Elements of Basic Category Theory
started 2/16/2004; 5:07:07 AM  last post 2/21/2004; 5:04:09 AM


Ehud Lamm  Elements of Basic Category Theory
2/16/2004; 5:07:07 AM (reads: 9096, responses: 24)



Andris Birkmanis  Re: Elements of Basic Category Theory
2/16/2004; 11:03:48 AM (reads: 846, responses: 1)


I liked the ability to define a diagram categorically, providing a visual intuition for functors at the same time (6.1.5).
Before this book, I saw only "visual" metaphor of natural transformations being "slides" from one functor to another (which was not very helpful without being able to easily visualize functor).
Now it's a snap.
[on edit  not that I never saw diagrams, but it never occurred to me that each diagram with a categorical scheme is a visualization of a functor  and vice versa]
One just needs to be aware that each author has slightly different definitions, e.g., many authors do not require schemes of diagrams to be categories themselves (just general graphs). This, of course, does not serve well to the theory in general, and undermines its use as a common dictionary for discussion of problems. The worst, probably, is unability to use intuition developed for definitions of one author working with text written by another :(
On the bright side, (I think) there are not many such divergence points in category theory.
Can you share your experience?


Andris Birkmanis  Re: Elements of Basic Category Theory
2/16/2004; 11:06:54 AM (reads: 841, responses: 0)


On the other hand, even a category itself cannot be easily visualized. The graph is not enough, as it loses information about composition... Sigh...
We definitely have cognitive problems here ;)


andrew cooke  Re: Elements of Basic Category Theory
2/16/2004; 12:05:29 PM (reads: 835, responses: 0)


martini doesn't emphasise the type of an arrow nearly as much as other things i have read. also, he (? i think alfio is a man's name) starts with set theory and functions so it's not clear just how much a category is like a set (do i need to be worried about whether i can construct categories (to avoid contradictions)  presumably not, if the contradictions depend on set memebership, which is not important in categories) or how much an arrow is like a function. this confused me about whether the type of an arrow specifies the domain and codomain, or whether it specifies single objects in the category (is *an* arrow a single mapping from *one* object to another?). more generally, i felt he was less rigorous than other introductions (i keep thinking "hmm, that was vague", but maybe i simply don't understand).
on the other hand, looking ahead, i think his emphasis on diagrams is a good thing that is missing from other stuff i have skimmed through. and he seems to cover more topics.
this started as trying to explain different approaches to category theory, but i think i don't yet understand enough to do so. maybe it is more useful as guidance to anyone else thinking of learning from this paper?
pure mathematics is very annoying. on the one hand, all the precision is tedious. on the other, as soon as some precision is missed, or i overlook it, everything becomes uncertain... physics is so much easier, because you can (nearly) always look out of the window and check that you are making sense by seeing the maths at work.


DocOlczyk  Re: Elements of Basic Category Theory
2/16/2004; 10:29:12 PM (reads: 725, responses: 1)


Categories are not like sets. They are a property
that describes sets, and a definition of functions
that map sets that match that property. Categories
are not sets, though they can be thought of as "collections"
of sets and associated morphism. They might be considered
sets of sets which satisfy a certain property, if not for Russel's
paradox.
Examples are:
groups and homorphisms,
topological spaces and contintuous functions,
vector spaces and linear functions.


Andris Birkmanis  Re: Elements of Basic Category Theory
2/16/2004; 11:51:29 PM (reads: 715, responses: 0)


I like to think about category theory as emphasizing specifications or outer features/behavior as opposed to set theory focusing on implementations or inner features/behavior.
While reasoning about categories you may abstract from concrete objects and their inner machinery. I think this power of abstraction, as well as ability to abstract multiple times (think functors and natural transformations) are next to main advantages of CT. The main one, though, I think is giving a common dictionary to discuss all these things (and a common intuition to manage all this complexity).


nraynaud  Re: Elements of Basic Category Theory
2/17/2004; 10:29:24 AM (reads: 585, responses: 2)


I am reading this paper, aiming at knowing a little about the matter (and trying to prove myself that all my concentration faclties are not gone with inactivity), but I think there is a little mistake on page 14 (according to the numbering at the bottom of pages), example 2.3.4 first element :
(...) is the partial function g o f : A o> B defined above.
I think it should have been :
(...) is the partial function g o f : A o> C defined above.
Anybody to confirm the mistake or I don't understand *anything* about functions and sets ?


