Timeline for Tensor product and category theory
Current License: CC BY-SA 2.5
6 events
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| Jul 14, 2010 at 19:48 | comment | added | Charles Matthews | @Peter: Yes, different types of students are going to be worried by the different questions (a) how do you manipulate this gadget, and (b) how do you know it isn't 0, anyway? Because you need to able to answer both to do any serious work, it is no good telling just half the story. But my point is that the "economy" of saying you can teach both at once is a false one. | |
| Jul 14, 2010 at 18:42 | comment | added | Neel Krishnaswami | I didn't understand what the universal property was for, and so didn't appreciate the construction, until the monoidal closure of vector spaces was pointed out to me. That's when it all fell into place -- I could now see why we wanted this particular universal property (so we could curry and uncurry linear maps a la functional programming). Then each piece of the construction with generators and relations made sense as the minimal construction to meet this requirement. | |
| Jul 14, 2010 at 18:26 | comment | added | Peter LeFanu Lumsdaine | I think this is definitely a “different strokes for different folks” issue. I was first taught it just in terms of the generators and relations (as a second-year undergraduate at Cambridge!) and while I was able to use it then, I didn't begin to grok it until someone pointed out the universal property to me, transforming it from something fiddly and ad hoc into something natural and tractable. I appreciate that some people may find it clearer to learn with generators and relations emphasised, but not all of us did! | |
| Jul 14, 2010 at 17:56 | comment | added | Charles Matthews | When I learned it all, the universal property came first: i.e. I learned the orthodox post-Bourbaki version of tensor products. But I only really felt I had understood it properly after I had taught it. I was particularly struck by a student's remark that generators and relations was "more sensible" than what they were taught in lectures (I wasn't lecturing, but picking up the pieces.) Now, you do need both sides. The real test came when I tried using tensor products of fields in Galois theory lectures ... another story. | |
| Jul 14, 2010 at 16:57 | comment | added | Neel Krishnaswami | Pedagogically, do you find your two sentences should go in the order you wrote them? For myself, I understood it the other way around. My mental story went "Wouldn't it be nice if we could somehow turn bilinear maps into linear maps, so we could reuse all our theorems? (ie, the desired universal property) Now let's go build a vector space to make this happen (ie, the generators and relations)." But I haven't ever taught this, so I'm curious which way around you found better. | |
| Jul 14, 2010 at 13:51 | history | answered | Charles Matthews | CC BY-SA 2.5 |