Timeline for What are the advantages of phrasing results in terms of exact sequences and commutative diagrams?
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| Apr 19, 2010 at 14:02 | comment | added | KConrad | It takes effort, but you need to aspire to understand the tensor product in terms of its universal mapping property. You may not need that to do certain calculations inside a particular tensor product space, but as soon as you try to relate one tensor product space to something else (specifically, writing down a linear map out of a tensor product space) things are much simpler if you are comfortable using the universal mapping property. See math.uconn.edu/~kconrad/blurbs/linmultialg/tensorprod.pdf | |
| Apr 18, 2010 at 23:41 | comment | added | teil | @KConrad: I think in terms of symbols rather than visually so that may be another reason for my difficulties. Another example for me is the tensor product: taking linear combinations and then defining an equivalence relation I can understand; but every bilinear map can be factored uniquely is much less clear to me. I by no means meant to criticize the language, I was trying to gain some understanding of what it was doing. From the answers, I gather that I haven't seen sufficiently advanced mathematics to appreciate this view. | |
| Apr 18, 2010 at 19:01 | comment | added | KConrad | The absolute value sign was not invented until the 1840s by Weierstrass (cf. Wikipedia page on absolute value). I find that late date rather incredible, although certainly you can get by without it for discussing analysis on the real line. Is the fact that earlier mathematicians didn't have the absolute value notation a sign that we shouldn't use it even in simple situations? | |
| Apr 18, 2010 at 18:58 | comment | added | KConrad | Can you point to some examples where you are seeing this point of view introduced without an apparent benefit? I would say that seeing homomorphisms explained from the viewpoint of commutative diagrams is nice, even though it's very elementary. It provides a nice visual image of the purely algebraic equation f(xy) = f(x)f(y), which doesn't really seem so vivid in the equation itself. | |
| Apr 18, 2010 at 16:33 | comment | added | S. Carnahan♦ | We can consider the shift a sign of positive progress. Other signs of positive progress include the absence of creepy anatomical remarks in recent issues of the Bulletin of the AMS (cf. that review you quoted). | |
| Apr 18, 2010 at 12:54 | history | answered | teil | CC BY-SA 2.5 |