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To the contrary, when I think "monoid" I think good old nonnegative integers and addition, or positive integers and multiplication, or any of a half-dozen reall

To the contrary, when I think "monoid" I think good old nonnegative integers and addition, or positive integers and multiplication, or any of a half-dozen really easy-to-understand examples. Tensors... what the heck are those? Like matrices, but _more_ complicated, you say? Thanks, but I'll stick with my basic monoids over here. :-) The confusion between monoids and monads is perhaps unfortunate (even if the similarity is quite intentional, because technically monads are a particular kind of monoid), but easily sorted out and not much worse than having to distinguish between the nuts in your pantry and the Nutella.

throwawaymath 8y ago
> Tensors... what the heck are those? Like matrices, but _more_ complicated, you say? Thanks, but I'll stick with my basic monoids over here. :-) A tensor isn't actually a multidimensional array, just as a linear map isn't actually a matrix. Defining tensors as the multidimensional abstraction of scalars, vectors and matrices to n indices is anti-pedagogical, though regrettably common because it's computationally useful. Learning tensors this way is like learning linear algebra primarily by doing matrix multiplication. It hollows out all of what's actually going on by focusing on notation and symbol pushing. What you're comparing here is the heavily symbol-laden, notation-encumbered definition of a tensor with the elegant abstraction of a momoid. If you'd like to compare them on the merits of their abstraction, an alternative definition of "tensor" would be better. The object is not its representation. For example: a tensor is a multilinear map between a product of vector spaces and another vector space. So T: V_{1} x V_{2} x ... x V_{n} --> W is a tensor. Furthermore, every tensor T is itself an element of a tensor product. An example of a tensor product is R^{nk}, which is the nk-dimensional real space you obtain by taking the tensor direct product of the n- and k- dimensional real spaces, R^{n} ⊗ R^{k}. In this way the tensor product is an abstraction of the familiar Cartesian product in which the dimension of two spaces is multiplied instead of the cardinality of two sets. This definition makes it easier to understand why the objects of a monoidal category can be considered tensors. A monoidal category is a category C where the functor ⊗ is defined such that C x C --> C. Note the similarity of this definition with my alternative definition of a tensor - no multidimensional arrays are required! You're just moving up one level of generality from linear maps and Cartesian products. Technically speaking, the monoidal category functor is actually an abstraction the tensor product, the way the tensor product is an abstraction of the Cartesian product. Whereas many monoidal categories admit a Cartesian product, some monoidal categories (like the category of vector spaces) admit a tensor product. Edit - Sorry for the bracket notation used in subscripts and superscripte, I was thinking in LaTeX...
ginnungagap 8y ago
You can think of a tensor as "an element of a tensor product" (just like a vector is "an element of a vector space") and look at the definition of a tensor product via its universal property, it basically turns bilinear maps into linear maps. This universal property is elegant and immediately gives the Hom-tensor adjunction, but to show that tensor products satisfying it actually exist you'll need an explicit construction, which can be done by quotienting a free module. This explicit construction might look unintuitive at first, but you're really quotienting by the minimal relations required for the "bilinearity" property of tensors to work out
- JadeNB 8y ago
  I think this is a fantastic definition of what tensors are for mathematicians, but I think it is likely to be not very helpful for someone whose perspective is: > Tensors... what the heck are those? Like matrices, but _more_ complicated, you say? Thanks, but I'll stick with my basic monoids over here. :-) If matrices are the limit of your tolerance for complication, bilinear maps probably aren't intuitive tools for you.
  - deleted 8y ago
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throwawaymath 8y ago

> Tensors... what the heck are those? Like matrices, but _more_ complicated, you say? Thanks, but I'll stick with my basic monoids over here. :-) A tensor isn't actually a multidimensional array, just as a linear map isn't actually a matrix. Defining tensors as the multidimensional abstraction of scalars, vectors and matrices to n indices is anti-pedagogical, though regrettably common because it's computationally useful. Learning tensors this way is like learning linear algebra primarily by doing matrix multiplication. It hollows out all of what's actually going on by focusing on notation and symbol pushing. What you're comparing here is the heavily symbol-laden, notation-encumbered definition of a tensor with the elegant abstraction of a momoid. If you'd like to compare them on the merits of their abstraction, an alternative definition of "tensor" would be better. The object is not its representation. For example: a tensor is a multilinear map between a product of vector spaces and another vector space. So T: V_{1} x V_{2} x ... x V_{n} --> W is a tensor. Furthermore, every tensor T is itself an element of a tensor product. An example of a tensor product is R^{nk}, which is the nk-dimensional real space you obtain by taking the tensor direct product of the n- and k- dimensional real spaces, R^{n} ⊗ R^{k}. In this way the tensor product is an abstraction of the familiar Cartesian product in which the dimension of two spaces is multiplied instead of the cardinality of two sets. This definition makes it easier to understand why the objects of a monoidal category can be considered tensors. A monoidal category is a category C where the functor ⊗ is defined such that C x C --> C. Note the similarity of this definition with my alternative definition of a tensor - no multidimensional arrays are required! You're just moving up one level of generality from linear maps and Cartesian products. Technically speaking, the monoidal category functor is actually an abstraction the tensor product, the way the tensor product is an abstraction of the Cartesian product. Whereas many monoidal categories admit a Cartesian product, some monoidal categories (like the category of vector spaces) admit a tensor product. Edit - Sorry for the bracket notation used in subscripts and superscripte, I was thinking in LaTeX...

ginnungagap 8y ago

You can think of a tensor as "an element of a tensor product" (just like a vector is "an element of a vector space") and look at the definition of a tensor product via its universal property, it basically turns bilinear maps into linear maps. This universal property is elegant and immediately gives the Hom-tensor adjunction, but to show that tensor products satisfying it actually exist you'll need an explicit construction, which can be done by quotienting a free module. This explicit construction might look unintuitive at first, but you're really quotienting by the minimal relations required for the "bilinearity" property of tensors to work out

JadeNB 8y ago
I think this is a fantastic definition of what tensors are for mathematicians, but I think it is likely to be not very helpful for someone whose perspective is: > Tensors... what the heck are those? Like matrices, but _more_ complicated, you say? Thanks, but I'll stick with my basic monoids over here. :-) If matrices are the limit of your tolerance for complication, bilinear maps probably aren't intuitive tools for you.
- deleted 8y ago
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