULD be easy) is due to the unnecessarily abstract presentation. I'm trying (without too much optimism but some hope) to find a better way.
I would argue that a.) if you have any experience with algebra, the definition of a category is extremely easy to internalize, and that b.) the presentation you continually advocate is significantly more abstract and less satisfying. The fact that you continually fail to understand both of these points leads me to believe that you do not have enough experience with algebra and the axiomatic method to appreciate what category theory is actually about. I hope this doesn't offend your sensibilities, but it seems to me that one of the main reasons you haven't made very much progress in the past few months is because you're fixated on doing things "your way" instead of "the ordinary way".
I would argue that a.) if you have any experience with algebra, the definition of a category is extremely easy to internalize
I don’t doubt the correctness of this statement, but I’d also say the converse is probably true:
If you have experience with category theory, algebra is easier to internalize.
From what I’ve seen (I could be wrong) I don’t think algebra is necessarily a prerequisite for category theory. Algebra is a special case of category theory. More than that, it could be that category theory is more accessible than algebra so studying it first could help you understand algebra and group theory and topology, etc.
As an example, I remember being COMPLETELY BAFFLED by the isomorphism theorems in abstract algebra. I would be interested in seeing these statements now in terms of category theory. I would be surprised if they weren’t just (co)limits somehow or universal arrows or something.
From what I've seen (I could be wrong) I don't think algebra is necessarily a prerequisite for category theory. Algebra is a special case of category theory.
Yes, this is where you are wrong. This is why you're having trouble. Even if it was true that algebra is a special case of category theory (it's not), that still wouldn't mean that learning category theory first would be a good idea.
More than that, it could be that category theory is more accessible than algebra so studying it first could help you understand algebra and group theory and topology, etc.
Sure, lots of things could be true but aren't. There could be a teapot flying around in the asteroid belt, but it's pretty obvious that there isn't.
As an example, I remember being COMPLETELY BAFFLED by the isomorphism theorems in abstract algebra. I would be interested in seeing these statements now in terms of category theory. I would be surprised if they weren't just (co)limits somehow or universal arrows or something.
No, and the fact that you don't understand the isomorphism theorems makes it even more clear that you're not ready to learn category theory.
thinking of morphisms as processes is a great analogy. So a category is a bunch of states (objects) and a bunch of processes going between these states, and two processes can be carried out after each other and this gives a composition operation on them that does not depend on the order of the composition. And for each state we kee in mind the trivial process that does nothing.
That's it. Isn't that very simple and also very intuitively accessible?
Thanks for your comments Ian. I’ve added Steve Awodey’s book to my Amazon wish list and will have a look at some of his papers available online.
one of the things I see as being crucial to getting more physicists on board is to find a way for category theory to be, for lack of a better term, “calculationally useful,” so-to-speak
Yeah, I’m not there yet on this one either. That is why I’m watching Urs and Domenico’s discussion in “path cobordisms” with interest. One thing I wish I understood better was “integration without integration” and “Kantization”. These are both examples of calculating stuff (integrals) using category theory.
@Eric, I hope that you realize that at the end of the day, the computations are still pretty much the same difficulty.
@Ian, the name "abstract algebra" just grates on my ears (one reason why is that it's the title of an awful textbook on the subject [Dummit and Foote]). It sounds amateurish. You should just call it algebra.
Speaking of isomorphism theorems, I always intuitively think of them as mappings. I’m sure the following analogy will make Harry cringe, but he doesn’t have to teach the average (and sometimes below average) undergraduate.
So, start from the simple idea that isomorphisms are one-to-one and onto, which everyone knows. The key to getting students past this relatively simple point, however, is making sure they really understand it. As I’ve discovered, many of them don’t, even though they may claim they do (or even appear that they do in some instances). There are a number of ways to get them to understand one-to-one that usually trigger an “aha!” moment. Simple counter-examples like $f: \mathbb{Z} \to \mathbb{Z}$ where $f(x)=x^{2}$ work well. There are also some semi-graphical techniques including some quirky examples I’ve used involving roads and maps. Onto is sometimes a slightly harder concept for elementary-level students to get, though. I often use examples like (for example) a bag of marbles where the function (or mapping or morphism or whatever) picks a bunch of marbles out of the bag (which serves as the codomain). If it can use all of the marbles (even if there are infinitely many), then it’s onto.
Huh, Ian? The isomorphism theorems have to do with the behavior of kernels and quotients under different categorical operations. You just explained the definition of a bijection.
(Meanwhile, I've always wondered how to define the product of subgroups of a group. Is it the pushout of the inclusions over the group?)
Meanwhile, I’ve always wondered how to define the product of subgroups of a group.
in what category? There are several available.
And as far as Eric’s request goes, almost any talk by John Baez on 2-bundles or higher parallel transport gives a nice conceptual introduction to categories. It’s the best I’ve seen at an informal level. Sorry for not providing a link, I’ve not much time online at the moment.
