Talk:Introduction to Category Theory

From Wikiversity
Jump to: navigation, search

I wonder if it wouldn't be an idea to introduce initial and terminal objects before all the other constructions, and then piggyback products, etc. on these. I'm not a mathematician (rather a linguist trying to get a better sense of what Joachim Lambek, Anne Preller, Carl Pollard and various other people are on about), but I have spent a fair amount of time with various kinds of introductions to Category Theory.

One thing is that, from my beginner's perspective, it's quite impressive that the set-theoretic property of having one member can be induced by the arrow-theoretic property of being a terminal object, and zero members by the property of an initial object. & these are the simplest universal properties, and really the basis for all of them, since (co-)products etc. can be described as (co-)initial objects in various 'spinoff categories' from the category of current interest. - Avery Andrews (The preceding unsigned comment was added by (talkcontribs) 22:31, 24 October 2007.)

I've updated the skeleton of lessons 4 to include initial and terminal objects. I'm not familiar with the term 'piggyback product', but pullback is a product that is constrained by a relation, usually written A \times_C B, is it the same thing? Tlep 09:10, 25 October 2007 (UTC)

I was using `piggyback' as a verb. So a product of A and B (in that order) is a terminal object in the category whose objects are 3-tuples (C, f, g) and so on as described by John Armstrong in the Unapologetic blog. So the idea was that if people first get the idea of terminal objects, then you can get the product by adding a bit more stuff to that. Coproducts will likwise be initial objects in another category. Well I really don't know if this would be more immediately intelligible for beginners than what seems to be the usual approach of presenting the whole thing in one chunk -- it's still pretty complicated (for beginners) when all the bits are filled in. Looking at different cat theory books, I don't get the impression that people really know the best order to present things in. - Avery Andrews (The preceding unsigned comment was added by (talkcontribs) 21:37, 25 October 2007.)

Yes terminal object of 3-tuple category (C, f, g) is (A \times B, \pi_1, \pi_2), but I think it's too hard for beginners to think about the cone category over the original category. But maybe it's a good thing to mention after easier universal mapping property definition. Tlep 22:21, 25 October 2007 (UTC)

I'm not entirely sure it would be too hard, on the basis that I had so much trouble with the product concept to begin with. Of course the general concept of cone doesn't belong here at all. And people might differ considerably in what kind of approach they found most intelligible at the beginning. Maybe it would be possible to set up alternate tracks thru the material so as to find out which ones people found most useful. A possibility of the web that doesn't get used much, it seems to me.

Hmm what if I set up something like an `experimental alternate ch 4' in the lesson plan, to just see what it would look like. It would be a while before it looked like anything :) (The preceding unsigned comment was added by AveryAndrews (talkcontribs) 00:33, 27 October 2007.)

Also, what is your conception of the difference between a `course' and a `textbook'? I would think of a course, in this context, as being more like a series of lectures, which I'd see as ideally less formal in style and comprehensive in direct coverage, tho with full linkage to textbook-like treatments.AveryAndrews 23:35, 27 October 2007 (UTC)

If you want to set up alternative paths, Be bold and do it, I feel all participation is good. I think we both know too much to say what beginners can or can't learn, we will have to wait for nooby category theory virgins to tell us. What's the difference between course/textbook? Why put this in wikiversity and not wikibooks? Books are for self-study, courses are for interaction between people. I plan to stay here for a while and see if this course gets any happy students. Tlep 19:50, 28 October 2007 (UTC)

Well I think I'll hold off a bit on alternate paths, til the main one is better established; I just made some edits to expand things a bit (but forget to log in).AveryAndrews

In the lesson on monoids I use functions Abbalize(), Hellolize(), and Canyoulize(). How does the '-lize' ending sound to a native english speaker? The most logical choice would be AbbaPlus(), HelloPlus(), and CanyouPlus(), but i feel that's too dry, I rather let the reader decrypt the meaning of the ending. What is you're opinion as a linguist? Tlep 11:31, 3 November 2007 (UTC)

-lize isn't really anything (-al+ize, most of the time), and doesn't sound very good to me as a native speaker (more authoratative than 'linguist', for these pruposes); I tried -ify at the end of that section set off with a hyphen, tho you might want to remove the hyphen.


Just a note from somebody just learning category theory: I found the two perspectives on the product (universal property and special kind of object in the "product-maker spinoff category") very enlightening. Thanks! :-)

Using concept maps[edit]

It seems that the most common prefered learning style (by representation system) is Visual. Something that might ease the learning curve on this subject is to construct a concept map of category theory using cmaps or something similar.

Good idea, can you draw one? Tlep 16:07, 11 November 2007 (UTC)

So long, at least for now[edit]

Don't know why Tlep bailed, but in my case, the reason is the belief that a project like this needs an active mathematician involved. AveryAndrews 01:31, 16 December 2007 (UTC)

Why category theory[edit]

It's really unclear to me what this would be good for. I'm also unclear on how this is significantly different from talking about rigid designators in possible worlds theory (which has the advantage of being philosophy with a rather long history). 00:01, 29 August 2008 (UTC)

I have no problem with people looking at category theory. However, I think there is a "social construction of reality" issue. Presumably category theory could be presented with lots of baby examples allowing us to interpret what was being said. However category theory is usually presented as something terribly unreadably except by the select few.

Nice tone[edit]

I've really only read a little bit of this so far, but I love the tone of this so far. Not just this page, but the next one. The style of speach, the simplicity of language, the slight humor. I wish more subjects could be written about in this style. (discuss) 18:25, 19 June 2014 (UTC)