/sci/ - Science & Math

I'm interested in algebraic groups of matrices, namely those over finite or countable degree transcendental extensions over Q.
I can make that less stringent by saying I'm interested in algebraic groups like GL(n), but not over R or C or a finite field.

My issue is that most books are either super general or, similarly, full of algebraic geometry that I think I don't need too much for my case. Does anybody have a reference, and at best an introductory one?
Humpreys or Hochschield don't seem to fit well for the above reasons.

>>10981308
If I had to guess, I'd say something along the lines that the algebra can represent all points (as ideals of functions that are zero at those points)

>>9881191
>Not much has been done after his death
The field of computer science "has been done".
Gödel did loads of research on primitive recursive function, leading up to Turing complete computation as such.

>>9646078
For a group considered as a concrete set with a binary function to model their binary operations, and the isomorphism considered as a function into intself, then yes, you're right in that regard.
No reason to through around insults, though. Sometimes I feel like people feel some sort of general stress release when they can attack someone online. Is that it?

Coming back, however, for a vector space over a field, to say that an element of the field IS also an element of the vector space sure is a stretch. Even if plain text notation might use one symbol for two object in the respective structure.

>>9395341
thx, didn't know that theorem

>categories are just hyper graphs
They are comleted mutligraphs with an algeraic law, associativity

>functors and natural equivalences are just isomorphisms
Functors are homomorphisms, natural transformations are homotopies

>topological spaces are just spaces contained by sets
Point set topology is one way to look at it, but the laws are a little more restrictive than that.
Of course, in many case you'll be able to argue that X is just sets with Y.

>groups are just symmetric spaces
I'll not try to make sense of that characterization, but there surely is an intimate tie between the two.

>God, I can't wait to find out what de Rahm cohomology "really" is...
Differential forms form a ring - itself a direct sum of several rings - and the exteriour differential d is a natural transformation mapping between them. Restricted to subspaces, d is really a chain of d's and the properties it has are descibed by cohomology. Homology theory also deals with chains of operations. You can draw both as arrows in categories, if ya like.

>>9323105
Category theory, just like graph theory, are extremely abstract. Don't conflate abstract with hard, nor with deep. The same tree graph can describe the hiarchy at a company, the evaluation structure of an arithmetic expression and the topology of a river on a map. Hard to deny graph theory, dealing with those structures without the semantics, is not abstract.
Every category theory book will tell you that there are hardly any theorems in raw category theory. They don't try to lure you into something.

You want it to be abstract, but earlier you also say it's too abstract as everything shares too much structure and is a copy of one another. Mate, is abstract good or bad?

Yes, there's structure shared and the category theorist set out to solve your dilemma by taking the common things and rewording all the similar things with one language that takes them as fundamental building blocks. But I'm sure you don't like that either.