Basic objects:
Structure on a set:
. groups
. rings / algebras
. fields
Partial structure: categories!
. groups -> categories
. rings -> abelian categories
More foreign:
. Modules are slightly foreign;
. non-associative: Lie algebras, Jordan algebras
. co: co-algebras, Hopf algebras
. chain complexes quite foreign.
(they require linear algebra to even define)
operads, monads etc. also more technical (higher level)
key concept:
action/something *over* something:
"group action"
"module over ring/algebra"
"vector space over field"
groups, categories, and their connections:
group semigroup
groupoid category
you might write: groupoid:category as group:semigroup
...but also:
semigroup:category as group:groupoid
"group of units; groupoid of units/isomorphisms"
familiar examples:
functions between sets -> category
integers (addition), (semi-)regular solids -> group
integers, polynomials -> ring
Q, R, C -> field
R^2, R^3 -> vector space
matrices -> non-commutative algebra
abelian groups ez; non-abelian hard & interesting
commutative rings interesting (alg. geom.!)
dunno much about (general) non-commutative rings
[BTW, polys in 1 var have composition, which is weird;
in several vars they don't; think of 'em as "functions on affine
space"]
logic and algebra:
(lattices, closure, implication;
also connected to topology via locales)
Connections:
[modules and Hopf algebras help you understand ab. grp, v.s., grp.;
general principle: understand algebra by more algebra (different
object)]
- actually, abelian groups are Z-modules!
- understand v.s. via theory of module over K[x]
- group representations (looking at associated Z[G]) useful
Applications:
. symmetries in chemistry (of molecules, of crystals)
. symmetries in physics (of laws of physics)
. "module over algebra" in function programming