4 ms·

I think it's "explained" when you go from slide 6 to 7. Slide 6: objects are points in space. Slide 7: arrows are ways of transforming objects into other object

I think it's "explained" when you go from slide 6 to 7. Slide 6: objects are points in space. Slide 7: arrows are ways of transforming objects into other objects. I'm no expert, but from my understanding, there is nothing more to be said (other than stating the composition / identity laws): you're at a level of abstraction where you cannot say anything more. Objects. Arrows. An arrow goes from an object to another. If it's any help, you can think of them as mathematical functions, where the origin point is the function's domain and its destination point the function's range (called co-domain in category theory, because why not). Bear in mind that while this example is correct, that's all it is: a concrete example of an abstract notion. If it feels simplistic, that's because it is. Arrows are a trivial notion. It gets more complicated when you study different kinds of arrows (homomorphism, isomorphism, whateverism), or arrows in "concrete" categories (functors are arrows in the category of categories, natural transformations are arrows between functors...).

ryanobjc 12y ago
But is this the best pedagogical approach to teaching this area? In the book "Conceptual Mathematics" the concrete example is finite sets and mapping between finite sets. That is way more understandable than 'objects' and 'how to go from one object to another'. It's just too abstract for a general audience I believe.
- nrinaudo 12y ago
  It's the only one that does not require knowledge of other fields - not everyone knows set theory (although yes, almost everyone here probably knows enough). Conceptual Mathematics needs concrete example to explain the algebra of function composition, and more importantly to prove certain properties and theorems. This presentation's goals are far more modest and the abstract approach allows it to not take up 700 slides. That being said, I don't think I'd have been able to go through this presentation in its entirety if I hadn't had some previous exposure to the field. So while I disagree with you for discussion's sake, I secretly think you're right.
  - ryanobjc 12y ago
    I found the 'requirement' of set theory in the book is very 'common sense' and doesnt really use any advanced knowledge. It depends greatly on finite sets, which is easily and intuitively drawable. The point of these instructional methods is to build an 'intuitive sense' or "system 1" knowledge of category theory. Having a lot of easy to grasp examples to help you start forming the theory in your head i find quite valuable.

ryanobjc 12y ago

But is this the best pedagogical approach to teaching this area? In the book "Conceptual Mathematics" the concrete example is finite sets and mapping between finite sets. That is way more understandable than 'objects' and 'how to go from one object to another'. It's just too abstract for a general audience I believe.

nrinaudo 12y ago
It's the only one that does not require knowledge of other fields - not everyone knows set theory (although yes, almost everyone here probably knows enough). Conceptual Mathematics needs concrete example to explain the algebra of function composition, and more importantly to prove certain properties and theorems. This presentation's goals are far more modest and the abstract approach allows it to not take up 700 slides. That being said, I don't think I'd have been able to go through this presentation in its entirety if I hadn't had some previous exposure to the field. So while I disagree with you for discussion's sake, I secretly think you're right.
- ryanobjc 12y ago
  I found the 'requirement' of set theory in the book is very 'common sense' and doesnt really use any advanced knowledge. It depends greatly on finite sets, which is easily and intuitively drawable. The point of these instructional methods is to build an 'intuitive sense' or "system 1" knowledge of category theory. Having a lot of easy to grasp examples to help you start forming the theory in your head i find quite valuable.