Andris Birkmanis  Re: Elements of Basic Category Theory
2/17/2004; 10:48:19 AM (reads: 581, responses: 1)


Anybody to confirm the mistake
Yes, that's a bug.
It's a pity mathematicians do not use literate programming for their texts ;)


Ehud Lamm  Re: Elements of Basic Category Theory
2/17/2004; 11:21:44 AM (reads: 578, responses: 0)


It's a pity mathematicians do not use literate programming for their texts
Aye. I think computer scientists esp. those working on programming languages tend to be much more rigorous than your average mathematician.


DocOlczyk  Re: Elements of Basic Category Theory
2/17/2004; 12:46:06 PM (reads: 552, responses: 1)


I think computer scientists esp. those working on programming languages tend to be much more rigorous than your average mathematician.
No.
In fact lambda calculus looks much less rigourous then other
branchs. Category theory is mainstream mathematics ( sort of
a lot of grad students try to avoid it as much as possible ).
Granted it seems more abstract, but then hey, it tries to abstract
the abstractions.


nraynaud  Re: Elements of Basic Category Theory
2/17/2004; 1:01:41 PM (reads: 539, responses: 0)


Thank you Andris.


Ehud Lamm  Re: Elements of Basic Category Theory
2/17/2004; 1:03:20 PM (reads: 546, responses: 0)


I am not talking about LC (which is quite an umbrela term anyway). I am thinking about the likes of Lamport who want detailed, computer verfiable, proofs. I understand the difficulty, and I may even agree that the benefits are marginal, but I am only willing to accept these factors when the majority of mathematicians finally accept that mathematical proofs are a social construction.


DocOlczyk  Re: Elements of Basic Category Theory
2/17/2004; 1:08:57 PM (reads: 539, responses: 1)



Ehud Lamm  Re: Elements of Basic Category Theory
2/17/2004; 1:16:29 PM (reads: 544, responses: 0)


You may be right, but as long as there's interest I don't see any reason why the discussion should end.


Andrei Formiga  Re: Elements of Basic Category Theory
2/17/2004; 8:19:35 PM (reads: 483, responses: 0)


What other introductions to category theory are freely available ? And how do these compare to available books ?


Andrei Formiga  Re: Elements of Basic Category Theory
2/17/2004; 8:25:15 PM (reads: 479, responses: 0)


Being lazy doesn't pay. After very little searching, I found this address which sort of answers my queries:
TUNES: Category Theory 101
Anyone has more links ?


Derek Ross  Re: Elements of Basic Category Theory
2/17/2004; 11:55:45 PM (reads: 462, responses: 0)



xeo_at_thermopylae  Re: Elements of Basic Category Theory
2/18/2004; 2:29:23 PM (reads: 365, responses: 0)



Frank Atanassow  Re: Elements of Basic Category Theory
2/19/2004; 12:41:52 PM (reads: 306, responses: 2)


Andris: One just needs to be aware that each author has slightly different definitions, e.g., many authors do not require schemes of diagrams to be categories themselves (just general graphs). This, of course, does not serve well to the theory in general, and undermines its use as a common dictionary for discussion of problems. The worst, probably, is unability to use intuition developed for definitions of one author working with text written by another :(
I think there is broad agreement on the basic concepts, although sometimes there are several names for the same concept (for example, source versus domain, or arrow versus morphism).
The issue you take exception with, namely whether diagrams are functors or graph morphisms, is not really an issue, though, only a question of presentation. The reason is that every graph morphism to a category extends uniquely to a functor from the path category generated by its domain; conversely, every functor is a graph morphism from the underlying graph of its domain. (This is actually an adjunction between Grf and Cat.) If you define a diagram as a functor, then, you just have to be a little more careful to make sure that it preserves paths.
If you feel you would benefit from a more careful treatment of diagrams, then I recommend Barr and Wells' Category Theory for Computer Science; they are very conscientious in their explanation of diagrams, and it really pays off when you have to discuss (co)limits and sketches.
ehud: Aye. I think computer scientists esp. those working on programming languages tend to be much more rigorous than your average mathematician.
I think computer scientists are more formal, not really more rigorous. (There is a difference.) Mathematicians, BTW, seem to see CT as a way to be more rigorous without being more formal;
computer scientists
tend to see it as a way of being both more rigorous and formal, without being longwinded.


