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Category Theory for Dummies - slides available

From the discussion group:
For those interested in a brief introduction to category theory, James Cheney has recently posted some PDF slides titled Category Theory for Dummies on his home page.

These slides are introductory, and don't contain advanced techniques, but they seem to be nicely done.

By Ehud Lamm at 2004-08-01 06:51 | Category Theory | other blogs | 15216 reads

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Slide 15

It seems that the diagram in the left of slide 15 is wrong, the objects should be [A, B, C, D] not [A, B, B, C] otherwise g is just the identity of that category.

Or am I wrong?

By Daniel Yokomizo at Sun, 2004-08-01 14:09 | login or register to post comments

Re: Slide 15

Why does g have to be the identity? It just says "gf = gf" and "hg = hg" and (hence) "h(gf) = (hg)f".

Relabelling with [A,B,C,D] would be OK too.

By Frank Atanassow at Sun, 2004-08-01 14:37 | login or register to post comments

Thanks, now I see it. I was c

Thanks, now I see it. I was confusing objects with values and categories with types, hence my confusion.

CT terminology is terribly confusing for me.

By Daniel Yokomizo at Sun, 2004-08-01 22:15 | login or register to post comments

simple questions

Since arrows are associative, why does slide 15 call it a commutive diagram? It seems like it should be called an associative diagram.

Also, it's unclear to me whether an "arrow" is a function A->B, or a single mapping from an element of A to an element of B. Slide 14 talks about a "collection of arrows".

By skybrian at Sun, 2004-08-01 19:01 | login or register to post comments

Simple answers

Since arrows are associative, why does slide 15 call it a commutive diagram? It seems like it should be called an associative diagram.

Associativity is an axiom. Commutativity of diagrams, on the other hand, is generally something that you need to prove.

A diagram commutes if the "loops" in it commute. Take, for example, a typical diagram:

 A ------------> B
 |       f       |
 |               |
 | j             | g
 |               |
 |               |
 V       k       V
 C ------------> D

If this diagram commutes, then g.f = k.j, that is, you can freely exchange g.f with k.j, which is a commutative law.

Also, it's unclear to me whether an "arrow" is a function A->B, or a single mapping from an element of A to an element of B. Slide 14 talks about a "collection of arrows".

While it's good to think of an arrow as a function from the point of view of analogy, an arrow is fundamental in category theory. Don't try to interpret it unless you're talking about a specific category (e.g. Set, where arrows are functions).

By Pseudonym at Mon, 2004-08-02 01:42 | login or register to post comments