0708-1300/Class notes for Tuesday, January 8

From Drorbn
Revision as of 22:59, 8 January 2008 by Megan (Talk | contribs)

(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)
Jump to: navigation, search

Algebraic Topology

Temporarily, at least, we will no longer assume that all structures are manifolds and that all functions are smooth. However, functions will still be assumed to be continuous.

We will also, temporarily, assume that all spaces are pointed spaces. That is, a space X will be assumed to have a distinguished point x, whether mentioned explicitely or not.

General idea of algebraic topology: to find "functors" from topology to algebra.

Informal definition: A category C consists of

1) a collection Obj(C) of objects

2) a class hom(C) of morphisms between these objects, so that for each X1 and X2 in Obj(C) we get a set mor{X1, X2}

3) Composition laws:

a) a binary function on hom(C): mor(X1, X2) x mor (X2, X3 --> mor (X1, X3)

b) Identities (for every X, there is an IX in mor(X,X))

4) Compatibility laws:

a) associativity of composition

b) identities are in fact identities.