I think Harry might mean the group generated by products coming from the two subgroups, and yes, one way of constructing it is as the pushout along the two inclusions from the intersection. A simpler definition is just by taking the intersection of all subgroups containing the given subgroups (and noticing that the poset of subgroups is closed under set-theoretic intersection, as a special case of abstract nonsense about closure operators).
@Todd: Yep, that's precisely what I meant.
Is there a name in general for the pushout over the canonical projections from the pullback? It seems like an awfully useful operation.
And I think this is precisely where you guys lose me. Yes, I described a bijection which happens to be a type of isomorphism. But if you want newbies who aren’t as advanced as you to intuitively understand what an isomorphism is, you need to start with a bijection. If they don’t get bijections they’ll never get more general isomorphisms.
That's not what Eric is talking about though. There are three isomorphism theorems for the common structures in algebra dealing with kernels. In the case of groups, these are normal subgroups. In the case of rings, these are two-sided ideals. For modules, these are submodules, etc.
I think my point is that this thread was about introducing categories and such to physicists. Eric expressed the fact that he used to have trouble with the isomorphism concept and Eric is a physicist. So I was making the point that you have to start with the “bare bones” with folks like me and Eric and others before you start jumping in with kernals and quotients and pullbacks (the former two I know fairly well, but the latter I know very little about).
Remember that you can’t assume that any physicist interested in category theory is necessarily going to even be familiar with groups, rings, and fields from the same language standpoint, so-to-speak. For some, their only exposure to groups, for instance, will be through a physics class on mathematical methods for physicists. If you look through typical texts for such a course, they don’t all delve into things like the three isomorphism theorems. But they do use the term ’isomorphism’ and usually treat it synonymously with bijection, at least in my experience.
OK, we’re crossing in cyberspace. I see now what you meant. Eric had trouble with, specifically, the three isomorphism theorems of groups, rings, fields, etc. In that case, nevermind.
the name “abstract algebra” just grates on my ears (one reason why is that it’s the title of an awful textbook on the subject [Dummit and Foote]).
That’s funny. The two references I used were (the official text for the class) Herstein “Abstract Algebra” and Dummit and Foote (since I found it more readable than Herstein) :)
From Herstein:
Theorem 2.7.1 (First Homomorphism Theorem). Let $\phi$ be a homomorphism of $G$ onto $G'$ with kernel $K$. Then $G'\cong G/K$, the isomorphism between these being effected by the map
$\psi: G/K\to G'$defined by $\psi(Ka) = \phi(a)$.
Theorem 2.7.3 (Second Homomorphism Theorem). Let $H$ be a subgroup of a group $G$ and $N$ a normal subgroup of $G$. Then $H N = \{h n | h\in H, n\in N\}$ is a subgroup of $G$, $H\cap N$ is a normal subgroup of $H$, and $H/(H\cap N)\cong (H N)/N$.
Theorem 2.7.4 (Third Homomorphism Theorem). If the map $\phi:G\to G'$ is a homomorphism of $G$ onto $G'$ with kernel $K$ then, if $N'\lhd G'$ and $N=\{a\in G | \phi(a)\in N'\}$, we conclude that $G/N\cong G'/N'$. Equivalently $G/N\cong (G/K)/(N/K)$.
I could (and pretty much expect) to be wrong, but I’d think these should all have some very nice “arrow theoretic” presentation and are all probably special cases of some more general category theoretic concept that applies to many categories.
The three books I mentioned are the only ones I would recommend to anyone.
I could (and pretty much expect) to be wrong, but I'd think these should all have some very nice "arrow theoretic" presentation and are all probably special cases of some more general category theoretic concept that applies to many categories.
Nope, not really. These have really intuitive interpretations, but they are not true in some general class of categories. The diagram for the first isomorphism theorem, for instance, is not really nice in any way.
When I taught it, I used Beachy & Blair’s “Abstract Algebra,” Wallace’s “Groups, Rings and Fields,” and Saracino’s “Abstract Algebra.”
There is nothing troubling to me about the definition of a category other than the fact that I can understand completely that it is not something others can easily latch on to.
I think if it is presented much like Urs mentioned above, it could become easier to latch onto. Visually, it falls into the “category” (no pun intended - really) of things I often draw as a meat grinder on a blackboard: something goes in and (usually) something else comes out.
These have really intuitive interpretations, but they are not true in some general class of categories.
Careful with your blanket assertions, Harry. I don’t understand the details, but I believe that the notion of semi-abelian category is intended to capture some of these at least.
I also don’t mind Dummit&Foote; it’s what I learned out of as an undergraduate.
I don't really agree that the class of semi-abelian categories is all that big (in particular, we need kernels to even begin to make sense of the isomorphism theorems, which throws out some massive portion of all categories).
Anyway, I feel like introducing Eric to semi-abelian categories before he understands the isomorphism theorems is missing the point that Semi-abelian categories are designed specifically to abstract the properties of the common algebraic categories (although strangely enough, I don't think that the category of unital rings (with unital homomorphisms) is even semi-abelian, since kernels of ring homomorphisms are not subrings.
