this part of my web-WP-place to interact algebraicaly. I mean, to talk about today’s algebra:
- estructures: groups, rings, fields, vector spaces, modules, Lie groups and algebras, universal algebra
- functors or arrows inter categories
- applications in analysis, geometry, topology:
- applications elsewhere:
A CATEGORY IN MATH IS A FASHION TO ESTRUTURE MATH THEMSELVES INTO A COLLECTION OF SETS CALLED THE OBJECTS OF THE CATEGORY AND A COLLECTION OF MORPHISMS (or ARROWS) WHICH ARE FUNCTIONS AMONG THE OBJECTS
We already had studied the
- abstraction lemma one:
Given a partition of a set there is an equivalence relation defined on such that the equivalence classes coincide with the components of the partition. That is . Conversely, an equivalence relation in a set determines a partition of it.
- which is a tool to say something about a set if it is this to big to handle.
- now, a categorification in the mother category of set and functions interacting with a equivalence relation is:
- consider the math-phenomena of relating sets by functions and anyone may ask: how one has to splitt a domain into disjoint subsets for a function to be locally constant?
- abstraction lemma two:
Given a epijective-map then can be factored as where is the projection defined as that is surjective, and defined as that is bijective
- first, one has to construct a binary relation in which is an equivalence relation:
- RE: since obviously for each
- SI: implies . Then , so
- TR: if and then and . So since
- the equivalence classes are . That is, subsets where the function is constant: if then
- So it makes sense to speak about the partition and the arrow via
- note that this assignation is surjective since then such that , so
- now, it is possible to construct a map simply by
- but still we have to check that this is indeed a function
- we must convince ourselves that this way of relate it does not depends in the form we choose to represent an equivalence class, for if then and then . Hence
- now let’s proof that is bijective:
- IY: let be two equivalence classes such that , then , so and hence
- SY: if is any element of by surjectivity of we can find such that , so applying , we get a class with
Después de PRELIMINARES: GRUPOS
- Definición, orden de elemento, producto de subconjuntos, conmutadores, presentaciones,…
- Subgrupos y subgrupos normales. Clases laterales. Teorema de Lagrange.
- Grupo cociente.
- Morfismos de grupos. Pequeño teorema de Fermat.
TÓPICOS de GRUPOS
- Grupo simétrico. Teorema de Cayley. es simple.
- Acciones de grupos en conjuntos. Ecuación de Clase. Orbitas e Isotropía.
- Teoremas de Cauchy y Sylow.
- Resoluciones. Teorema fundamental de grupos abelianos.
- Schreier systems. Teorema de Schreier-Nielsen.
- Productos libres. El ejemplo
- HNN-extensiones y