Andris Birkmanis  Re: Elements of Basic Category Theory
2/20/2004; 12:19:43 AM (reads: 280, responses: 0)


Frank: If you feel you would benefit from a more careful treatment of diagrams, then I recommend Barr and Wells' Category Theory for Computer Science
It's not available online, is it?
I remember looking for it... I might end up supporting imperialistic Amazon by buying it (the book, not Amazon) :)


Ehud Lamm  Re: Elements of Basic Category Theory
2/20/2004; 2:56:00 AM (reads: 276, responses: 0)


I think computer scientists are more formal, not really more rigorous. (There is a difference.)
This is a good distinction and worth keeping in mind, nad in general I'd have to agree with it. But I think this is only part of the story. The trend towards careful axiomatization and fanatical rigor in math is quite recent. NonEuclidean geometry is quite a recent idea. Same goes for the axiomtization of set theory etc.
Due to the historical context, computer science began after this trend has already started, and from the get go was aware of the importance of formal reasoning as tool for achieving more rigorous results.
I am not sure whether this is the place to go into details concerning the philosophy of math (a personal hobby horse), but let me just say that where as most mathematicians decide that the formalist agenda has run its course, and work from a platonist framework, computer science remain true to the formal philosophy.


Frank Atanassow  Re: Elements of Basic Category Theory
2/20/2004; 6:21:31 AM (reads: 256, responses: 2)


Andris: It's not available online, is it?
Nope, unfortunately not. Their other book, Toposes, Triples and Theories is, and is available from PLT Online, but it's much harder going. I haven't read it, and I dunno how it handles diagrams offhand.
Ehud: The trend towards careful axiomatization and fanatical rigor in math is quite recent.
Yes. If by `recent' you mean since the beginning of the twentieth century... :) (I'm not trying to be sarcastic!)
computer science remain true to the formal philosophy
Since programs are elements of a formal language, this is somewhat unavoidable. Of course, when you are discussing the metatheory, nothing stops you from employing nonconstructive (`Platonic') methods, but it's more economical just to use the same sort of logic at the metalevel that you use at the objectlevel.
Most proofs in CS are pretty easy, anyway, compared to ones in general mathematics, and don't seem to benefit much from employing nonconstructive arguments. Also, if you work in a constructive metatheory, there is always the tantalizing possibility that you can internalize a metaresultconstructive proofs are programs.
Hm, I wonder how we (I?) got from `formal' to `constructive'; they are not the same.


Ehud Lamm  Re: Elements of Basic Category Theory
2/20/2004; 6:31:07 AM (reads: 260, responses: 0)


Yes, by recent I meant 20th C. And I agree with most of what you have to say regarding CS.
Hm, I wonder how we (I?) got from `formal' to `constructive'; they are not the same.
Not by a long shot. But it is true that CS is not only more formal, but also more constructive.


DocOlczyk  Re: Elements of Basic Category Theory
2/21/2004; 5:04:09 AM (reads: 226, responses: 0)


Hmm.
There seem to be two different books by Barr and Wells:
Category Theory for Computing Science
and
Category Theory for Computer Science
at least according to several places I visited.
What is the difference? Or are these perhaps
different editions of the same book.


Andris Birkmanis  Re: Elements of Basic Category Theory
2/21/2004; 6:42:42 AM (reads: 229, responses: 0)


These Lecture Notes describe diagrams pretty much like Frank did.
Frank: Of course, when you are discussing the metatheory, nothing stops you from employing nonconstructive (`Platonic') methods
And when discussing metacircular theory, you must employ nonconstructive methods? Or am I treating Hoedel's theorem too liberally?



