Timeline for Categories First Or Categories Last In Basic Algebra?
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| Jan 4, 2015 at 21:05 | comment | added | Andreas Blass | @Vÿska In order to properly appreciate the unifying power of category theory, students should know at least a couple of the left adjoints that I listed in my answer, and a right adjoint or two, like the universal covering space. It would be good to know something about partially ordered sets and about groups. In general, the more concrete examples the student already knows, the better. I don't think I'd say any one example is indispensable; others can usually be used instead. But a student who knows none of the examples is likely to see category theory as just juggling weird abstractions. | |
| Jan 4, 2015 at 15:23 | comment | added | Red Banana | @AndreasBlass What are these important examples that must be introduced before diving deeper in the study of categories? | |
| Oct 5, 2010 at 13:15 | comment | added | JBorger | Note that by this principle, you should not teach a class on category theory unless you're comfortable with 2-categories and know many natural examples. | |
| Oct 5, 2010 at 13:14 | comment | added | JBorger | On the other hand, I think it is good for the teacher to be comfortable at one higher level of abstraction. So they should be experts on the collection of all objects of type $X$, but they should also be comfortable with examples of whatever such a collection is an example of. So if you're teaching groups, say, you should understand the category of all groups of course, but you should also be comfortable with categories. This makes it much clearer which concepts are need to be emphasized, which is essentially what Andreas Blass said above. | |
| Oct 5, 2010 at 13:07 | comment | added | JBorger | ...the commutativity of multiplication of numbers, that the product of two negatives is a positive, that zero times anything is zero, and so on. And it is also probably good to give them names to attach to the concepts, even if you never really use them in a serious way. But it usually the collection of all objects of type $X$ is a much bigger leap in abstraction and will usually be lost on the students. (continued) | |
| Oct 5, 2010 at 13:00 | comment | added | JBorger | I agree very much with the comments above. I just want add that I think it is nothing about categories in particular but applies equally well at every level of abstraction. The question is how much of the general theory or language of all objects of type $X$ should be introduced when students have seen only a small number of examples. The discussion above is when $X$='category' and the students have only seen vector spaces and groups, say. The situation when $X$='ring' occurs much earlier in ones education. It is important to draw attention to facts like (continued) | |
| Oct 4, 2010 at 23:47 | comment | added | Terry Tao | I agree. One can also sneak in a little bit of categorical notation into other courses without having to develop any abstract category theory (e.g. note that homomorphisms can also be called group morphisms, linear transformations can be called linear morphisms, etc.; we already use "isomorphism" in this manner in non-category-theoretic contexts, so why not "morphism"?). One can certainly plant the idea that there is something in common to group theory, topology, linear algebra, algebraic geometry, etc. well before one gets to see the abstract formalism that encodes this commonality. | |
| Oct 4, 2010 at 22:23 | history | answered | Andreas Blass | CC BY-SA 2.5 |