Categories from scratch

http://staff.science.uva.nl/~poss/categories-from-scratch.html

by mmastrac • 12 years ago

133 20 12 years ago

staff.science.uva.nl

solomatov• 12 years ago

This post is useless. The most important notions of category theory, namely universal mapping properties, natural transformations and adjoints aren't even mentioned.

bronxbomber92• 12 years ago

It's useful for intuitively conveying the notion of what a category actually is. I think many people don't even get past this point. So that in and of itself seems useful to me.

vv311• 12 years ago

I fully agree after reading the article. I might even read up on category theory after that, something I have avoided before.

tluyben2• 12 years ago

Exactly; even though I understood it before; it's hard to convey this to people in 'normal' language. I think this page does that well.

daffodil22• 12 years ago

The article argues against overly formal approaches, but I actually think it errs too much in that direction. When I'm studying a math thing, at some point I need you to shut up with the English words and simply describe the object in terms of sets, functions, relations and properties. Categories stopped being mysterious for me once I realized they were just multigraphs with a certain algebraic operation defined on edges (composition) such that edges with certain properties (identities) were stipulated to exist.

kybernetikos• 12 years ago

I suspect that that explanation only works on people with a fairly specific background. I think the hope of the article is to target a wider range of people who could benefit from understanding the formalism.

nilkn• 12 years ago

I feel that you have to be careful with that definition insofar as the objects in a category may not form a set at all but a proper class.

rfrey• 12 years ago

"Categories are just multigraphs with composition and stipulated identities on edges, what's the problem?"

Just kidding, I appreciate the trailhead. :)

sanderjd3• 12 years ago

Ha, I actually read his last sentence as sarcasm initially and thought "haven't I heard this joke..." only to find that you had already referenced the one I was thinking of:

"A monad is just a monoid in the category of endofunctors, what's the problem?"

nmrm2• 12 years ago

More specifically, the common explanatory route to approach categories is usually: “here is a formal specification of what a category is; then look at these known things from maths and theoretical computer science, and admire how they can be described using the notions of category theory.” In practice, quite a few people only adopt conceptual objects by abstracting from two or more contexts where the concepts are applicable, instead.

(Note the instead.)

I understand the author's point, and perhaps these examples are easier to follow for non-math people.

However, to be fair, this approach is the approach taken by pretty much every single description of Category Theory I've read. One of the first things MacLane does is provide a bunch of relevant and familiar examples (groups, rings, etc).

sanderjd3• 12 years ago

Just depends on your audience! Groups and rings aren't familiar at all to lots of programmers. It's fine if category theory is actually only accessible to those to whom that kind of math is familiar, but the author of this article doesn't seem to think that's the case. His bottom-up approach tailored to a very specific programming audience seems fairly unique and I think it works pretty well!

freyrs33• 12 years ago

The examples the author chose are kind of hand-wavy, unix pipes in full are obviously not categories. But I recognize how hard is to come up with examples that aren't contrived and that programmers can relate to without knowing much mathematics.

That's the inherent difficulty in discussing category theory solely in terms of programming, it's hard to see the forest for the trees if you don't have a bunch of well-defined categories you can draw examples from and interrelate.

In an undergraduate course one would typically start with examples from algebraic topology which has a lot of great cross-categorical relations, like how the homology group of a topological space is an example of a functor from the category of groups to the category of topological spaces.

nmrm2• 12 years ago

That's the inherent difficulty in discussing category theory solely in terms of programming

Worse, I don't see any point in categorical models if you want to think solely in terms of programming.

one would typically start with examples from algebraic topology

My initial exposure was along these lines as well, although the idea of a "course" in category theory seems a bit odd (not in the sense of wrong or bad, just not common-place).

Talking about categories in the context of everyday functional programming (even where the connections are correct) always seemed abstruse and a bit silly. If you're not using categorical models, what's the point (other than giving fancy names to things)? And if you don't have some background in algebra or topology and a specific research goal, why build the models?

RBerenguel• 12 years ago

When we were having the Categories subject in Algebra II (maths degree) half-jokingly with a few friends we built a category (with operations, universal properties and objects and all that funny and weird stuff) based on the columns of the university courtyard (this one: http://blocmat.files.wordpress.com/2011/03/matefestub-ba.jpg)

acbart• 12 years ago

+1 for actually providing meaningful contexts! This is right in line with Situated Learning Theory, to think about how your audience is full of Engineers that want relevant examples rather than abstract math.

freyrs33• 12 years ago

For those interested in a little more rigor, there's a great series of lectures on the subject that was just posted online from Steve Awodey, who wrote one of the best introductory texts on the subject.

https://www.youtube.com/watch?v=BF6kHD1DAeU

hardwaresofton• 12 years ago

This is fantastic. I just read through it and it is a fantastic diversion from the usual descriptions of category theory

I scoffed at the "we'll explain it in computer terms" notes at the beginning, but that caveat really helped.

Categories from scratch

Related Stories