![Tensor_product_of_modules[1].png](https://i.imgur.com/dtS2xNz.png "Tensor_product_of_modules[1].png")
What's the point of tensor products, anyway? We have an R-module N, and we want to extend the scalars to find a natural embedding into some S-module, and the S-module S(x) N is the nicest module with this property. What's the motivation?
Tensor products are the coproduct in the category of rings, which is useful when taking products in geometry.
Extending scalars is a good way of killing torsion in abelian groups. Higher homotopy groups of spheres are hard to find, but tensoring with [math]\mathbb Q[/math] lets us get some idea of what they should look like.
Number theorists are particularly interested in extending scalars along field extensions.
>>7943283
Tensor products are how you glue modules together without making independent copies of the ring of scalars but instead only allowing a shared copy of them. This is the kind of thing that multilinear algebra is about. Note that the determinant is still well defined if instead of giving it nxn matrices you give it objects in the tensor product of n copies of the n-dimensional vectors. Hence the tensor product definitely should show up in geometry contexts (and I know it shows up in general relativity.) As for the category theory, the universal property resembles the one for subobjects, so you could think of it as the smallest subobject where you can still commute with these maps you care about.
>>7943283
Non maths person here. What is that diagram you're posting called? What course do you start using them in?
>>7944239
Wait, shit, I can't remember whether the universal property you've got resembles the subobject one or the quotient object one (which is the subobject one with the arrows turned around.) Quotient would make more sense since I know the tensor product can be defined as a quotient of the regular product and I can't quite see the inclusion. Well, good luck.
>>7944255
it's aptly called a diagram. it's used in algebra and the theory of categories in general, so you might see it in other fields when dealings with morphisms
>>7944270
So its basically a roadmap for morphisms comprised of one way streets?
>>7944276
yeah. it lets you picture a group of morphisms as a graph and say things about them. like when a diagram "commutes":
https://en.wikipedia.org/wiki/Commutative_diagram
>>7944255
It is a commutative diagram as used in category theory. Very useful notation when all that is relevant to the present discussion is special sets you care about, functions preserving structure between them, and how function composition works on those functions. The definition is more general and abstract though.
>>7944276
It's okay to think of the situation of a category as a bunch of things, some arrows between the things, and going down one arrow and then another arrow always equaling some particular third arrow, of potentially many possibilities.
>>7944255
>>7944324
Very Nice. Thank you
>>7944265
the tensor product is a quotient, so it is a oclimit.
the subobject stuff is a comma category, which means it is a limit.
https://ncatlab.org/nlab/show/tensor+product
https://ncatlab.org/nlab/show/subobject
https://ncatlab.org/nlab/show/comma+category
>>7944334
begin with the book by lawere, since it expresses set theory into category theory. then move on to borceux's
>>7944340
Will do, thanks for the rec.
there is also https://ncatlab.org/nlab/show/HomePage which is a wiki for category theory. very useful to check a definition quickly
>>7943283
Hold up; while they do turn out to be the categorical products, you don't need category theory, outside of maybe the idea of universal map, to get a tensor of modules.
It's just the `smallest' module where where scalar multiplication and addition is bilinear. It's like a `free' construction on linear combinations of (m,n) over R, which you can do without category theory, held to the axioms of bilinearity.
It effectively tells you the best way to be able to form `products' of elements from M and N, sort of like taking a semidirect product of groups.
In an abelian context like this, one way to think of it is formalizing the laws of multiplication, in that (rm,n)=(m,rn), thought of as rmn=mrn, ie. just "commutivity", and (m+m',n), thought of as (m+m')n is equal to (m,n)+(m',n), thought of as mn+m'n, i.e. its just the "distributive law".
It has uses in finding annihilators, and its status as the coproduct of modules has nice properties. In particular, its also the coproduct of commutative rings, which means its the product in the category of affine schemes.
>>7945792
where coproduct, if you are not familiar, means it is the universal "best" ring that both R and S map into, like the "best" set that two sets map into is their disjoint union. However, disjoint unions of rings are obviously not rings, and the tensor product tells you how to fix it (freely generate an algebra and subject it to bilinearity)
>>7943283
Quantized version of logical 'and'.
