Timeline for Taking a theorem as a definition and proving the original definition as a theorem
Current License: CC BY-SA 4.0
22 events
| when toggle format | what | by | license | comment | |
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| Sep 29, 2021 at 0:50 | comment | added | Zach Teitler | I’m a little taken aback. There are several ways that people think about tensors, I listed a few. | |
| Sep 28, 2021 at 23:55 | comment | added | David Roberts♦ | @LSpice because I recognise the physicists notion of tensor [field] when I see it, because that was my first education. Also, when does someone say 'coordinates' instead of 'basis' in the purely algebraic setting that Will is trying to tease out? | |
| Sep 28, 2021 at 23:53 | comment | added | Will Sawin | @LSpice In my original comment I intended to use a theorem like "To produce a functor from a groupoid to another category, it suffices to fix the value of that functor on one object in each equivalence class together with its automorphisms." Since you mentioned based vector spaces i thought it might be helpful to mention the category of based vector spaces with vector space isomorphisms and explain why it is relevant here. (Though I guess, among other things, I should probably have said why the category of based vector spaces with basis-preserving maps is not relevant.) | |
| Sep 28, 2021 at 23:44 | comment | added | LSpice | @WillSawin, I guess this should be the last I say, because it's clear I'm misunderstanding. You didn't say based vector spaces at all; I did (and I meant the more restricted class of morphisms, though I didn't say so). Did you mean to? Or is your point that it's irrelevant? | |
| Sep 28, 2021 at 23:42 | comment | added | Will Sawin | @LSpice: I didn't say the category of finite-dimensional based vector spaces and vector space isomorphisms respecting the choice of basis! I said the category of finite-dimensional based vector spaces and vector space isomorphisms. So there is no need to respect the choice of basis. | |
| Sep 28, 2021 at 23:40 | comment | added | LSpice | @WillSawin, I'm sorry to be so obtuse, but I think that that cannot be part of an equivalence of categories. The functor needs to assign different isomorphisms to the two isomorphisms $1, -1 : k \to k$ to be part of an equivalence of categories; but two different isomorphisms cannot both respect the choice of a basis of the 1D vector space $k$. | |
| Sep 28, 2021 at 23:29 | comment | added | Will Sawin | @LSpice Just write down all the vector spaces in a list and choose a basis for each, applying the axiom of choice. (Without the axiom of choice you need to use anafunctors to make this make sense.) | |
| Sep 28, 2021 at 23:13 | comment | added | LSpice | @DavidRoberts, how do you know that @WillSawin's suggestion isn't what was meant? | |
| Sep 28, 2021 at 23:11 | comment | added | LSpice | … isomorphisms. You refer to "the $\operatorname{GL}_n$ representation"; is it possible that you intended one of the categories to carry the additional datum of a $\operatorname{GL}_n$ representation in some way? | |
| Sep 28, 2021 at 23:11 | comment | added | LSpice | @WillSawin, I am sorry; I don't understand how these categories are equivalent. To exhibit an equivalence involves in particular a functor from the category of vector spaces with isomorphisms, to the category of based vector spaces with isomorphisms. I do not think it is possible functorially to pick a basis of a vector space, even if we require only that the functor respect … | |
| Sep 28, 2021 at 22:44 | comment | added | David Roberts♦ | @Will that may be, but it's not what was meant. | |
| Sep 28, 2021 at 22:13 | comment | added | Will Sawin | @LSpice A perhaps slightly different approach would be to define a tensor as a function from the set of bases of the vector space to the set of multidimensional arrays of numbers that satisfies some transformation rules. | |
| Sep 28, 2021 at 22:11 | comment | added | Will Sawin | @LSpice If two categories are equivalent, a functor from one (up to isomorphism) is the same data as a functor from the other. The category of finite-dimensional vector spaces and vector space isomorphisms is equivalent to the category of based finite-dimensional vector spaces and vector space isomorphisms. In concrete terms, you just pick a set of coordinates arbitrarily and define tensors using those coordinates, and the $GL_n$ representation tells you how to deal with changes of coordinates. | |
| Sep 28, 2021 at 21:18 | comment | added | LSpice | @WillSawin, shouldn't the objects of your category be based vector spaces? I may be misunderstanding, but precisely the fact that there are multiple choices of basis for a vector space seems to prevent me from thinking of (5) as a definition in the way you propose: if tensors are specified in coördinates, how to attach one to a non-coördinatised vector space? | |
| Sep 28, 2021 at 14:16 | comment | added | Will Sawin | @DavidRoberts If you interpret "change of coordinates" as "automorphisms of $V_1,\dots, V_n$", (5) seems to me like a perfectly reasonable definition of a tensor. It's equivalent, I would say, to defining the tensor product as a functor from the $n$-fold product of the category of finite-dimensional vector spaces and isomorphisms, by observing that each object in this category is isomorphic to $F^m$ for some $m$ and defining an explicit representation of $GL_{m_1} (F) \times \dots \times GL_{m_n}(F)$. | |
| Sep 28, 2021 at 5:54 | comment | added | Zach Teitler | @DavidRoberts I've edited my answer and I hope that it is less bothersome now. | |
| Sep 28, 2021 at 5:52 | history | edited | Zach Teitler | CC BY-SA 4.0 |
added 112 characters in body
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| Sep 28, 2021 at 5:44 | comment | added | David Roberts♦ | The problem is, a tensor in the sense of 1-4 is an entirely different object to 5. OK, so I've got a span of some basic tensors. how does this transform under "changes of coordinates". Really the issue is that physicists are slack (and I come from that culture...) and say 'tensor' when they mean 'tensor field', or 'tensor-valued function', and give unhelpful definitions like "A tensor is a thing that transforms like a tensor". Mathematically, we have separate words for these things... | |
| Sep 28, 2021 at 4:13 | comment | added | Zach Teitler | I concede that I'm blurring lines. The lists for vector spaces, and for bundles, would have been mostly the same, so I kind of mashed them together. It's true, I'm abusing the ambiguity of the term "tensor" by importing a notion that people use for bundles into the vector spaces setting. Nevertheless it's an idea that some people use, and surely that makes a profound difference in how they think of tensors, compared to (1)-(4). | |
| Sep 28, 2021 at 0:18 | comment | added | David Roberts♦ | 5 is different to the others, since it is talking about sections of tensor powers of a vector bundle and their coordinates with respect to local frames. The others are purely algebraic. So something like 'tensor-valued functions'. No one confuses a real number (Cauchy seq or Dedekind cut or...) with a function. | |
| S Sep 28, 2021 at 0:09 | history | answered | Zach Teitler | CC BY-SA 4.0 | |
| S Sep 28, 2021 at 0:09 | history | made wiki | Post Made Community Wiki by Zach Teitler |