Straw man again, Harry. I didn’t say there were a lot of semi-abelian categories (although they include all abelian categories, to start with, which is a fair number already), but I do claim they are “a general class” of categories in which the isomorphism theorems hold. I think what you originally said gives the impression that the isomorphism theorems are not at all categorial, and I don’t think that’s true. The theorems as stated are certainly category-theoretic properties of the category of groups, despite the fact that it seems to be hard to identify precisely what categorical conditions are required for an arbitrary category to have similar properties.
The first isomorphism theorem, by the way, is the most categorial. Rephrased a bit, it says that any regular epimorphism which has a kernel pair is the quotient of that kernel pair, which is a very natural thing. So I’m surprised you say it’s not nice. It’s the other two that I think are difficult and lead to all the semiabelian complications.
I do agree that one should understand the isomorphism theorems for groups before trying to think about semiabelian categories however!
I do agree that one should understand the isomorphism theorems for groups before trying to think about semiabelian categories however!
Yes, this is the important point. I'm glad we agree, then.
Also, my statement about "general class of categories" was in response to Eric's:
I could (and pretty much expect) to be wrong, but I'd think these should all have some very nice "arrow theoretic" presentation and are all probably special cases of some more general category theoretic concept that applies to many categories.
which I was responding to. By a "is not true for a general class of categories", I meant something closer to "is not a general category-theoretic concept". Here, by "general", I meant on a scale similar to what Eric was looking for (namely for some very weak conditions like the existence of some kinds of limits/colimits as he mentioned above) rather than just "a general class", which includes the much less general classes of abelian and semi-abelian categories. It was my mistake, and I apologize for it.
Eric wrote:
I’ve added Steve Awodey’s book to my Amazon wish list and will have a look at some of his papers available online.
Awodey’s book is a very nice exposition, and it is written for people who don’t have much background in (abstract) algebra, so I recommend that you get it now :-)
Eric wrote:
This conversation reminds me of the frustration suffered by me and many others before me trying to convince practitioners in computational physics that differentials forms were much more appropriate than vector calculus.
The argument that I heard was: “All we want to do is calculate stuff, and for calculating we will have to introduce coordinates anyway”. Which is true, but not a reason not to use differential forms: With differential forms many formulas become much more elegant (unless one has fallen in love with the baroque index texture of tensor analysis), and it is manifest what is independent of coordinate systems. I would not try to convince anyone that calculating anything with differential forms is easier…which is a point, I think, that Urs has made over and over again.
Ian wrote:
… if you want to win over (and perhaps you don’t) physicists…
and Harry wrote:
I don’t see why we should be catering to people whose eyes glaze over when reading the axioms for a group.
Another point that Urs has made over and over again is that history shows that once obscure and difficult to learn topics become simplified over time, so that almost everyone can learn about them, although only geniuses could when they were invented. Think of complex numbers or analysis. Once upon a time it was a secret knowledge, mastered only by a couple of people in Europe, how to differentiate a polynom. Category theory should go down the same path.
Another point that Urs has made over and over again is that history shows that once obscure and difficult to learn topics become simplified over time, so that almost everyone can learn about them, although only geniuses could when they were invented. Think of complex numbers or analysis. Once upon a time it was a secret knowledge, mastered only by a couple of people in Europe, how to differentiate a polynom. Category theory should go down the same path.
This is an awful comparison: the axiomatic method was invented in the interim, and unlike analysis or the complex numbers, there is no prerequisite material for algebra. Category theory is an important part of mathematics, but it is an algebraic theory with an axiomatic definition.
If you cannot understand basic group theory, you have no business learning category theory! I am happy to help people with algebra, but I can't stand people who keep trying to get ahead of themselves by reading some abstract nonsense that does nothing more than confuse them.
Category theory has no place in education before a first algebra course (or at least a hard linear algebra course focused on abstraction).
Now some of you may say, "But category theory is fundamental to mathematics", but I think that to appreciate this point, one absolutely needs to see the motivation from algebra. Perhaps you guys have students who can understand category theory with no motivation and no experience in algebra, but I can say for sure that this is not working for Eric, and you're doing active harm by encouraging him to continue on this path.
This is an awful comparison: the axiomatic method was invented in the interim, and unlike analysis or the complex numbers, there is no prerequisite material for algebra.
I’m sure one can learn algebra without any background in anything, and I do not think that the axioms defining a category will be simplified much in the future :-)
…you’re doing active harm by encouraging him to continue on this path.
I did? Sorry. Eric: learn some algebra! You can use Awodey’s book for that, too, because, although it is intended for an audience without much knowledge of algebra, it still explains the connections :-)
I did? Sorry. Eric: learn some algebra! You can use Awodey's book for that, too, because, although it is intended for an audience without much knowledge of algebra, it still explains the connections :-)
I wasn't singling you out. I was talking to the "some of you" from the beginning of the paragraph (I didn't have you in mind, TvB).
=).
As a note to those who like Awodey’s book, apparently there’s a new edition coming out in July with a new section on monoidal categories! Plus it will be in paperback (as good as it is, the hardcover wasn’t worth $130 or whatever outrageous amount they were charging for it).
1 to 37 of 37